Video Content by Date

2021

Dec, 1: Skeleta for Monomial Quiver Relations
Speaker: Jesse Huang
Abstract:

I will introduce a skeleton obtained directly from monomial relations in a finite quiver without cycles, and relate the construction to some classical examples in mirror symmetry. This is work in progress with David Favero.

Nov, 18: Z_2 harmonic spinors in gauge theory
Speaker: Rafe Mazzeo
Abstract:

Gauge-theoretic moduli spaces are often noncompact, and various techniques have been introduced to study their asymptotic features. Seminal work by Taubes shows that in many situations where the failure of compactness for sequences of solutions is due to the noncompactness of the gauge group, diverging sequences of solutions lead to what he called Z_2 harmonic spinors. These are multivalued solutions of a twisted Dirac equation which are branched along a codimension two subset. This leads to a number of new problems related to these Z_2 harmonic spinors as interesting geometric objects in their own right. I will survey this subject and talk about some recent work in progress with Haydys and Takahashi to compute the index of the associated deformation problem.

Speaker Biography

Rafe Mazzeo is an expert in PDEs and Microlocal analysis. He did his PhD at MIT, and was then appointed as Szegő Assistant Professor at Stanford University, where he is now Professor and Chair of the Department of Mathematics. He has served the mathematical community in many important ways, including as Director of the Park City Mathematics Institute.

Abstract:

In this work, improved swarm intelligent algorithms, namely, Salp Swarm Optimization algorithm, whale optimization, and Grasshopper Optimization Algorithm are proposed for data clustering. Our proposed algorithms utilize the crossover operator to obtain an improvised version of the existing algorithms. The performance of our suggested algorithms is tested by comparing the proposed algorithms with standard swarm intelligent algorithms and other existing algorithms in the literature. Non-parametric statistical test, the Friedman test, is applied to show the superiority of our proposed algorithms over other existing algorithms in the literature. The performance of our algorithms outperforms the performance of other algorithms for the data clustering problem in terms of computational time and accuracy.

Nov, 16: An Overview of Knots and Gauge Theory
Speaker: Edward Witten
Abstract:

The Jones polynomial of a knot, discovered in 1983, is a very
subtle invariant that is related to a great deal of mathematics and
physics. This talk will be an overview of quantum field theories in
dimensions 2, 3, 4 and 5 that are intimately related to the Jones
polynomial of a knot and a more contemporary refinement of it that is known
as Khovanov homology.

Nov, 15: Differential Equations and Algebraic Geometry - 5
Speaker: Andreas Malmendier
Abstract:

This is a guest lecture in the PIMS Network Wide Graduate Course in Differential Equations in Algebraic Geometry.

Abstract:

This is a guest lecture in the PIMS Network Wide Graduate Course in Differential Equations in Algebraic Geometry.

Nov, 3: Differential Equations and Algebraic Geometry - 3
Speaker: Adrian Clingher
Abstract:

This is a guest lecture in the PIMS Network Wide Graduate Course in Differential Equations in Algebraic Geometry.

Oct, 30: Differential Equations and Algebraic Geometry - 2
Speaker: Hossein Movasati
Abstract:

This is a guest lecture in the PIMS Network Wide Graduate Course in Differential Equations in Algebraic Geometry.

Abstract:

In this talk, we consider epidemic models with a continuum of agents where the evolution of the epidemics is represented by an ODE system. In contrast with most of the existing literature, we allow the agents to make decisions and we incorporate game theoretical ideas in the model such as the notion of Nash equilibrium. When the population is homogeneous, this leads to a continuous time, finite state mean field game. We consider, from a mathematical viewpoint, mainly two questions: (1) How to find optimal public policies to reduce the impact of the epidemics while taking into account the agents' rational choices? (2) How to handle heterogeneities among the population while keeping a continuum of agents? For the first point, we use a Stackelberg mean field game model, while for the second point, we rely on the framework of graphon games. In each case, we develop numerical methods based on machine learning tools to efficiently compute approximately optimal solutions.

Oct, 29: On nonlocal interactions in mean field games - Part 2
Speaker: Levon Nurbekyan
Abstract:

Numerous applications of mean-field games theory assume nonlocal interactions between agents. Although somewhat simpler from a mathematical analysis perspective, nonlocal models are often challenging for numerical solutions. Indeed, direct discretizations of mean-field interaction terms yield dense systems that are not economical from computational and memory perspectives. In this talk, I will discuss several options to mitigate the challenges above by importing methods from Fourier analysis and kernel methods in machine learning.

To go back to part 1 of the talk, click here: https://mathtube.org/lecture/video/nonlocal-interactions-mean-field-game...

Oct, 29: On nonlocal interactions in mean field games - Part 1
Speaker: Levon Nurbekyan
Abstract:

Numerous applications of mean-field games theory assume nonlocal interactions between agents. Although somewhat simpler from a mathematical analysis perspective, nonlocal models are often challenging for numerical solutions. Indeed, direct discretizations of mean-field interaction terms yield dense systems that are not economical from computational and memory perspectives. In this talk, I will discuss several options to mitigate the challenges above by importing methods from Fourier analysis and kernel methods in machine learning.

Part 2 of this talk continues here: https://mathtube.org/lecture/video/nonlocal-interactions-mean-field-game...

Abstract:

We study a class of linear-quadratic stochastic differential games in which each player interacts directly only with its nearest neighbors in a given graph. We find a semi-explicit Markovian equilibrium for any transitive graph, in terms of the empirical eigenvalue distribution of the graph’s normalized Laplacian matrix. This facilitates large-population asymptotics for various graph sequences, with several sparse and dense examples discussed in detail. In particular, the mean field game is the correct limit only in the dense graph case, i.e., when the degrees diverge in a suitable sense. Even though equilibrium strategies are nonlocal, depending on the behavior of all players, we use a correlation decay estimate to prove a propagation of chaos result in both the dense and sparse regimes, with the sparse case owing to the large distances between typical vertices. Without assuming the graphs are transitive, we show also that the mean field game solution can be used to construct decentralized approximate equilibria on any sufficiently dense graph sequence. This is joint work with Daniel Lacker.

Abstract:

Graphon control (CDC 17-18-19, IEEE TAC 20, Gao and Caines) and graphon mean field games (CDC18, CDC19, Caines and Huang) were used to address decision problems on very large-scale networks by employing graphons to represent arbitrary size graphs, from, respectively centralized and decentralized perspectives. Graphon couplings may be considered as a generalization of mean-field couplings with network heterogeneity. Such couplings may appear in states, controls and cost, and may be represented by different graphons in each case. In this talk, I will present the use of graphon spectral decomposition in graphon control and graphon mean field games in a linear quadratic setting. The complexity of the method does not directly depend on the number of agents or number of nodes, instead, it depends on the dimension of the characterizing graphon invariant subspace shared by the coupling operators.

Abstract:

The complexity of a dense graph increases combinatorically as its size increases. One approach to alleviate this complexity is to use graphon analysis to find an approximation of a very large graph’s adjacency matrix. Standard graphons are defined as functions on the unit square, but mapping nodes of a graph onto the unit interval may entail the loss of information. To account for this, a type of random graph is introduced called a featured graph which is a graph where each vertex has meaningful attributes determining connectivity. Featured graphons also provide an approach to the problems arising with graphs embedded in higher dimensional spaces. It is shown that in an appropriate norm the adjacency matrix operator converges to the associated featured graphon. Convergence is illustrated numerically with an SIR epidemic model generalized to multiple communities.

Abstract:

n this paper, we study the optimal control problem arising from the mean-field game formulation of the collective decision-making in honeybee swarms. A population of homogeneous players (the honeybees) has to reach consensus on one of two options. We consider three states: the first two represent the available options (or strategies), and the third one represents the uncommitted state. We formulate the continuous-time discrete-state mean-field game model. The contributions of this paper are the following: i) we propose an optimal control model where players have to control their transition rates to minimize a running cost and a terminal cost, in the presence of an adversarial disturbance; ii) we develop a formulation of the micro-macro model in the form of an initial-terminal value problem (ITVP) with switched dynamics; iii) we study the existence of stationary solutions and the mean-field Nash equilibrium for the resulting switched system; iv) we show that under certain assumptions on the parameters, the game may admit periodic solutions; and v) we analyze the resulting microscopic dynamics in a structured environment where a finite number of players interact through a network topology.

Abstract:

In this short talk, we study the solvability of Linear Quadratic Gaussian Graphon Mean Field Games (LQG-GMFGs). We motivate and define critical nodes to be those nodes at which the value function is stationary with respect to its index. We present an example of such nodes for LQG-GMFGs with the uniform attachment graphon and present some numerical simulations.

Abstract:

The Graphon Mean Field Game equations consist of a collection of parameterized Hamilton-Jacobi-Bellman equations, and a collection of parameterized Fokker-Planck-Kolmogorov equations coupled through a given graphon. In this talk, we will discuss the sensitivity of the gradient of HJB solutions with respect to the coefficients, which can be used for the solvability of Graphon Mean Field Game equation. It's based on the joint work with Peter Caines, Daniel Ho, Minyi Huang, and Jiamin Jian, see https://arxiv.org/pdf/2009.12144.pdf.

Oct, 28: Graphon Mean Field Games and the GMFG Equations
Speaker: Peter Caines
Abstract:

The existence of Nash equilibria in the Mean Field Game (MFG) theory of large non-cooperative populations of stochastic dynamical agents is established by passing to the infinite population limit. Individual agent feedback strategies are obtained via the MFG equations consisting of (i) a McKean-Vlasov-Hamilton-Jacobi-Bellman equation generating the Nash values and the best response control actions, and (ii) a McKean-Vlasov-Fokker-Planck-Kolmogorov equation for the probability distribution of the state of a generic agent in the population, otherwise known as the mean field. The applications of MFG theory now extend from economics and finance to epidemiology and physics.

In current work, MFG and MF Control theory is extended to Graphon Mean Field Game (GMFG) and Graphon Mean Field Control (GMFC) theory. Very large scale networks linking dynamical agents are now ubiquitous, with examples being given by electrical power grids, the internet, financial networks and epidemiological and social networks. In this setting, the emergence of the graphon theory of infinite networks has enabled the formulation of the GMFG equations for which we have established the existence and uniqueness of solutions. Applications of GMFG and GMFC theory to systems on particular networks of interest are being investigated and computational methods developed. As in the case of MFG theory, it is the simplicity of the infinite population GMFG and GMFC strategies which, in principle, permits their application to otherwise intractable problems involving large populations on complex networks. Work with Minyi Huang

Abstract:

TBC

Abstract:

Large-scale systems comprising of multiple subsystems connected over a network arise in a number of applications including power systems, traffic networks, communication networks, and some economic systems. A common feature of such systems is that the state evolutions and costs are coupled, i.e., the state evolution and local cost of one subsystems depend not only on its own state and control action, but also on the state and control actions of other subsystems in the network.

We consider the problem of designing control strategies for such systems when some of the parameters of the system model are not known. Due to the unknown parameters, the control problem is also a learning problem. Directly using existing reinforcement learning algorithms on such network coupled subsystems would incur $O(n^{1.5} \sqrt{T})$ regret over a horizon $T$, where the $n$ is the number of subsystems. This superlinear dependence on $n$ is prohibitive in large scale networked systems because the regret per subsystem (which is $O(\sqrt{nT})$ grows with the size of the network.

We consider networks where the dynamics coupling may be represented by a symmetric matrix (e.g., the adjacency or Laplacian matrix corresponding to a undirected weighted graph) and the cost coupling matrix have the same eigenvalues as the dynamics coupling. We use spectral decomposition of the coupling matrices to decompose the system into (L + 1) systems which are only coupled through the noise, where L is the rank of the coupling matrix. We show that, when the system model is known, the optimal control input at each subsystem can be computing by solving (L+1) decoupled Riccati equations.

Using this structure of the planning solution, we propose a new Thompson sampling algorithm and show that its regret is bounded by $O(n\sqrt{T})$, which increases linearly in the size of the network. We present numerical simulations to illustrate our results on mean-field control and general low-rank networks.

Joint work with Shuang Gao, Sagar Sudhakara Ashutosh Nayyar, and Ouyang Yi

Abstract:

The purpose of this work is to introduce a notion of weak solution to the master equation of a potential mean field game and to prove that existence and uniqueness hold under quite general assumptions. Remarkably, this is shown to hold true without any monotonicity constraint on the coefficients. The key point is to interpret the master equation in a conservative sense and then to adapt to the infinite dimensional setting earlier arguments for hyperbolic systems deriving from a HJB equation. Here, the master equation is indeed regarded as an infinite dimensional system set on the space of probability measures. To make the analysis easier, we assume that the coefficients are periodic and accordingly that the probability measures are defined on the torus. This allows to represent probability measures through their Fourier coefficients. Most of the analysis then consists in rewriting the master equation and the corresponding HJB equation for the mean field control problem lying above the mean field game as PDEs set on the Fourier coefficients themselves.

Joint work with A. Cecchin (Ecole Polytechnique, France)

Abstract:

Real data collected from different applications that have additional topological structures and connection information are amenable to be represented as a weighted graph.
Considering the node labeling problem, Graph Neural Networks (GNNs) is a powerful tool, which can mimic experts' decisions on node labeling.
GNNs combine node features, connection patterns, and graph structure by using a neural network to embed node information and pass it through edges in the graph.
We want to identify the patterns in the input data used by the GNN model to make a decision and examine if the model works as we desire.
However, due to the complex data representation and non-linear transformations, explaining decisions made by GNNs is challenging.
In this work, we propose new graph features' explanation methods to identify the informative components and important node features. Besides, we propose a pipeline to identify the key factors used for node classification. We use four datasets (two synthetic and two real) to validate our methods. Our results demonstrate that our explanation approach can mimic data patterns used for node classification by human interpretation and disentangle different features in the graphs. Furthermore, our explanation methods can be used for understanding data, debugging GNN models, and examine model decisions.

Abstract:

A two-stage enrichment design is a type of adaptive design, which extends a stratified design with a futility analysis on the marker negative cohort at the first stage, and the second stage can be either a targeted design with only the marker positive stratum, or still the stratified design with both marker strata, depending on the result of the interim futility analysis.

In this talk we consider the situation where the marker assay and the classification rule are possibly subject to error. We derive the sequential tests for the global hypothesis as well as the component tests for the overall cohort and the marker-positive cohort. We discuss the power analysis with the control of the type-I error rate and show the adverse impact of the misclassification on the powers. We also show the enhanced power of the two-stage enrichment over the one-stage design, and illustrate with examples of the recent successful development of immunotherapy in non-small-cell lung cancer.​

Abstract:

Modeling and understanding crowd evacuation dynamics has been a long-standing problem. Most realistic models involve nonlinear effects to capture individual velocity decrease with crowd density increase. This leads to useful but essentially intractable partial differential equation-based models. We consider here for tractability purposes, a class of linear quadratic large-scale evacuation games where velocity can be improved through crowd avoidance. This is simulated in the agent cost functions through negative costs which accrue when agents drift away from variously defined population mean trajectories, in a multi-exit situation. The presence of negative cost components generically induces a finite escape time phenomenon if the time horizon is not adequately bounded. We formulate two types of models for which we provide sufficient time horizon upper bounds for agent cost convergence and establish existence of limiting mean field game equilibria as well as their ε-Nash property. This is joint work with Noureddine Toumi and Jérôme Le Ny.

Abstract:

This work introduces a new N-player dynamic routing game that extend current Markovian traffic static assignment model.
It extends the N-player dynamic routing game to the corresponding mean field routing game, which models congestion in its dynamics.
Therefore, this new mean field routing game does not need to model congestion in the player cost function as done in the existing literature.
Both games are implemented in the open source library OpenSpiel.
The mean field game is used to solve the N-player dynamic game which leads to efficient computation of a approximate dynamic user equilibrium of the dynamic routing game.

Oct, 27: Differential Equations and Algebraic Geometry - 1
Speaker: Hossein Movasati
Abstract:

This is a guest lecture in the PIMS Network Wide Graduate Course in Differential Equations in Algebraic Geometry.

Oct, 27: Quantum operations as resources
Speaker: Thomas Theurer, UCalgary
Abstract:

Protocols and devices that exploit quantum mechanical effects can outperform their classical counterparts in certain tasks ranging from communication and computation to sensing. Intuitively speaking, the reason for this is that different physical laws allow for different technological applications. Therefore, the question where quantum mechanics differs from classical physics is not only of foundational or philosophical interest but might have technological implications too. To address it in a systematic manner, so-called quantum resource theories were developed. These are mathematical frameworks that emerge from (physically motivated) restrictions that are put on top of the laws of quantum mechanics and single out specific aspects of quantum theory as resources. A widely studied example would be the restriction to local operations and classical communication, which leads to the resource theory of entanglement. It is then investigated how these restrictions influence our abilities to do certain tasks (e.g., communicate securely), how these restrictions can be overcome, and how the resulting resources can be quantified. Historically, resource theories were mainly focused on the resources present in quantum states. After an introduction to the general topic, I will speak about my recent research on how these concepts can be extended to quantum operations and why this is of interest.

Abstract:

This talk will be a fairly high-level one, addressing various current issues in mean field games (MFGs), the underlying challenges primarily with regard to robustness, learning, and incentivization, and paths toward their resolution. Among these are: (i) use of multi-agent reinforcement learning for the computation of mean-field equilibrium (MFE) with state samples drawn from an unmixed Markov chain, and studying the performance of the associated (actor-critic) algorithms; (ii) adversarial MFGs on multi-graphs where agents interact with their neighbors, with such interactions propagating from neighborhoods to the entire network, and with an adversary counteracting the consensus formation process among the agents; and (iii) MFGs with a decision hierarchy, where the agent at the top of the hierarchy (leader) aims at designing incentive strategies (as in mechanism design) to induce a high population of agents at the lower level (followers) to act rationally toward a globally optimal solution in spite of their non-cooperative behavior. The talk will also identify several fruitful directions of research in this domain.

Abstract:

This talk is an overview of a recent and ongoing line of work on large sparse networks of interacting diffusion processes. Each process is associated with a vertex in a graph and interacts only with its neighbors. When the graph is complete and the size grows to infinity, the system is well-approximated by its mean field limit, which describes the behavior of one typical process. For general graphs, however, the mean field approximation can fail, most dramatically when the graph is sparse. Nevertheless, if the underlying graph is locally tree-like (as is the case for many canonical sparse random graph models), we show that a single process and its nearest neighbors are characterized by an autonomous evolution which we call the "local dynamics." This can be viewed as a sparse counterpart of the usual McKean-Vlasov equation. The structure of the local dynamics depend heavily on the symmetries of the underlying graph and the conditional independence structure of the solution process. In the time-stationary case, the local dynamics take a particular tractable form. Based on joint works with Kavita Ramanan, Ruoyu Wu, and Jiacheng Zhang.

Abstract:

We consider heterogeneously interacting diffusive particle systems and their large population limit. The interaction is of mean field type with random weights characterized by an underlying graphon. The limit is given by a graphon particle system consisting of independent but heterogeneous nonlinear diffusions whose probability distributions are fully coupled. Under suitable convexity/dissipativity assumptions, we show the exponential ergodicity for both systems, establish a uniform-in-time law of large numbers for the empirical measure of particle states, and introduce the uniform-in-time Euler approximation. The precise rate of convergence of the Euler approximation is provided. We also provide uniform-in-time exponential concentration bounds for the rate of the LLN convergence under additional integrability conditions. Based on joint works with Erhan Bayraktar and Suman Chakraborty.

Abstract:

The study of large populations of interacting agents connected via a non-trivial network of connections represents a field of growing interest in Applied Mathematics. The relatively recent theory of graphons turns out to be well-adapted to model the emergence of complex networks and has been applied in several contexts by now, including mean-field systems. In this talk, we discuss how some of the key properties of graphon objects, e.g., exchangeability and labelling, are related to the study of interacting particle systems. Hopefully, this will shed some light on the behavior of systems described by partial differential equations and graphons, as the Graphon Mean-Field Game equations.

Abstract:

We consider heterogeneously interacting diffusive particle systems and their large population limit. The interaction is of mean field type with random weights characterized by an underlying graphon. The limit is given by a graphon particle system consisting of independent but heterogeneous nonlinear diffusions whose probability distributions are fully coupled. A law of large numbers result is established as the system size increases and the underlying graphons converge. Under suitable additional assumptions, we show the exponential ergodicity for the system, establish the uniform in time law of large numbers, and introduce the uniform in time Euler approximation. The precise rate of convergence of the Euler approximation is provided.

Based on joint works with Suman Chakraborty and Ruoyu Wu.

Abstract:

Network dynamical systems add an additional challenge of scale to optimal control schemes. There are many options of overcoming it, such as approximations and heuristics based on mean field games, neural networks, or reinforcement learning, or the actual structure of the networks, each with its own advantages and tradeoffs.

Metapopulation epidemic models, where each population is an entity on a map, such as a city or a district, are a convenient option for benchmarking varying optimal control schemes: these can be designed with varying number of nodes (dimension), have a natural per-node optimal control, e.g. the “lockdown level,” and a straightforward visualization option of choropleth maps.

In this talk, we will describe a procedure for generating plausible instances of such models with from 1 to circa 64,000 nodes based on publicly available census data for the contiguous U.S., each with the network of short-range travel (commute) and long-range travel (airplane), the latter derived from publicly available passenger flight statistics---along with a formal aggregation routine enabling a view of the same geography at different resolutions.

As a showcase, we designed a “baseline” optimal control scheme for three instances covering Oregon and Washington states: a 2-node instance on state level, a 75-node on county level, and a 2,072-node instance made of “atomic” population units, the census tracts, which are put through a metapopulation SIR model with per-node “lockdown level” optimal control on a 180-day time horizon, with the objective of minimizing the cumulative number of infections and the square of this lockdown control; the results are compared with the “no-lockdown” model.
The optimal control was derived through the Pontryagin Maximum Principle and numerically computed by the forward-backward sweep method, which converges within 5 seconds on the 2- and 75-node instances and within 40 seconds on the 2,072-node one.

Oct, 26: Controlling Human Microbiota
Speaker: Yang-Yu Liu
Abstract:

We coexist with a vast number of microbes—our microbiota—that live in and on our bodies, and play an important role in human physiology and diseases. Many scientific advances have been made through the work of large-scale, consortium-driven metagenomic projects. Despite these advances, there are still many fundamental questions regarding the dynamics and control of microbiota to be addressed. Indeed, it is well established that human-associated microbes form a very complex and dynamic ecosystem, which can be altered by drastic diet change, medical interventions, and many other factors. The alterability of our microbiome offers opportunities for practical microbiome-based therapies, e.g., fecal microbiota transplantation and probiotic administration, to restore or maintain our healthy microbiota. Yet, the complex structure and dynamics of the underlying ecosystem render the quantitative study of microbiome-based therapies extremely difficult. In this talk, I will discuss our recent theoretical progress on controlling human microbiota from network science, dynamical systems, and control theory perspectives.

Oct, 22: 2021 PIMS-UBC Math Job Forum for Postdoctoral Fellows & Graduate Students
Speaker: Moderator: Stephanie van Willigenburg
Abstract:

The PIMS-UBC Math Job Forum is an annual Forum to help graduate students and postdoctoral fellows in Mathematics and related areas with their job searches. The session is divided in two parts: short presentations from our panel followed by a discussion.

Learn the secrets of writing an effective research statement, developing an outstanding CV, and giving a winning job talk. We will address questions like: Who do I ask for recommendation letters? What kind of jobs should I apply to? What can I do to maximize my chances of success?

Panelists:

Dan Coombs, Head of Mathematics, UBC

Pamela Harris, Associate Professor, Williams College, and Faculty Fellow of the Office of Institutional Diversity, Equity and Inclusion

Eugene Li, Chair of Mathematics and Statistics, Langara

Luis Serrano, Quantum AI Research Scientist at Zapata Computing, and ex-Google, ex-Apple

Abstract:

Generative models such as Generative Adversarial Nets (GANs), Variational Autoencoders and Normalizing Flows have been very successful in the unsupervised learning task of generating samples from a high-dimensional probability distribution. However, the task of conditioning a high-dimensional distribution from limited empirical samples has attracted less attention in the literature but it is a central problem in Bayesian inference and supervised learning. In this talk we will discuss some ideas in this direction by viewing generative modelling as a measure transport problem. In particular, we present a simple recipe using block-triangular maps and monotonicity constraints that enables standard models such as the original GAN to perform conditional sampling. We demonstrate the effectiveness of our method on various examples ranging from synthetic test sets to image in-painting and function space inference in porous medium flow.

Oct, 14: The Connection Between RDEs and PDEs
Speaker: Louigi Addario-Berry
Abstract:

Recursive distributional equations (RDEs) are ubiquitous in probability. For example, the standard Gaussian distribution can be characterized as the unique fixed point of the following RDE

$$
X = (X_1 + X_2) / \sqrt{2}
$$

among the class of centered random variables with standard deviation of 1. (The equality in the equation is in distribution; the random variables and must all be identically distributed; and and must be independent.)

Recently, it has been discovered that the dynamics of certain recursive distributional equations can be solved using by using tools from numerical analysis, on the convergence of approximation schemes for PDEs. In particular, the framework for studying stability and convergence for viscosity solutions of nonlinear second order equations, due to Crandall-Lions, Barles-Souganidis, and others, can be used to prove distributional convergence for certain families of RDEs, which can be interpreted as tree- valued stochastic processes. I will survey some of these results, as well as the (current) limitations of the method, and our hope for further interplay between these two research areas.

Abstract:

Synthetic biology offers bottom-up engineering strategies that intends to understand complex systems via design-build-test cycles. In development, gene regulatory networks emerge into collective cellular behaviors with multicellular forms and functions. Here, I will introduce a synthetic developmental biology approach for tissue engineering. It involves building developmental trajectories in stem cells via programmed gene circuits and network analysis. The outcome of our approach is decoding our own development and to create programmable organoids with both natural or artificial designs and augmented functions.

Abstract:

In this presentation, we discuss the Variational AutoEncodeur (VAE): a latent variable model emerging from the machine learning community. To begin, we introduce the theoretical foundations of the model and its relationship with well-established statistical models. Then, we discuss how we used VAEs to solve two widely different problems. First, we tackled a classic statistical problem, survival analysis, and then a classic machine learning problems, image analysis and image generation. We conclude with a short discussion of our latest research project where we establish a new metric for the evaluation or regularization of latent variable models such a Gaussian Mixture Models and VAEs.​

Abstract:

We develop a general framework for identifying phase reduced equations for finite populations of coupled oscillators that is valid far beyond the weak coupling approximation. This strategy represents a general extension of the theory from [Wilson and Ermentrout, Phys. Rev. Lett 123, 164101 (2019)] and yields coupling functions that are valid to higher-order accuracy in the coupling strength for arbitrary types of coupling (e.g., diffusive, gap-junction, chemical synaptic). These coupling functions can be used to understand the behavior of potentially high-dimensional, nonlinear oscillators in terms of their phase differences. The proposed formulation accurately replicates nonlinear bifurcations that emerge as the coupling strength increases and is valid in regimes well beyond those that can be considered using classic weak coupling assumptions. We demonstrate the performance of our approach through two examples. First, we use diffusively coupled complex Ginzburg-Landau (CGL) model and demonstrate that our theory accurately predicts bifurcations far beyond the range of existing coupling theory. Second, we use a realistic conductance-based model of a thalamic neuron and show that our theory correctly predicts asymptotic phase differences for non-weak synaptic coupling. In both examples, our theory accurately captures model behaviors that weak coupling theories can not.

Speaker Biography

Youngmin Park, Ph.D., is currently a PIMS Postdoc at the University of Manitoba under the supervision of Prof. Stéphanie Portet. He received his PhD in Mathematics from the University of Pittsburgh in 2018, where he applied dynamical systems methods to problems in neuroscience. His first postdoc involved auditory neuroscience research at the University of Pennsylvania in the Department of Otorhinolaryngology, before moving on to his next postdoc researching molecular motor dynamics in the Department of Mathematics at Brandeis University. He is now at Manitoba, continuing to apply dynamical systems methods to biological questions related to molecular motor transport and neural oscillators.

Abstract:

Motivated by recent work with biologists, I will showcase some mathematical results on Turing instabilities in complex domains. This is scientifically related to understanding developmental tuning in a variety of settings such as mouse whiskers, human fingerprints, bat teeth, and more generally pattern formation on multiple scales and evolving domains. Some of these problems are natural extensions of classical reaction-diffusion models, amenable to standard linear stability analysis, whereas others require the development of new tools and approaches. These approaches also help close the vast gap between the simple theory of diffusion-driven pattern formation, and the messy reality of biological development, though there is still much work to be done in validating even complex theories against the rich pattern dynamics observed in nature. I will emphasize throughout the role that Turing's 1952 paper had in these developments, and how much of our modern progress (and difficulties) were predicted in this paper. I will close by discussing a range of open questions, many of which fall well beyond the extensions I will discuss, but at least some of which were known to Turing.

Abstract:

In longitudinal studies, we measure the same variables at multiple time-points to track their change over time. The exact data collection schedules (i.e., time of participants' visits) are often pre-determined to accommodate the ease of project management and compliance. Therefore, it is common to schedule those visits at equally spaced time intervals. However, recent publications based on simulated experiments indicate that the power of studies and the precision of model parameter estimators is related to the participants' visiting scheme. So, in this work, we investigate how to schedule participants' visits to better study the accelerated cognitive decline of senior adults, where a broken-stick model is often applied. We formulate this optimal design problem on scheduling participants' visiting into a high- dimensional optimization problem and derive its approximate solution by adding reasonable constraints. Based on this approximation, we propose a novel design of the visiting scheme that aims to maximize the power (i.e. reduce the variance of estimators) in identifying the onset of accelerated decline. Using both simulation studies and evidence from real data, we demonstrate that our design outperforms the standard equally-spaced one when we have strong prior knowledge on the change-points. This novel design helps researchers plan their longitudinal studies with improved power in detecting pattern change without collecting extra data. Also, this individual-level scheduling system helps monitor seniors' cognitive function and, therefore, benefits the development of personal level treatment for cognitive decline, which agrees with the trend of the health care system.

Sep, 30: Finite sample rates for optimal transport estimation problems
Speaker: Jan-Christian Hütter
Abstract:

The theory of optimal transport (OT) gives rise to distance measures between probability distributions that take the geometry of the underlying space into account. OT is often used in the analysis of point cloud data, for example in domain adaptation problems, computer graphics, and trajectory analysis of single-cell RNA-Seq data. However, from a statistical perspective, straight-forward plug-in estimators for OT distances and couplings suffer from the curse of dimensionality in high dimensions. One way of alleviating this problem is to employ regularized statistical procedures, either by changing the transport objective or exploiting additional structure in the underlying probability distributions or ground truth couplings. In this talk, I will outline the problem and give an overview of recent solution approaches, in particular those employing entropically regularized optimal transport or imposing smoothness assumptions on the ground truth transport map.

Sep, 29: Topological Data Analysis of Collective Behavior
Speaker: Dhananjay Bhaskar
Abstract:

Active matter systems, ranging from liquid crystals to populations of cells and animals, exhibit complex collective behavior characterized by pattern formation and dynamic phase transitions. However, quantitative classification is challenging for heterogeneous populations of varying size, and typically requires manual supervision. In this talk, I will demonstrate that a combination of topological data analysis (TDA) and machine learning can uniquely identify the spatial arrangement of agents by keeping track of clusters, loops, and voids at multiple scales. To validate the approach, I will present 3 case studies: (1) data-driven modeling and analysis of epithelial-mesenchymal transition (EMT) in mammary epithelia, (2) unsupervised classification of cell sorting, and self-assembly patterns in co-cultures, and (3) parameter recovery from animal swarming trajectories.

Sep, 29: Unsolved Problems in Number Theory
Speaker: Ben Green
Abstract:

Richard Guy's book "Unsolved Problems in Number Theory" was one of the first mathematical books I owned. I will discuss a selection of my favorite problems from the book, together with some of the progress that has been made on them in the 30 years since I acquired my copy.

Speaker Biography

Ben Green was born and grew up in Bristol, England. He was educated at Trinity College, Cambridge and has been the Waynflete Professor of Pure Mathematics at Oxford since 2013.

About the Series

The Richard & Louise Guy Lecture Series, presented from Louise Guy to Richard in recognition of his love of mathematics and his desire to share his passion with the world, celebrates the joy of discovery and wonder in mathematics for everyone.

Abstract:

One of the challenges of the accurate simulation of turbulent flows is that initial data is often incomplete. Data assimilation circumvents this issue by continually incorporating the observed data into the model. A new approach to data assimilation known as the Azouani-Olson-Titi algorithm (AOT) introduced a feedback control term to the 2D incompressible Navier-Stokes equations (NSE) in order to incorporate sparse measurements. The solution to the AOT algorithm applied to the 2D NSE was proven to converge exponentially to the true solution of the 2D NSE with respect to the given initial data. In this talk, we present our tests on the robustness, improvements, and implementation of the AOT algorithm, as well as generate new ideas based off of these investigations. First, we discuss the application of the AOT algorithm to the 2D NSE with an incorrect parameter and prove it still converges to the correct solution up to an error determined by the error in the parameters. This led to the development of a simple parameter recovery algorithm, whose convergence we recently proved in the setting of the Lorenz equations. The implementation of this algorithm led us to provide rigorous proofs that solutions to the corresponding sensitivity equations are in fact the Fréchet derivative of the solutions to the original equations. Next, we present a proof of the convergence of a nonlinear version of the AOT algorithm in the setting of the 2D NSE, where for a portion of time the convergence rate is proven to be double exponential. Finally, we implement the AOT algorithm in the large scale Model for Prediction Across Scales - Ocean model, a real-world climate model, and investigate the effectiveness of the AOT algorithm in recovering subgrid scale properties.

Speaker Biography

Elizabeth Carlson, is a homeschooler turned math PhD! She grew up in Helena, MT, USA, where she also graduated from Carroll College with a Bachelor's in mathematics and minor in physics. She became interested in fluid dynamics as an undergraduate, and followed this interest through her graduate work at the University of Nebraska - Lincoln in Lincoln, NE, USA, where she just earned my PhD in May 2021. Her research focus is in fluid dynamics, focusing on the well-posedness of systems of partial differential equations and numerical computations and analysis in fluid dynamics. In her free time, she enjoys hiking, playing piano, reading, and martial arts.

Read more about Elizabeth Carlson on our PIMS Medium blog here.

Abstract:

The intrinsic infinite-dimensional nature of functional data creates a bottleneck in the application of traditional classifiers to functional settings. These classifiers are generally either unable to generalize to infinite dimensions or have poor performance due to the curse of dimensionality. To address this concern, we propose building a distance-weighted discrimination (DWD) classifier on scores obtained by projecting data onto one specific direction. We choose this direction by minimizing, over a reproducing kernel Hilbert space, an empirical risk function containing the DWD classifier loss function. Our proposed classifier avoids overfitting and enjoys the appealing properties of DWD classifiers. We further extend this framework to accommodate functional data classification problems where scalar covariates are involved. In contrast to previous work, we establish a non-asymptotic estimation error bound on the relative misclassification rate. Through simulation studies and a real-world application, we demonstrate that the proposed classifier performs favourably relative to other commonly used functional classifiers in terms of prediction accuracy in finite-sample settings.

Abstract:

Large systems of interacting particles (or agents) are widely used to investigate self-organization and collective behavior. They frequently appear in modeling phenomena such as biological swarms, crowd dynamics, self-assembly of nanoparticles and opinion formation. Similar particle models are also used in metaheuristics, which provide empirically robust solutions to tackle hard optimization problems with fast algorithms. In this talk I will start with introducing some generic particle models and their underlying mean-field equations. Then we will focus on a specific particle model that belongs to the class of Consensus-Based Optimization (CBO) methods, and we show that it is able to perform essentially as good as ad hoc state of the art methods in challenging problems in signal processing and machine learning.

Speaker Biography

Hui Huang, Ph.D., is currently a PIMS Postdoc at the University of Calgary under the supervision of Prof. Jinniao Qiu. Before moving to Calgary, he worked as a postdoctoral researcher in the Chair for Applied Numerical Analysis at the Technical University of Munich, Germany. Prior to being at TUM he was an Alan Mekler Postdoctoral Fellow in the Department of Mathematics at Simon Fraser University. In 2017, he received his PhD in Mathematics from Tsinghua University. His doctoral dissertation was conducted in consultations with Prof. Jian¬-Guo Liu from Duke University, where he studied as a joint PhD student from 2014 to 2016. His research has been focused on complex dynamical systems and their related kinetic equations.

Read more about Hui Huang on the PIMS Medium blog.

Aug, 26: Branes, Quivers, and BPS Algebras 4 of 4
Speaker: Miroslav Rapčák
Abstract:

This series of lectures covers a brief introduction into some algebro-geometric techniques used in the construction of BPS algebras. The starting point of our construction is a physical picture of D0-branes bound to D-branes of higher dimension. Using methods of the derived category of coherent sheaves, we are going to derive a framed quiver with potential describing supersymmetric quantum mechanics capturing the low-energy behavior of such D0-branes. For a large class of quivers, we are going to identify the space of BPS states with different melted-crystal configurations. Finally, by employing correspondences, we are going to construct an action of a BPS algebra known as the affine Yangian on the space of BPS states. The action of the affine Yangian factors through the action of various vertex operator algebras, Cherednik algebras, and more. This construction leads to an enormously rich interplay between physics, geometry and representation theory.

  1. Lecture 1
  2. Lecture 2
  3. Lecture 3
  4. Lecture 4
Aug, 26: Geometry of N=2 Supersymmetry 4 of 4
Speaker: Andy Neitzke
Abstract:

Coulomb branches of N=2 supersymmetric field theories in four dimensions support a rich geometry. My aim in these lectures will be to explain some aspects of this geometry, and its relation to the physics of the N=2 theories themselves.

I will first describe various constructions of N=2 theories and the corresponding Coulomb branches. In this story the main geometry visible is that of a complex integrable system, fibered over the Coulomb branch; one nice class of examples is associated to the Hitchin integrable system (moduli space of Higgs bundles over a Riemann surface). Fundamental objects in the N=2 theory (local operators, line operators, surface operators) all have geometric counterparts in the integrable system, as I will explain.

Next I will discuss a deformation of the story, which arises in physics from the Nekrasov-Shatashvili Omega-background. In this deformation, the Coulomb branch is replaced by a closely related space; for instance, the base of the Hitchin integrable system is replaced by a space parameterizing opers over the Riemann surface. One can use this deformation to give a concrete picture of the space of opers; in so doing one meets Stokes phenomena which are governed by the BPS indices (Donaldson-Thomas invariants) of the N=2 theory. This turns out to be closely related to the "exact WKB method" in analysis of ODEs. It is also connected to Riemann-Hilbert problems of a sort recently investigated by Bridgeland, as I will describe if time permits.

  1. Lecture 1
  2. Lecture 2
  3. Lecture 3
  4. Lecture 4
Aug, 26: Derived Geometry in Twists of Gauge Theories 4 of 4
Speaker: Tudor Dimofte
Abstract:

These lectures will review and develop methods in algebraic geometry (in particular, derived algebraic geometry) to describe topological and holomorphic sectors of quantum field theories. A recurring theme will be the interaction of local and extended operators, and of QFT's in different dimensions. The main examples will come from twists of supersymmetric gauge theories, and will connect to a large body of recent and ongoing work on 3d Coulomb branches, 3d mirror symmetry (and geometric Langlands), logarithmic VOA's and non-semisimple TQFT's, and categorified cluster algebras.

The basic plan for the lectures is:

  • Lecture 1 (2d warmup): categories of boundary conditions, interfaces, and Koszul duality
  • Lectures 2 and 3 (3d): twists of 3d N=2 and N=4 gauge theories; vertex algebras, chiral categories, and braided tensor categories; d mirror symmetry; quantum groups at roots of unity and derived non-semisimple 3d TQFT's (compared and contrasted with Chern-Simons theory)
  • Lecture 4 (4d): line and surface operators in 4d N=2 gauge theory, the coherent Satake category, and relations to Schur indices and 4d N=2 vertex algebras
  1. Lecture 1
  2. Lecture 2
  3. Lecture 3
  4. Lecture 4
Abstract:

Throughout my lectures I will explain the geometry of elliptic fibration which can gave rise to understanding the spectra and anomalies in lower-dimensional theories from the Calabi-Yau compactifications of F-theory. I will first explain what elliptic fibration is and explain Kodaira types, which gives rise an ADE classification. Utilizing Weierstrass model of elliptic fibrations, I will discuss Tate’s algorithm and Mordell-Weil group. By considering codimension one and two singularities and studying the geometry of crepant resolutions, we can define G-models that are geometrically-engineered models from F-theory. I will discuss the dictionary between the gauge theory and the elliptic fibrations and how to incorporate this to learn about topological invariants of the compactified Calabi-Yau that is one of the ingredient to understand spectra in the compactified theories. I will explain the more refined connection to understand the Coulomb branch of the 5d N=1 theories and 6d (1,0) theories and their anomalies from this perspective.

  1. Lecture 1
  2. Lecture 2
  3. Lecture 3
  4. Lecture 4
Aug, 25: Branes, Quivers, and BPS Algebras 3 of 4
Speaker: Miroslav Rapčák
Abstract:

This series of lectures covers a brief introduction into some algebro-geometric techniques used in the construction of BPS algebras. The starting point of our construction is a physical picture of D0-branes bound to D-branes of higher dimension. Using methods of the derived category of coherent sheaves, we are going to derive a framed quiver with potential describing supersymmetric quantum mechanics capturing the low-energy behavior of such D0-branes. For a large class of quivers, we are going to identify the space of BPS states with different melted-crystal configurations. Finally, by employing correspondences, we are going to construct an action of a BPS algebra known as the affine Yangian on the space of BPS states. The action of the affine Yangian factors through the action of various vertex operator algebras, Cherednik algebras, and more. This construction leads to an enormously rich interplay between physics, geometry and representation theory.

  1. Lecture 1
  2. Lecture 2
  3. Lecture 3
  4. Lecture 4
Aug, 25: Geometry of N=2 Supersymmetry 3 of 4
Speaker: Andy Neitzke
Abstract:

Coulomb branches of N=2 supersymmetric field theories in four dimensions support a rich geometry. My aim in these lectures will be to explain some aspects of this geometry, and its relation to the physics of the N=2 theories themselves.

I will first describe various constructions of N=2 theories and the corresponding Coulomb branches. In this story the main geometry visible is that of a complex integrable system, fibered over the Coulomb branch; one nice class of examples is associated to the Hitchin integrable system (moduli space of Higgs bundles over a Riemann surface). Fundamental objects in the N=2 theory (local operators, line operators, surface operators) all have geometric counterparts in the integrable system, as I will explain.

Next I will discuss a deformation of the story, which arises in physics from the Nekrasov-Shatashvili Omega-background. In this deformation, the Coulomb branch is replaced by a closely related space; for instance, the base of the Hitchin integrable system is replaced by a space parameterizing opers over the Riemann surface. One can use this deformation to give a concrete picture of the space of opers; in so doing one meets Stokes phenomena which are governed by the BPS indices (Donaldson-Thomas invariants) of the N=2 theory. This turns out to be closely related to the "exact WKB method" in analysis of ODEs. It is also connected to Riemann-Hilbert problems of a sort recently investigated by Bridgeland, as I will describe if time permits.

  1. Lecture 1
  2. Lecture 2
  3. Lecture 3
  4. Lecture 4
Aug, 25: Derived Geometry in Twists of Gauge Theories 3 of 4
Speaker: Tudor Dimofte
Abstract:

These lectures will review and develop methods in algebraic geometry (in particular, derived algebraic geometry) to describe topological and holomorphic sectors of quantum field theories. A recurring theme will be the interaction of local and extended operators, and of QFT's in different dimensions. The main examples will come from twists of supersymmetric gauge theories, and will connect to a large body of recent and ongoing work on 3d Coulomb branches, 3d mirror symmetry (and geometric Langlands), logarithmic VOA's and non-semisimple TQFT's, and categorified cluster algebras.

The basic plan for the lectures is:

  • Lecture 1 (2d warmup): categories of boundary conditions, interfaces, and Koszul duality
  • Lectures 2 and 3 (3d): twists of 3d N=2 and N=4 gauge theories; vertex algebras, chiral categories, and braided tensor categories; d mirror symmetry; quantum groups at roots of unity and derived non-semisimple 3d TQFT's (compared and contrasted with Chern-Simons theory)
  • Lecture 4 (4d): line and surface operators in 4d N=2 gauge theory, the coherent Satake category, and relations to Schur indices and 4d N=2 vertex algebras
  1. Lecture 1
  2. Lecture 2
  3. Lecture 3
  4. Lecture 4
Abstract:

Throughout my lectures I will explain the geometry of elliptic fibration which can gåve rise to understanding the spectra and anomalies in lower-dimensional theories from the Calabi-Yau compactifications of F-theory. I will first explain what elliptic fibration is and explain Kodaira types, which gives rise an ADE classification. Utilizing Weierstrass model of elliptic fibrations, I will discuss Tate’s algorithm and Mordell-Weil group. By considering codimension one and two singularities and studying the geometry of crepant resolutions, we can define G-models that are geometrically-engineered models from F-theory. I will discuss the dictionary between the gauge theory and the elliptic fibrations and how to incorporate this to learn about topological invariants of the compactified Calabi-Yau that is one of the ingredient to understand spectra in the compactified theories. I will explain the more refined connection to understand the Coulomb branch of the 5d N=1 theories and 6d (1,0) theories and their anomalies from this perspective.

  1. Lecture 1
  2. Lecture 2
  3. Lecture 3
  4. Lecture 4
Aug, 24: Branes, Quivers, and BPS Algebras 2 of 4
Speaker: Miroslav Rapčák
Abstract:

This series of lectures covers a brief introduction into some algebro-geometric techniques used in the construction of BPS algebras. The starting point of our construction is a physical picture of D0-branes bound to D-branes of higher dimension. Using methods of the derived category of coherent sheaves, we are going to derive a framed quiver with potential describing supersymmetric quantum mechanics capturing the low-energy behavior of such D0-branes. For a large class of quivers, we are going to identify the space of BPS states with different melted-crystal configurations. Finally, by employing correspondences, we are going to construct an action of a BPS algebra known as the affine Yangian on the space of BPS states. The action of the affine Yangian factors through the action of various vertex operator algebras, Cherednik algebras, and more. This construction leads to an enormously rich interplay between physics, geometry and representation theory.

  1. Lecture 1
  2. Lecture 2
  3. Lecture 3
  4. Lecture 4
Aug, 24: Geometry of N=2 Supersymmetry 2 of 4
Speaker: Andy Neitzke
Abstract:

Coulomb branches of N=2 supersymmetric field theories in four dimensions support a rich geometry. My aim in these lectures will be to explain some aspects of this geometry, and its relation to the physics of the N=2 theories themselves.

I will first describe various constructions of N=2 theories and the corresponding Coulomb branches. In this story the main geometry visible is that of a complex integrable system, fibered over the Coulomb branch; one nice class of examples is associated to the Hitchin integrable system (moduli space of Higgs bundles over a Riemann surface). Fundamental objects in the N=2 theory (local operators, line operators, surface operators) all have geometric counterparts in the integrable system, as I will explain.

Next I will discuss a deformation of the story, which arises in physics from the Nekrasov-Shatashvili Omega-background. In this deformation, the Coulomb branch is replaced by a closely related space; for instance, the base of the Hitchin integrable system is replaced by a space parameterizing opers over the Riemann surface. One can use this deformation to give a concrete picture of the space of opers; in so doing one meets Stokes phenomena which are governed by the BPS indices (Donaldson-Thomas invariants) of the N=2 theory. This turns out to be closely related to the "exact WKB method" in analysis of ODEs. It is also connected to Riemann-Hilbert problems of a sort recently investigated by Bridgeland, as I will describe if time permits.

  1. Lecture 1
  2. Lecture 2
  3. Lecture 3
  4. Lecture 4
Aug, 24: Derived Geometry in Twists of Gauge Theories 2 of 4
Speaker: Tudor Dimofte
Abstract:

These lectures will review and develop methods in algebraic geometry (in particular, derived algebraic geometry) to describe topological and holomorphic sectors of quantum field theories. A recurring theme will be the interaction of local and extended operators, and of QFT's in different dimensions. The main examples will come from twists of supersymmetric gauge theories, and will connect to a large body of recent and ongoing work on 3d Coulomb branches, 3d mirror symmetry (and geometric Langlands), logarithmic VOA's and non-semisimple TQFT's, and categorified cluster algebras.

The basic plan for the lectures is:

  • Lecture 1 (2d warmup): categories of boundary conditions, interfaces, and Koszul duality
  • Lectures 2 and 3 (3d): twists of 3d N=2 and N=4 gauge theories; vertex algebras, chiral categories, and braided tensor categories; d mirror symmetry; quantum groups at roots of unity and derived non-semisimple 3d TQFT's (compared and contrasted with Chern-Simons theory)
  • Lecture 4 (4d): line and surface operators in 4d N=2 gauge theory, the coherent Satake category, and relations to Schur indices and 4d N=2 vertex algebras
  1. Lecture 1
  2. Lecture 2
  3. Lecture 3
  4. Lecture 4
Abstract:

Throughout my lectures I will explain the geometry of elliptic fibration which can give rise to understanding the spectra and anomalies in lower-dimensional theories from the Calabi-Yau compactifications of F-theory. I will first explain what elliptic fibration is and explain Kodaira types, which gives rise an ADE classification. Utilizing Weierstrass model of elliptic fibrations, I will discuss Tate’s algorithm and Mordell-Weil group. By considering codimension one and two singularities and studying the geometry of crepant resolutions, we can define G-models that are geometrically-engineered models from F-theory. I will discuss the dictionary between the gauge theory and the elliptic fibrations and how to incorporate this to learn about topological invariants of the compactified Calabi-Yau that is one of the ingredient to understand spectra in the compactified theories. I will explain the more refined connection to understand the Coulomb branch of the 5d N=1 theories and 6d (1,0) theories and their anomalies from this perspective.

  1. Lecture 1
  2. Lecture 2
  3. Lecture 3
  4. Lecture 4
Aug, 23: Branes, Quivers, and BPS Algebras 1 of 4
Speaker: Miroslav Rapčák
Abstract:

This series of lectures covers a brief introduction into some algebro-geometric techniques used in the construction of BPS algebras. The starting point of our construction is a physical picture of D0-branes bound to D-branes of higher dimension. Using methods of the derived category of coherent sheaves, we are going to derive a framed quiver with potential describing supersymmetric quantum mechanics capturing the low-energy behavior of such D0-branes. For a large class of quivers, we are going to identify the space of BPS states with different melted-crystal configurations. Finally, by employing correspondences, we are going to construct an action of a BPS algebra known as the affine Yangian on the space of BPS states. The action of the affine Yangian factors through the action of various vertex operator algebras, Cherednik algebras, and more. This construction leads to an enormously rich interplay between physics, geometry and representation theory.

  1. Lecture 1
  2. Lecture 2
  3. Lecture 3
  4. Lecture 4
Aug, 23: Geometry of N=2 Supersymmetry 1 of 4
Speaker: Andy Neitzke
Abstract:

Coulomb branches of N=2 supersymmetric field theories in four dimensions support a rich geometry. My aim in these lectures will be to explain some aspects of this geometry, and its relation to the physics of the N=2 theories themselves.

I will first describe various constructions of N=2 theories and the corresponding Coulomb branches. In this story the main geometry visible is that of a complex integrable system, fibered over the Coulomb branch; one nice class of examples is associated to the Hitchin integrable system (moduli space of Higgs bundles over a Riemann surface). Fundamental objects in the N=2 theory (local operators, line operators, surface operators) all have geometric counterparts in the integrable system, as I will explain.

Next I will discuss a deformation of the story, which arises in physics from the Nekrasov-Shatashvili Omega-background. In this deformation, the Coulomb branch is replaced by a closely related space; for instance, the base of the Hitchin integrable system is replaced by a space parameterizing opers over the Riemann surface. One can use this deformation to give a concrete picture of the space of opers; in so doing one meets Stokes phenomena which are governed by the BPS indices (Donaldson-Thomas invariants) of the N=2 theory. This turns out to be closely related to the "exact WKB method" in analysis of ODEs. It is also connected to Riemann-Hilbert problems of a sort recently investigated by Bridgeland, as I will describe if time permits.

  1. Lecture 1
  2. Lecture 2
  3. Lecture 3
  4. Lecture 4
Aug, 23: Derived Geometry in Twists of Gauge Theories 1 of 4
Speaker: Tudor Dimofte
Abstract:

These lectures will review and develop methods in algebraic geometry (in particular, derived algebraic geometry) to describe topological and holomorphic sectors of quantum field theories. A recurring theme will be the interaction of local and extended operators, and of QFT's in different dimensions. The main examples will come from twists of supersymmetric gauge theories, and will connect to a large body of recent and ongoing work on 3d Coulomb branches, 3d mirror symmetry (and geometric Langlands), logarithmic VOA's and non-semisimple TQFT's, and categorified cluster algebras.

The basic plan for the lectures is:

  • Lecture 1 (2d warmup): categories of boundary conditions, interfaces, and Koszul duality
  • Lectures 2 and 3 (3d): twists of 3d N=2 and N=4 gauge theories; vertex algebras, chiral categories, and braided tensor categories; d mirror symmetry; quantum groups at roots of unity and derived non-semisimple 3d TQFT's (compared and contrasted with Chern-Simons theory)
  • Lecture 4 (4d): line and surface operators in 4d N=2 gauge theory, the coherent Satake category, and relations to Schur indices and 4d N=2 vertex algebras
  1. Lecture 1
  2. Lecture 2
  3. Lecture 3
  4. Lecture 4
Abstract:

Throughout my lectures I will explain the geometry of elliptic fibration which can give rise to understanding the spectra and anomalies in lower-dimensional theories from the Calabi-Yau compactifications of F-theory. I will first explain what elliptic fibration is and explain Kodaira types, which gives rise an ADE classification. Utilizing Weierstrass model of elliptic fibrations, I will discuss Tate’s algorithm and Mordell-Weil group. By considering codimension one and two singularities and studying the geometry of crepant resolutions, we can define G-models that are geometrically-engineered models from F-theory. I will discuss the dictionary between the gauge theory and the elliptic fibrations and how to incorporate this to learn about topological invariants of the compactified Calabi-Yau that is one of the ingredient to understand spectra in the compactified theories. I will explain the more refined connection to understand the Coulomb branch of the 5d N=1 theories and 6d (1,0) theories and their anomalies from this perspective.

  1. Lecture 1
  2. Lecture 2
  3. Lecture 3
  4. Lecture 4
Abstract:

We discuss deformation theory of rational curves and Mori’s famous Bend and Break techniques as well as their applications to Geometric Manin’s Conjecture. The lecture series contain introductory components as well as problem sessions and they aim for graduate students and postdocs.

Abstract:

Growing plant shoots exhibit circumnutations, namely, oscillations that draw three-dimensional trajectories, whose projections on the horizontal plane generate pendular, elliptical, or circular orbits. A large body of literature has followed the seminal work by Charles Darwin in 1880, but the nature of this phenomena is still uncertain and a long-lasting debate produced three main theories: the endogenous oscillator, the exogenous feedback oscillator, and the two-oscillator model. After briefly reviewing the three existing hypotheses, I will discuss a possible interpretation of these spontaneous oscillations as a Hopf-like bifurcation in a growing morphoelastic rod.

Abstract:

We cover Brauer classes, how they arise as obstructions on moduli spaces of sheaves, and how they can be used to obstruct rational points, highlighting recent links between the two.

Abstract:

We discuss deformation theory of rational curves and Mori’s famous Bend and Break techniques as well as their applications to Geometric Manin’s Conjecture. The lecture series contain introductory components as well as problem sessions and they aim for graduate students and postdocs.

Abstract:

The Pacific Rim Mathematical Association Congress meets in December 2022. A number of summer schools will take place prior to the main event at the end of the year. This summer school is part of the PRIMA Special Session on Arithmetic geometry: theory and computation. In this summer school, we cover two topics:(1) Brauer classes in moduli problems and arithmetic and (2) theory of rational curves and its arithmetic applications.

Abstract:

We discuss deformation theory of rational curves and Mori’s famous Bend and Break techniques as well as their applications to Geometric Manin’s Conjecture. The lecture series contain introductory components as well as problem sessions and they aim for graduate students and postdocs.

Aug, 2: Brauer classes in moduli problems and arithmetic: Lecture 1
Speaker: Nicholas Addington, Sara Frei
Abstract:

We cover Brauer classes, how they arise as obstructions on moduli spaces of sheaves, and how they can be used to obstruct rational points, highlighting recent links between the two.

Jul, 28: Environmental Escape from the Prisoner's Dilemma
Speaker: Jaye Sudweeks
Abstract:

During reproduction, viruses manufacture products that diffuse within the host cell. Because a virus does not have exclusive access to its own gene products, coinfection of multiple viruses allows for strategies of cooperation and defection— cooperators produce large amounts of gene product while defectors produce less product but specialize in appropriating a larger share of the common pool. Experimental data shows that, under conditions where coinfection is common, bacteriophage $\Phi$6 becomes trapped in a Prisoner’s dilemma, with defectors spreading to fixation, causing lowered population fitness. However, these experiments did not allow for fluctuation in the density of the external viral population. Here, I’ll discuss a model formulated to see if environmental feedback can free $\Phi$6 from the Prisoner’s dilemma. I’ll also discuss the concept of the Effective Game, which incorporates the frequency and density of different viral types in the environment.

Abstract:

n this talk, I will discuss random walks on Gromov hyperbolic spaces. Due
to the hyperbolicity of the spaces, random walks exhibit behaviors that
differ from the classic (Euclidean) ones. These behaviors include the
escape to infinity, central limit theorems when centered at the escape
rate, and geodesic tracking. I will explain how one can sharpen these
behaviors based on the recent observations by Gouëzel and Baik-Choi-Kim. If
time allows, I will also explain how one can implement this theory on
(non-hyperbolic) Teichmüller spaces.

Abstract:

The principal function of the nucleus is to facilitate storage, retrieval, and maintenance of the genetic information encoded into DNA and RNA sequences. A unique feature of nucleoplasm—the fluid of the nucleus—is that it contains chromatin (DNA) and RNA.

In contrast to other important biological polymer hydrogels, such as mucus and extracellular matrix, the nucleic acid polymers have a sequence. Recent experiments have shown that during the growth phase of the cell cycle, chromatin condenses in a sequence specific manner into regions within the nucleoplasm, possibly so that functionally related genes are grouped together spatially even though they might be far apart in terms of sequence distance.

At the same time, we are becoming increasingly aware of the role of liquid-liquid phase separation (LLPS) in cellular processes in the nucleus and the cytoplasm. Complex molecular interactions over a wide range of timescales can cause large biopolymers (RNA, protein, etc) to phase separate from the surrounding nucleoplasm into distinct biocondensates (spherical droplets in the simplest cases).

I will discuss recent work modelling the role of nuclear biocondensates in neurodegenerative disease and several ongoing projects related to
modelling and microscopy image analysis.

Jul, 7: Epidemic arrivals and Antibiotic Calenders
Speaker: Alastair Jamieson-Lane
Abstract:

Here I give two tiny talks on some of my research from the past couple years. In the first half of the talk I re-examine some popular heuristics for epidemic "time of spread" through the world airline network, and use hitting times and branching processes to explore the mathematical underpinnings of these observations. In the second half of the talk, we zoom in to exploring how antibiotics spreads through a single hospital, the various models and their conflicting recommendations. Mostly just some straightforward dynamical systems, with the opportunity for some cute asymptotic arguments on the side.

Abstract:

Understanding how cells change their identity and behaviour over time in living systems is a key question in many fields of biology. Measurement of cell states is inherently destructive, and so the relationship of the current state of a cell to some future state, or ‘fate’, cannot be observed experimentally. Trajectory inference refers to the general problem of trying to estimate various aspects of the state-fate relationship. We discuss optimal transport as a useful analytical tool for trajectory inference, and we develop a mathematical framework for recovering trajectories in both non-equilibrium as well as equilibrium systems.

Abstract:

The plasma membrane contains a wide array of glycans and glycolipids, many of which are capped by sialic acids (also called neuraminic acid). As a result, sialic acids are front-line mediators of interactions between the extracellular surface and the external environment. Examples include host-pathogen interactions (e.g. influenza) and the recognition of host cells by leukocytes (white blood cells). Thus, the composition of sialosides in the membrane can influence receptor-receptor interactions critical to immunity and cellular function. Our group is investigating the influence of sialic acid on the function of adhesion and immune receptors through the development of tools that alter catabolism of membrane sialosides. The human neuraminidases (NEU) are a family of four isoenzymes (NEU1, NEU2, NEU3, and NEU4) which have a range of substrate preferences as well as cellular and tissue localization. Our group has developed a panel of selective inhibitors, many with nanomolar potency, are being used to investigate how degradation of sialosides influences the function of cellular receptors. We use fluorescence microscopy to measure the size of receptor clusters and lateral mobility of receptors. These biophysical methods provide critical insight into the influence of NEU activity on membrane receptor organization. We have examined the role of NEU enzymes on the function and organization of leukocyte adhesion receptors. We find that specific NEU enzymes can modulate integrin adhesion and affect leukocyte transmigration. In related work, we have examined the influence of synthetic glycoconjugates and inhibitors of NEU on the organization of the CD22 receptor of B cells. We propose that understanding the specific roles of NEU isoenzymes will identify new therapeutic strategies for autoimmunity, inflammation, and cancer.

Abstract:

Growing plant shoots exhibit circumnutations, namely, oscillations that draw three-dimensional trajectories, whose projections on the horizontal plane generate pendular, elliptical, or circular orbits. A large body of literature has followed the seminal work by Charles Darwin in 1880, but the nature of this phenomena is still uncertain and a long-lasting debate produced three main theories: the endogenous oscillator, the exogenous feedback oscillator, and the two-oscillator model. After briefly reviewing the three existing hypotheses, I will discuss a possible interpretation of these spontaneous oscillations as a Hopf-like bifurcation in a growing morphoelastic rod.

Abstract:

Amyotrophic lateral sclerosis (ALS) is a fatal neurodegenerative disease primarily impacting motor neurons. Mutations in superoxide dismutase 1 (SOD1) are the second most common cause of familial ALS. Several of these mutations lead to misfolding or toxic gain of function in the SOD1 protein. Recently, we reported that misfolded SOD1 interacts with TNF receptor-associated factor 6 (TRAF6) in the SOD1-G93A rat model of ALS. Further, we showed in cultured cells that several mutant SOD1 proteins, but not wild type SOD1 protein, interact with TRAF6 via the MATH domain. Here, we sought to uncover the structural details of this interaction through molecular dynamics (MD) simulations of a dimeric model system, coarse grained using the AWSEM force field. We used direct MD simulations to identify buried residues, and predict binding poses by clustering frames from the trajectories. Metadynamics simulations were also used to deduce preferred binding regions on the protein surfaces from the potential of the mean force in orientation space. Well-folded SOD1 was found to bind TRAF6 via co-option of its native homodimer interface. However, if loops IV and VII of SOD1 were disordered, as typically occurs in the absence of stabilizing Zn2+ ion binding, these disordered loops now participated in novel interactions with TRAF6. On TRAF6, multiple interaction hot-spots were distributed around the equatorial region of the MATH domain beta barrel. Expression of TRAF6 variants with mutations in this region in cultured cells demonstrated that TRAF6 residue T475 facilitates interaction with different SOD1 mutants. These findings contribute to our understanding of the disease mechanism and uncover potential targets for the development of therapeutics.

Abstract:

Cytoplasmic streaming is the persistent circulation of the fluid contents of large eukaryotic cells, driven by the action of molecular motors moving along cytoskeletal filaments, entraining fluid. Discovered in 1774 by Bonaventura Corti, it is now recognized as a common phenomenon in a very broad range of model organisms, from plants to flies and worms. This talk will discuss physical approaches to understanding this phenomenon through a combination of experiments (on aquatic plants, Drosophila, and other active matter systems), theory, and computation. A particular focus will be on streaming in the Drosophilaoocyte, for which I will describe a recently discovered "swirling instability" of the microtubule cytoskeleton.

Abstract:

Effectively scaffolding epitopes on immunogens, in order to raise conformationally selective antibodies through active immunization, is a central problem in treating protein misfolding diseases, particularly neurodegenerative diseases such as Alzheimer's disease or Parkinson's disease. We seek to selectively target conformations enriched in toxic, oligomeric propagating species while sparing healthy forms of the protein which are often more abundant. To this end, we scaffolded cyclic peptides by varying the number of flanking glycines, to best mimic a misfolding-specific conformation of an epitope of alpha-synuclein enriched in the oligomer ensemble, as characterized by a region most readily disordered and solventexposed in a stressed, partially denatured protofibril. We screen and rank the cyclic peptide scaffolds of alpha-synuclein in silico based on their ensemble overlap properties with the fibril, oligomer-model, and isolated monomer ensembles. We introduce a method for screening against structured off-pathway targets in the human proteome, by selecting scaffolds with minimal conformational similarity between their epitope and the same primary sequence in structured human proteins. Ensemble comparison and overlap was quantified by the Jensen-Shannon Divergence, and a new measure introduced here---the embedding depth, which determines the extent to which a given ensemble is subsumed by another ensemble, and which may be a more useful measure in sculpting the conformational-selectivity of an antibody.

Abstract:

The swimming sperm of many external fertilizing marine organisms face complex fluid flows during their search for egg cells. Aided by chemotaxis, relatively weak flows and marine turbulence enhance spermegg fertilization rates through hydrodynamic guidance and mixing. However, strong flows can mechanically inhibit flagellar motility through elastohydrodynamic interactions - a phenomenon that remains poorly understood. We explore the effects of flow on the buckling dynamics of sperm flagella in an extensional flow through detailed numerical simulations, which are informed by microfluidic experiments and high-speed imaging. Compressional fluid forces lead to rich buckling dynamics of the sperm flagellum beyond a critical dimensionless sperm number, Sp, which represents the ratio of viscous force to elastic force. For non-motile sperm, the maximum buckling curvature and the number of buckling locations, or buckling mode, increase with increasing sperm number. In contrast, motile sperm exhibit an intrinsic flagellar curvature due to the propagation of bending waves along the flagellum. In compressional flow, this preexisting curvature acts as a precursor for buckling, which enhances local curvature without creating new buckling modes and leads to asymmetric beating. However, in extensional flow, flagellar beating remains symmetric with a smaller head yawing amplitude due to tensile forces. We also explore sperm motility in different shear flows. In the presence of Poiseuille flow, the sperm moves downstream or upstream depending on the flow strength along with net movement toward the centerline.

Abstract:

Motor-driven intracellular transport of organelles, vesicles, and other molecular cargo is a highly collective process. An individual cargo is often pulled by a team of transport motors, with numbers ranging from only a few to over 200. We explore the behaviour of these systems using a stochastic model for motordriven transport of molecular cargo by N motors which we solve analytically. We investigate the Ndependence of important quantities such as the velocity, precision of forward progress, energy flows between different system components, and efficiency; these properties obey simple scaling laws with N in two opposing regimes. Finally, we explore performance bounds and trade-offs as N is varied, providing insight into how different numbers of motors might be well-matched to different types of systems depending on which performance metrics are prioritized.

Abstract:

The reaction coordinate describing a transition between reactant and product is a fundamental concept in the theory of chemical reactions. Within transition-path theory, a quantitative definition of the reaction coordinate is found in the committor, which is the probability that a trajectory initiated from a given microstate first reaches the product before the reactant. Here we demonstrate an information-theoretic origin for the committor, show how it naturally arises from selecting out the transition-path ensemble from the equilibrium ensemble, and prove that the resulting entropy production is fully determined by committor dynamics. Our results provide parallel stochastic-thermodynamic and information-theoretic measures of the relevance of any system coordinate to the reaction, each of which are maximized by the committor, providing further support for its status as the ‘true’ reaction coordinate.

Abstract:

Spin-lattice (T1) relaxation is widely used in NMR to characterize chemical structure, molecular dynamics, and to provide a contrast mechanism for in-vivo imaging. When tissue is heterogeneous and multicompartment like brain tissue, however, it becomes difficult to model and assign physiological meaning to T1 relaxation due to the transfer of magnetization between pools during relaxation. Using wood as a model system, we explore the deviation from a standard exponential in the relaxation component stemming from this transfer. Fractional calculus offers a generalized exponential function to fit relaxation data from which a potentially unique parameter associated with the sample’s inhomogeneity results. We show the improved fit to the data of the fractional model compared to standard exponentials in wood as well as a lipid bilayer system and posit a white matter mapping technique based on the added fractional fit parameter.

Abstract:

The prevalence of intrinsically disordered polypeptides (IDPs) and protein regions in structural biology has prompted the recent development of molecular dynamics (MD) force fields for the more realistic representations of such systems. Using experimental NMR backbone scalar 3J-coupling constants of the intrinsically disordered proteins alpha-synuclein and amyloid-beta in their native aqueous environment as a metric, we compare the performance of four recent MD force fields, namely AMBER ff14SB, CHARMM C36m, AMBER ff99SB-disp, and AMBER ff99SBnmr2, by partitioning the polypeptides into an overlapping series of heptapeptides for which a cumulative total of 276 us MD simulations are performed. The results show substantial differences between the different force fields at the individual residue level. Except for ff99SBnmr2, the force fields systematically underestimate the scalar 3J(HN,Ha) couplings, due to an underrepresentation of beta-conformations and an overrepresentation of either alpha- or PPII conformations. The study demonstrates that the incorporation of coil library information in modern molecular dynamics force fields, as shown here for ff99SBnmr2, provides substantially improved performance and more realistic sampling of local backbone phi,psi dihedral angles of IDPs as reflected in good accuracy of computed scalar 3J(HN,Ha)-couplings with < 0.5 Hz error. Such force fields will enable a better understanding how structural dynamics and thermodynamics influence IDP function. Although the methodology based on heptapeptides used here does not allow the assessment of potential intramolecular long-range interactions, its computational affordability permits well-converged simulations that can be easily parallelized. This should make the quantitative validation of intrinsic disorder observed in MD simulations of polypeptides with experimental scalar J-couplings widely applicable.

Jun, 16: PIMS EDI Panel: Effective Allyship in STEM
Speaker: Sophie MacDonald,, Shirou Wang, Bobby Wilson, Douglas Farenick, Greg Martin
Abstract:

In recent months, PIMS has been actively engaging in conversations on diversity, equity, and inclusion. Following the Panel on Women in STEM held in May, this next event looks at ways in which effective allyship can build a better and stronger community in the Mathematical Sciences. Being an ally involves much more than passively accepting someone's rights. It is a conscious engagement and active advocacy for those whose voices may be stifled, unheard, or underappreciated. Our panelists look at actionable steps we can take to be better champions in academia.

Jun, 11: Connes fusion of the free fermions on the circle
Speaker: Peter Kristel
Abstract:

A conformal net on $S^1$ is an assignment $\mathcal{A}:\left\{\textrm{open subsets of } S^1\right\} \to \left\{\mbox{von Neumann algebras acting on } \mathcal{F}\right\}$, which satisfies a slew of axioms motivated by quantum field theory. In this talk, I will consider the free fermionic conformal net. In this case, the Hilbert space $\mathcal{F}$ is the Fock space generated by the positive energy modes of square-integrable spinors on the circle $𝐿^2(𝑆^1,\mathbb{S})$; and the von Neumann algebras are Clifford algebras generated by those elements of $𝐿^2(𝑆^1,\mathbb{S})$ whose support lies in $𝐼\subset 𝑆^1$. After going over this construction, I will argue that given an open interval $𝐼\subset 𝑆^1$, one can equip $\mathcal{F}$ with the structure of $\mathcal{A}(I)-\mathcal{A}(I)$-bimodule. I will then outline the construction of a canonical isomorphism of bimodules $\mathcal{F}\boxtimes_{\mathcal{A}(I_\_)}\mathcal{F}\to\mathcal{F}$ where $\boxtimes_{\mathcal{A}(I_\_)}$ stands for the Connes fusion product over the algebra assigned to the lower semi-circle $I_\_$. If time permits, I will discuss some (anticipated) applications of this isomorphism, for example in string geometry, or in the construction of the free fermion extended topological field theory.

Jun, 11: SU(2) hadrons on a quantum computer
Speaker: Jinglei Zhang
Abstract:

Lattice gauge theories are relevant in many fields of physics, and simulations with quantum computers can become a powerful tool to study them, especially in regimes inaccessible to classical numerical methods. In particular, non-Abelian gauge theories, which among other things describe fundamental particles’ interactions, are of great interest. In this talk I will discuss the first quantum simulation of a non-Abelian lattice gauge theory that includes dynamical matter. I will show how the theory is formulated in order to include colour degrees of freedom, and how this allows for the existence of baryons in the model, which do not exist in Abelian theories. A quantum computation of the low-lying spectrum of the model is performed on an IBM superconducting platform using a variational quantum eigensolver. This proof-of-concept demonstration was made possible by a resource-efficient approach in the design of the quantum algorithm, and lays out the foundation for further development of the field. This talk is based on arXiv:2102.08920.

Abstract:

Harmonic analysis on the multiplicative group of positive rational numbers (ℚ+) has not been part of the common quantum-theoretic toolkit. In this talk, I will discuss how it lends itself to the analysis of operators in ℓ2(ℕ), in some cases leading to spectacular new insights into their spectral properties. I will also discuss its application in a study of the Bose-Hubbard model, i.e. a model of an array of bosons with the nearest-neighbour interactions. The Fourier transform on ℚ+ uncovers the model's unobvious symmetries and surprising connections with other structures. In addition, I will report a rigorous, albeit computer-assisted, proof of the existence of quantum phase transitions in finite quantum systems of this type. The study of the Bose-Hubbard model has been carried out in collaboration with Prof. Jonas Fransson (Department of Physics and Astronomy, University of Uppsala).

Jun, 11: Secure Software Leasing Without Assumptions
Speaker: Sébastien Lord
Abstract:

Quantum cryptography is known for enabling functionalities that are unattainable using classical information alone. Recently, Secure Software Leasing (SSL) has emerged as one of these areas of interest. Given a circuit 𝐶 from a circuit class, SSL produces an encoding of 𝐶 that enables a recipient to evaluate 𝐶 and also enables the originator of the software to later verify that the software has been returned, meaning that the recipient has relinquished the possibility to further use the software. Such a functionality is unachievable using classical information alone, since it is impossible to prevent a user from keeping a copy of the software. Recent results have shown the achievability of SSL using quantum information for compute-and-compare functions (a generalization of point functions). However, these prior works all make use of setup or computational assumptions. We show that SSL is achievable for compute-and-compare circuits without any assumptions.
We proceed by studying quantum copy-protection, which is a notion related to SSL, but where the encoding procedure inherently prevents a would-be quantum software pirate from splitting a single copy of an encoding for 𝐶 into two parts each allowing a user to evaluate 𝐶. Using quantum message authentication codes, we show that point functions can be copy-protected without any assumptions against one honest and one malicious evaluator. We then show that a generic honest-malicious copy-protection scheme implies SSL. By prior work, this yields SSL for compute-and-compare functions.

This is joint work with Anne Broadbent, Stacey Jeffery, Supartha Podder, and Aarthi Sundaram.

Abstract:

A central question in quantum information theory is to determine physical resources required for quantum computational speedup. In the model of quantum computation with magic states classical simulation algorithms based on quasi-probability distributions, such as discrete Wigner functions, are used to study this question. For quantum systems of odd local dimension it has been known that negativity in the Wigner function can be seen as a computational resource. The case of qubits, however, resisted a similar approach for some time since the nice properties of Wigner functions for odd dimensional systems no longer hold for qubits. In our recent work we construct a hidden variable model, which replaces the Wigner function representation, for qubit systems where any quantum state can be represented by a probability distribution over a finite state space and quantum operations correspond to Bayesian update of the probability distribution. When applied to the model of quantum computation with magic states the size of the state space only depends on the number of magic states. This is joint work with Michael Zurel and Robert Raussendorf; Phys. Rev. Lett. 125, 260404 (2020).

Jun, 11: Entanglement of Free Fermions on Graphs
Speaker: Luc Vinet
Abstract:

The entanglement of free fermions on Hamming graphs will be discussed. This will be used to showcase how tools of algebraic combinatorics such as the Terwilliger algebra are well suited for this analysis. The usefulness of a Heun operator generalization will also be stressed and extensions to other association schemes will be mentioned.

Jun, 11: Topological superconductivity in quasicrystals
Speaker: Kaori Tanaka
Abstract:

Majorana fermions -- charge-neutral spin-1/2 particles that are their own antiparticles -- have been detected in one- and two-dimensional topological superconductors. Due to the non-Abelian exchange statistics that they obey, Majorana fermions open the door to new and powerful methods of quantum information processing. Motivated by the recent experimental discovery of superconductivity in a quasicrystal, we study the possible occurrence of non-Abelian topological superconductivity (TSC) in two-dimensional quasicrystals by the same mechanism as in crystalline counterparts. We show that the TSC phase can be realised in Penrose and Ammann-Beenker quasicrystals, where the Bott index is unity. Furthermore, we confirm the existence of Majorana zero modes along the surfaces and in a vortex at the centre of the system, consistently with the bulk-boundary correspondence.

Abstract:

I will describe a way to compute anomalies in general (2+1)D fermionic topological phases. First, a mathematical characterization of symmetry fractionalization for (2+1)D fermionic topological phases is presented, and then this data will be used to define a (3+1)D state sum for a topologically invariant path integral that depends on a generalized spin structure and G bundle on a 4-manifold. This path integral is a cobordism invariant and describes a (3+1)D fermion symmetry-protected topological state (SPT). The special case of time-reversal symmetry with 𝑇2=−1𝐹 gives a ℤ16 invariant of the 4D Pin+ smooth bordism group, and gives an example of a state sum that can distinguish exotic smooth structure.

Please note, the last 3 minutes of the talk are missing from the video

Abstract:

I will discuss recent results in the theory of symmetry-enriched topological phases, with a focus on the (2+1) case. I will review the classification of symmetry-enriched topological order and present general formula to compute relative 't Hooft anomaly for bosonic topological phases. I will also discuss partial results for fermionic topological phases and open questions.

Jun, 9: Classification of topological orders
Speaker: Theo Johnson-Freyd
Abstract:

Topological orders have a mathematical axiomatization in terms of their higher fusion categories of extended operators; the characterizing property of these higher fusion categories is that they are satisfy a nondegeneracy condition. After overviewing some of the higher category theory that goes into this axiomatization, I will describe what we do and don't know about the classification of topological orders in various dimensions.

Jun, 9: Hyperbolic band theory
Speaker: Joseph Maciejko
Abstract:

The notions of Bloch wave, crystal momentum, and energy bands are commonly regarded as unique features of crystalline materials with commutative translation symmetries. Motivated by the recent realization of hyperbolic lattices in circuit QED, I will present a hyperbolic generalization of Bloch theory, based on ideas from Riemann surface theory and algebraic geometry. The theory is formulated despite the non-Euclidean nature of the problem and concomitant absence of commutative translation symmetries. The general theory will be illustrated by examples of explicit computations of hyperbolic Bloch wavefunctions and bandstructures.

Abstract:

The COVID-19 pandemic has passed its initial peak in most countries in the world, making it ripe to assess whether the basic reproduction number (R0) is different across countries and what demographic, social, and environmental factors other than interventions characterize vulnerability to the virus. In this talk, I will show the association (linear and non-linear) between COVID-19 R0 across countries and 17 demographic, social and environmental variables obtained using a generalized additive model. Moreover, I will present a mathematical model of COVID-19 that we designed and used to explore the effects of adopting various vaccination and relaxation strategies on the COVID-19 epidemiological long-term projections in Ontario. Our findings are able to provide public health bodies with important insights on the effect of adopting various mitigation strategies, thereby guiding them in the decision-making process.

May, 27: The extremal length systole of the Bolza surface
Speaker: Didac Martinez Granado
Abstract:

Extremal length is a conformal invariant that plays an important
role in Teichmueller theory. For each essential closed curve on a Riemann
surface, it furnishes a function on the Teichmueller space. The extremal
length systole of a Riemann surface is defined as the infimum of extremal
lengths of all essential closed curves. Its hyperbolic analogue is the
hyperbolic systole: the infimum of hyperbolic lengths of all essential
closed curves. While the latter has been studied profusely, the extremal
length systole remains widely unexplored. For example, it is known that in
genus 2, the hyperbolic systole has a unique global maximum: the Bolza
surface. In this talk we introduce the extremal length systole and show
that in genus two it attains a strict local maximum at the Bolza surface,
where it takes the value square root of 2. This is joint work with Maxime
Fortier Bourque and Franco Vargas Pallete.

May, 20: The Manhattan Curve and Rough Similarity Rigidity
Speaker: Ryokichi Tanaka
Abstract:

For every non-elementary hyperbolic group, we consider the Manhattan curve, which was originally introduced by M. Burger (1993), associated to any pair of (say) word metrics. It is convex; we show that it is continuously differentiable and moreover is a straight line if and only if the corresponding two metrics are roughly similar, that is, they are within bounded distance after multiplying by a positive constant.

I would like to explain how it is related to central limit theorem for uniform counting measures on spheres, to ergodic theory of topological flows built on general hyperbolic groups, and to multifractal structure of Patterson-Sullivan measures. Furthermore I will present some explicit examples including a hyperbolic triangle group and compute the exact value of the mean distortion for a pair of word metrics by using automatic structures of the group.

Joint work with Stephen Cantrell (University of Chicago).

May, 19: Stochastic Organization in the Mitotic Spindle
Speaker: Christopher Miles
Abstract:

For cells to divide, they must undergo mitosis: the process of spatially organizing their copied DNA (chromosomes) to precise locations in the cell. This procedure is carried out by stochastic components that manage to accomplish the task with astonishing speed and accuracy. New advances from our collaborators in the New York Dept of Health provide 3D spatial trajectories of every chromosome in a cell during mitosis. Can these trajectories tell us anything about the mechanisms driving them? The structure and context of this cutting-edge data makes utilizing classical tools from data science or particle tracking challenging. I will discuss my progress with Alex Mogilner on developing analysis for this data and mathematical modeling of emergent phenomena.

May, 19: Data accuracy for risk management in changing climate
Speaker: Chandra Rujalapati
Abstract:

The decade of the 2010s was the hottest yet in more than 150 years of global mean temperature measurements. The key climate change signatures include intensifying extreme events such as widespread droughts, flooding and heatwaves, severe impacts on human health, food security, ecology, and species biodiversity. Climate has been changing from ice-age and is expected to change in future, yet the rate of change is alarming. Data plays a crucial role in developing risk management, mitigation and adaptation strategies under changing climate conditions. This talk focuses on uncertainties in hydrological data and the subsequent effect on extreme events like floods, droughts and heatwaves. Projected changes along with apparent biases in the global climate models, tools available for understanding future climate, are discussed. Importance of understanding uncertainties in observations and simulations and the need to probabilistically evaluate simulations to identify those that agree with observations is emphasized. Finally, the effect of data accuracy and incorporating uncertainty in informed decisions and risk management strategies is highlighted through a case study.

Speaker Biography

Chandra Rajulapati is a GWF-PIMS PDF, working with Dr. Simon Papalexiou at the Global Institute for Water Security (GIWS), University of Saskatchewan, on the Global Water Futures (GWF) project. She obtained her doctoral degree from the Indian Institute of Science (IISc) Bangalore, India, under the supervision of Prof. Pradeep Mujumdar. Her research focuses on understanding historical and future changes in hydroclimatic variables like precipitation and temperature at different scales, estimating risk due to extreme events like floods, droughts and heatwaves, and developing sustainable water management systems, risk assessment, adaptation and mitigation strategies.

May, 14: Changing the Culture Panel Discussion: How has Coronavirus changed the teaching of Mathematics?
Speaker: Kseniya Garaschuk, Dan Laitsch, Cameron Morland, Rob Lovell
Abstract:

The title for the panel discussion at this year's Changing the Culture conference was "How has Coronavirus changed the teaching of Mathematics?". In the video, each of our panelists addresses that question from their perspective. Following these opening remarks, the panelists respond to questions posed by the Changing the Culture community.

May, 14: PIMS Education Prize 2021: Bruce Dunham
Speaker: Bruce Dunham
Abstract:

PIMS is pleased to announce that the winner of the 2021 Education Prize is Dr. Bruce Dunham, Professor of Teaching in the Statistics Department of the University of British Columbia.

Dr. Dunham is an internationally respected expert in statistics education, and has contributed to education in the mathematical sciences by developing and providing resources for evidence-based teaching. He has also provided training and expert advice on statistics teaching and curriculum. He has served in a range of leadership roles at UBC and at the provincial and national level.

Dr. Dunham has served on the British Columbia Committee on the Undergraduate Program in Mathematics (BCCUPMS) since 2006 and has been the chair of the BCCUPMS Statistics sub-committee since that time. He has played a major role in the new BC Statistics 12 high school course, from defining the vision of the course, to the development of the curriculum and currently, in his continued role in teacher support and training, including offering five training workshops for teachers. At the national level, Dr. Dunham has served in various roles in the Statistical Society of Canada. He has served on the executive committee of the Society’s Education Section, having previously been secretary and president and currently president-elect. He has served on the Society’s Education Committee.

The evaluation committee was particularly impressed by the direct public impact of his curriculum work in the BC school system, and the development of free software for the community. Dr. Dunham is a tremendous advocate for mathematics and statistics, his leadership contributes to public awareness, fostering communication among various groups concerned with mathematical training. We are very pleased to celebrate him, and his achievements with the PIMS Education prize.

Dr. Dunham's prize was awarded as part of the 2021 Changing the Culture event.

Abstract:

Come with something floppy in hand--a string, a shoelace, a tie, or perhaps a floppy zucchini. Not only will we fold the object into strange fractional lengths, but we’ll also see how folding it into fractions leads to famous unsolved mathematics! Can you solve an unsolved problem?

May, 13: Towards optimal spectral gaps in large genus
Speaker: Michael Lipnowski
Abstract:

I'll discuss recent joint work with Alex Wright (arXiv:
2103.07496
) showing that typical large genus hyperbolic surfaces have first
Laplacian eigenvalue at least 3/16−ϵ.

Abstract:

Narrow escape (NE) problems are concerned with the calculation of the mean first passage time (MFPT) for a Brownian particle to escape a domain whose boundary contains N small windows (traps). NE problems arise in escape kinetics modeling in chemistry and cell biology, including receptor trafficking in synaptic membranes and RNA transport through nuclear pores. The related Narrow capture (NC) problems are characterized by the presence of small traps within the domain volume; such traps may be fully absorbing, or have absorbing and reflecting boundary parts. The MFPT of Brownian particles traveling in domains with traps is commonly modeled using a linear Poisson problem with Dirichlet-Neumann boundary conditions. We provide an overview of recent analytical and numerical work pertaining to the understanding and solution of different variants of NE and NC problems in three dimensions. The discussion includes asymptotic MFPT expressions in in the limit of small trap sizes, the cases of spherical and non-spherical domains, same and different trap sizes, the dilute trap fraction limit and MFPT scaling laws for N 1 traps, and the global optimization of trap positions to seek globally and locally optimal MFPT-minimizing trap arrangements. We also present recent comparisons of asymptotic and numerical solutions of NE problems to results of full numerical Brownian motion simulations, in the usual case of constant diffusivity, as well as considering more realistic anisotropic diffusion near the domain boundary.

Abstract:

Cellular adhesion is one of the most important interaction forces between cells and other tissue components. In 2006, Armstrong, Painter and Sherratt introduced a non-local PDE model for cellular adhesion, which was able to describe known experimental results on cell sorting and pattern formation. The pattern formation arises through non-local attractive interactions of the cells. In this talk I will analyse the underlying symmetries and bifurcations that lead to the observed patterns. (joint work with A. Buttenschoen).

May, 13: Patterns, waves and bufurcations in cell migration
Speaker: Leah Edelstein-Keshet , Andreas Buttenschoen
Abstract:

Cell migration plays a central roles in embryonic development, wound healing and immune surveillance. In 2008, Yoichiro Mori, Alexandra Jilkine and LEK published a reaction-diffusion model for the initial step of cell migration, the front-back chemical polarization that sets a cell's directionality. (More detailed mathematical properties of this model were described by the same group in 2011.) Since then, progress has been made in investigating how that simple "wave-pinning" mechanism is shaped and tuned by feedback from other proteins, from the cell's environment (extracellular matrix), from interplay with larger signaling networks, and from cell-cell interactions. In this talk, we will describe some of this progress and mathematical questions that arise. In particular, AB will demonstrate how his numerical PDE bifurcation analysis has helped us to understand how cells repolarize to reverse their direction of motion.

May, 12: Picture A Scientist Panel Discussion: Lilian Eva (Quan) Dyck
Speaker: Lilian Eva (Quan) Dyck
Abstract:

This video shows the speaker's response to a question asked as part of the PIMS Women in Mathematics Day: Panel discussion for Picture as Scientist.

Speaker Biography

Born in N. Battleford, Saskatchewan (1945), member of the Gordon First Nation in Saskatchewan and a first generation Chinese Canadian, the Honorable Dr. Lillian Eva Quan Dyck is well-known for her extensive work in the senate on Missing & Murdered Aboriginal Women. She was the first female First Nations senator and first Canadian born Chinese senator. Prior to being summoned to the senate by the Rt. Hon. Paul Martin in 2005, she was a Full Professor in the Department of Psychiatry and Associate Dean, College of Graduate Studies & Research at the University of Saskatchewan.

She earned a BA, MSc in Biochemistry and Ph.D. in Biological Psychiatry, all from the University of Saskatchewan. She was conferred a Doctor of Letters, Honoris Causa by Cape Breton University in 2007. She has also been recognized in a number of ways, such as: A National Aboriginal Achievement Award for Science & Technology in 1999 and most recently the YWCA Saskatoon Women of Distinction Lifetime Achievement Award in 2019. She has been presented three eagle feathers by the Indigenous community.

Abstract:

Cells receive chemical signals at localized surface receptors, process the data and make decisions on where to move or what to do. Receptors occupy only a small fraction of the cell surface area, yet they exhibit exquisite sensory capacity. In this talk I will give an overview of the mathematics of this phenomenon and discuss recent results focusing on receptor organization. In many cell types, receptors have very particular spatial organization or clustering - the biophysical role of which is not fully understood. In this talk I will explore how the number and configuration of receptors allows cells to deduce directional information on the source of diffusing particles. This involves a wide array of mathematical techniques from asymptotic analysis, homogenization theory, computational PDEs and Bayesian statistical methodologies. Our results show that receptor organization plays a large role in how cells decode their environmental situation and infer the location of distant sources.

Abstract:

A two species interacting system motivated by the density functional theory for triblock copolymers contains long range interaction that affects the two species differently. In a two species periodic assembly of discs, the two species appear alternately on a lattice. A minimal two species periodic assembly is one with the least energy per lattice cell area. There is a parameter b in [0,1] and the type of the lattice associated with a minimal assembly varies depending on b. There are several thresholds defined by a number B=0.1867... If b is in [0, B), a minimal assembly is associated with a rectangular lattice; if b is in [B, 1-B], a minimal assembly is associated with a square lattice; if b is in (1-B, 1], a minimal assembly is associated with a rhombic lattice. Only when b=1, this rhombic lattice is a hexagonal lattice. None of the other values of b yields a hexagonal lattice, a sharp contrast to the situation for one species interacting systems, where hexagonal lattices are ubiquitously observed.

Abstract:

The phase-field model, also known as the conserved Swift-Hohenberg equation, provides a useful model of crystallization that is derivable from the more accurate dynamical density functional theory. I will survey the properties of this model focusing on spatially localized structures and their organization in parameter space. I will highlight the role played by conserved mass and discuss the role played by these structures in the thermodynamic limit in both one and two spatial dimensions. I will then discuss dynamic crystallization via a propagating crystallization front. Two types of fronts can be distinguished: pulled and pushed fronts, with different properties. I will demonstrate, via direct numerical simulation, that the crystalline structures deposited by a rapidly moving front are not in thermodynamic equilibrium and so become disordered as they age. I will conclude with a discussion of a two-wavelength generalization of the model that exhibits quasicrystalline order in both two and three dimensions and of the associated spatially localized structures with different quasicrystalline motifs. The possible role of metastable spatially localized structures in nucleating crystallization will be highlighted.

Abstract:

The interplay between 1D traveling pulses with oscillatory tails (TPO) and heterogeneities of bump type is studied for a generalized three-component FitzHugh-Nagumo equation. We first present that stationary pulses with oscillatory tails (SPO) forms a “snaky" structure in homogeneous space, then TPO branches take a form of "figure-eight-like stack of isolas" located close to the snaky structure of SPO. Here we adopt voltage-difference as a bifurcation parameter. A drift bifurcation from SPO to TPO can be found by introducing another parameter at which these two solution sheets merge. As for the heterogeneous problem, in contrast to monotone tail case, there appears a nonlocal interaction between the TPO and the heterogeneity that creates infinitely many saddle solutions. The response of TPO shows a variety of dynamics including pinning and depinning processes in addition to penetration and rebound. Stable/unstable manifolds of these saddles interact with TPO in a complex way, which causes a subtle dependence on the initial condition and a difficulty to predict the behavior after collision even in one-dimensional space. Nevertheless, for 1D case, a systematic global exploration of solution branches (HIOP) induced by heterogeneities, and the reduction method to finite-dimensional ODEs allow us to clarify such a subtle dependence of initial condition and detailed mechanism of the transitions from penetration to pinning and pinning to rebound from dynamical system view point. It turns out that the basin boundary between two different outputs against the heterogeneities forms an infinitely many successive reconnections of heteroclinic orbits among those saddles as the height of the bump is changed, which causes the subtle dependence of initial condition. This is a joint work with Takeshi Watanabe.

Abstract:

Complex natural systems at times manifest transitions between disparate diffusive regimes. Efforts to devise measurement techniques capable of identifying the cross-over moments have recently borne fruit, however interpretation of findings remains contentious when the bigger picture is considered. This study generalises the 1D Gierer-Meinhardt reaction – diffusion model to a system that permits transitions between regular diffusive regimes with distinct diffusivities as well as sub-diffusion of a variable order. This is a sufficiently general, yet tractable description for the dynamics of a pattern qualitatively redolent of molecular clusters subject to transient anomalous diffusion mechanisms. The resulting system of equations substantiates the difficulties encountered when attempting to distinguish between various diffusive regimes in experimental settings: a non-monotonic dependence of the pat-tern’s evolution on parameters defining the diffusion mechanism is a common occurrence, as is a non-injective mapping between a given sequence of diffusion regimes and ensuing drift behaviour.

Abstract:

How organ size is controlled during development has been a subject of scientific study for centuries, but the growth control mechanisms are still poorly understood. The Drosophila wing imaginal disc has widely been used as a model system to study the regulation of growth. Growth control in the Drosophila wing disc involves various local signals, including signaling pathways, mechanical signals, etc. We developed a model of the Hippo pathway, which is the core regulatory pathway that mediates cell proliferation and apoptosis in Drosophila and mammalian cells, and contains a core kinase mechanism that affects control of the cell cycle and growth. We investigated the regulatory role of two upstream components Fat and Ds on the downstream mediator Yki of the pathway, and provide explanations to some of the seemingly contradictory experimental results. We found that a number of non-intuitive experimental results can be explained by subtle changes in the balances between inputs to the Hippo pathway. Since signal transduction and growth control pathways are highly con-served across species and directly involved in tumor growth, much of what is learned about Drosophila will have relevance to growth control in mammalian systems. Our recent work on morphogen transport in the wing disc will also be discussed.

Abstract:

We will give a brief overview of results in localized pattern formation and narrow escape problems that have been achieved through hybrid asymptotic-numerical methods. We will then briefly discuss how we have used these methods to extend results to surfaces with variable curvature.

May, 11: Extreme first passage times
Speaker: Sean Lawley
Abstract:

The first passage time (FPT) of a diffusive searcher to a target determines the timescale of many physical, chemical, and biological processes. While most studies focus on the FPT of a given single searcher, another important quantity in some scenarios is the FPT of the first searcher to find a target from a large group of searchers. This fastest FPT is called an extreme FPT and can be orders of magnitude faster than the FPT of a given single searcher. In this talk, we will explain recent results in extreme FPT theory and give special attention to the case of extreme FPTs to small targets. Time permitting, we will also explain results on extreme FPTs of subdiffusion modeled by a fractional time derivative and superdiffusion modeled by a fractional Laplacian.

Abstract:

We consider a reaction-diffusion system with two activators and one inhibitor. We prove rigorous results on the existence and stability of spiky patterns. We show that for certain conditions on the parameters these solutions can be stable. The approach is based on analytical methods such as elliptic estimates, Liapunov-Schmidt reduction and nonlocal eigenvalue problems. This is joint work with Weiwei Ao and Juncheng Wei.

Abstract:

Collective cell movement occurs throughout biology and medicine and there are many common features shared across different areas. I will review work we have carried out over the past few years on (i) systematically deriving a PDE model for tumour angiogenesis from a discrete formulation and comparing this model with the classical, phenomenological snail-trail model; (ii) agent-based models for cranial neural crest cell migration in a collabo-ration with experimental biologists that has revealed a number of new biological insights.

Abstract:

We propose an extension of the well-known Klausmeier model of vegetation to two plant species that consume water at different rates. Rather than competing directly, the plants compete through their intake of water, which is a shared resource between them. In semi-arid regions, the Klausmeier model produces vegetation spot patterns. We are interested in how the competition for water affects co-existence and stability of patches of different plant species. We consider two plant types: a thirsty species and a frugal species, that only differ by the amount of water they consume, while being identical in all other aspects. We find that there is a finite range of precipitation rate for which two species can co-exist. Outside of that range, (when the rate is either sufficiently low or high), the frugal species outcompetes the thirsty species. As the precipitation rate is decreased, there is sequence of stability thresholds such that thirsty plant patches are the first to die off, while the frugal spots remain resilient for longer. The pattern consisting of only frugal spots is the most resilient. The next-most-resilient pattern consists of all-thirsty patches, with the mixed pattern being less resilient than either of the homogeneous patterns. We also examine numerically what happens for very large precipitation rate. We find that for sufficiently high rate, the frugal plant takes over the entire range, outcompeting the thirsty plant.

Abstract:

The singularly perturbed Gierer-Meinhardt system has been a prototypical reaction diffusion system for the analysis of localized multi spike solutions. Motivated by recent interest in bulk-surface coupled systems, in this talk we address the structure and linear stability of multi spike solutions in the presence of inhomogeneous boundary conditions. Such inhomogeneities are shown to lead to the formation of both stable symmetric and asymmetric boundary bound spike solutions in one-dimensional domains and analogous solutions in higher dimensions.

Abstract:

The stability and dynamic properties of spike-type solutions to the Gierer- Meinhart equations are well understood. We examine the effect of adding noise to the equations on the spike-dynamics. We derive a stochastic ordinary differential equation for the motion of a single spike as well as the distribution of spike location from the associated Fokker-Plank equation. With sufficiently large amplitude noise, it is possible for the spike to reach the boundary of the domain and become effectively trapped for some time. In this case, we calculate the expected time to capture.

May, 10: A ring of spikes for the Schnakenberg model
Speaker: Theodore Kolokolnikov
Abstract:

Consider N spikes on located along a ring inside a unit disk. This highly symmetric configuration corresponds to an equilibrium of a two-dimensional Schnakenberg model; the ring radius can be characterized in terms of the modified Green’s function. We study the stability of such a ring with respect to both small and large eigenvalues (corresponding to spike position and spike height perturbations, respectively), and characterize the instability thresholds. For sufficiently large feed rate, we find that a ring of 8 or less spikes is stable with respect to both small and large eigenvalues, whereas a ring of 9 spikes is unstable with respect to small eigenvalues. For 8 spikes or less, as the feed rate is decreased, a small eigen-value instability is triggered first, followed by large eigenvalue instability. For 8 spikes, this instability appears to be supercritical, and deforms a ring into a square-type configuration. The main tool we use is circulant matrices and an analogue of the floquet theory.

Abstract:

The Phase-Field-Crystal (PFC) model is a simple yet surprisingly useful model for successfully capturing the phenomenology of grain growth in polycrystalline materials. PFC models are variational with a free energy functional which is very similar (in some cases, identical) to the well-known Swift-Hohenberg free energy. In this talk, we will discuss the simplest PFC functional and its gradient flow.

The first part of the talk will focus on large scales and address the model’s uncanny ability to o capture certain features of grain growth. We introduce a novel atom-based grain extraction and measurement method, and then use it to provide a comparison of multiple statistical grain metrics between (i) PFC simulations, (ii) experimental thin film data for aluminum, and (iii) simulations from the Mullins model.

In the second part of the talk, we investigate the PFC energy landscape at small scales (the local arrangement of atoms). We address patterns which are numerically observed as steady states via the framework of the modern theory of rigorous computations. In doing so, we make rigorous conclusions on the existence of similar states. In particular, we show that localized patterns and grain boundaries are critical and not simply metastable states. Finally, we present preliminary work on connections and parameter continuation in the PFC system. This talk consists of work from the PhD thesis of Gabriel Martine La Boissoniere at McGill. Parts of the talk also involve joint work with S. Esedoglu (Michigan), K. Barmak (Columbia) and J.-P. Lessard (McGill).

Abstract:

Motivated by recent work with biologists, I will showcase some mathematical results on Turing instabilities in complex domains. This is scientifically related to understanding developmental tuning in a variety of settings such as mouse whiskers, human fingerprints, bat teeth, and more generally pattern formation on multiple scales and evolving domains. Such phenomena are typically modelled using reaction-diffusion systems of morphogens, and one is often interested in emergent spatial and spatiotemporal patterns resulting from instabilities of a homogeneous equilibrium, which have been well-studied. In comparison to the well-known effects of how advection or manifold structure impacts unstable modes in such systems, I will present results on instabilities in heterogeneous systems, as well as those arising in the set-ting of evolving manifolds. These contexts require novel formulations of classical dispersion relations, and may have applications beyond developmental biology, such as in population dynamics (e.g. understanding colony or niche formation of populations in heterogeneous environments). These approaches also help close the vast gap between the simple theory of diffusion-driven pattern formation, and the messy reality of biological development, though there is still much work to be done in validating even complex theories against the rich dynamics observed in nature. I will close by discussing a range of open questions, many of which fall well beyond the extensions I will discuss.

Abstract:

Systems of activator-inhibitor reaction-diffusion equations posed on an infinite line are studied using a variety of analytical and numerical methods. A canonical form is considered that contains all known models with simple cubic autocatalytic nonlinearity and arbitrary constant and linear kinetics. Restricting attention at first to models that have a unique homogeneous equilibrium, this class includes the classical Schnakenberg and Brusselator models, as well as other systems proposed in the literature to model morphogenesis. Such models are known to feature Turing instability, when activator diffuses more slowly than inhibitor, leading to stable spatially periodic patterns. Conversely in the limit of small feed rates, semi-strong interaction asymptotic analysis as introduced by Michael Ward and his collaborators shows existence of isolated spike-like patterns.

Connecting these two regions, a certain universal two-parameter state diagram is revealed in which the Turing bifurcation becomes sub-critical, leading to the onset of homoclinic snaking. This regime then morphs into the spike regime, with the outer-fold being predicted by the semi-strong asymptotics. A rescaling of parameters and field concentrations shows how this state diagram can be studied independently of the diffusion rates. Temporal dynamics is found to strongly depend on the diffusion ratio though. A Hopf bifurcation occurs along the branch of stable spikes, which is subcritical for small diffusion ratio, leading to collapse to the homogeneous state. As the diffusion ratio increases, this bifurcation typically becomes supercritical, interacts with the homoclinic snaking and also with a supercritical homogeneous Hopf bifurcation, leading to complex spatio-temporal dynamics. The details are worked out for a number of different models that fit the theory using a mixture of weakly nonlinear analysis, semi-strong asymptotics and different numerical continuation algorithms.

The theory is extended include models, such as Gray-Scott, with bistability of homogeneous equilibria. A homotopy is studied that takes a Schnakenberg-like glycolysis model for r = 0 to the Gray-Scott model for r = 1. Numerical continuation is used to understand the complete sequence of transitions to two-parameter bifurcation diagrams within the localised pattern parameter regime as r varies. Several distinct codimension-two bifurcations are dis-covered including cusp and quadruple zero points for homogeneous steady states, a degenerate heteroclinic connection and a change in connectedness of the homoclinic snaking structure. The analysis is repeated for the Gierer-Meinhardt system, which lies outside the canonical framework. Similar transitions are found under homotopy between bifurcation diagrams for the case where there is a constant feed in the active field, to it being in the inactive field. Wider implications of the results are discussed for other kinds of pattern-formation systems as well as to distinguishing between different kinds of observed behaviour in the natural world.

Abstract:

Localized patterns in singularly perturbed reaction-diffusion equations typically consist of slow parts – in which the associated solution follows an orbit on a slow manifold in a reduced spatial dynamical system – alternated by fast excursions – in which the solution jumps from one slow manifold to another, or back to the original slow manifold. In this talk we consider the existence and stability of localized slow patterns that do not exhibit such jumps, i.e. that are completely embedded in a slow manifold of the singularly perturbed spatial dynamical system. These patterns have rarely been considered in the literature, for two reasons: (i) in the classical Gray-Scott/Gierer-Meinhardt type models that dominate the literature, the flow on the slow manifold is linear and thus cannot exhibit homoclinic pulse or heteroclinic front solutions; (ii) the slow manifolds occurring in the literature are typically trivial, or ‘vertical’ – i.e. given by u ≡ u_0, where u is the fast variable – so that the stability problem is determined by a simple (decoupled) scalar equation. The present talk is motivated by several explicit ecosystem models (of singularly perturbed reaction-diffusion type) that do give rise to nontrivial normally hyperbolic slow manifolds on which the flow may exhibit both homoclinic and heteroclinic orbits – that correspond to either stationary or traveling localized patterns. The associated spectral stability problems are at leading order given by a nonlinear, but scalar, eigenvalue problem with Sturm-Liouville type characteristics and we establish that homoclinic pulse patterns are typically unstable, while heteroclinic fronts can either be stable or unstable. However, we also show that homoclinic pulse patterns that are asymptotically close to a heteroclinic cycle may be stable. This result is obtained by explicitly determining the leading order approximations of 4 critical asymptotically small eigenvalues. Through this somewhat subtle analysis – that involves several orders of magnitude in the small parameter – we also obtain full control over the nature of the bifurcations – saddle-node, Hopf, global, etc. – that determine the existence and stability of the heteroclinic fronts and/or homoclinic pulses. Finally, we show that heteroclinic orbits may correspond to stable (slow) interfaces (in 2-dimensional space), while the homoclinic pulses must be unstable as localized stripes –even when they are stable in 1 space dimension.

May, 6: Random Hyperbolic Surfaces Via Flat Geometry
Speaker: Aaron Calderon
Abstract:

Mirzakhani gave an inductive procedure to build random hyperbolic surfaces by gluing together smaller random pieces along curves. She proved that as the length of the gluing curve grows, these families equidistribute in the moduli space of hyperbolic surfaces. In this talk, I’ll explain how the conjugacy (exposited in James’s talk) between the earthquake and horocycle flows provides a template for translating equidistribution results for flat surfaces into equidistribution results for hyperbolic ones. Using this correspondence, we address Mirzakhani’s twist torus conjecture and exhibit new limiting distributions for hyperbolic surfaces built out of symmetric pieces. This is joint work (in progress) with James Farre.

Abstract:

Entropic optimal transport has received a lot of attention in recent years and has become a popular framework for computational optimal transport thanks to the Sinkhorn scaling algorithm. In this talk, I will discuss the multi-marginal case which arises in different applied contexts in physics, economics and machine learning. I will show in particular that the multi-marginal Schrödinger system is well posed (joint work with Maxime Laborde) and that the multi-marginal Sinkhorn algorithm converges linearly.

Abstract:

Cells in tissue can communicate short-range via direct contact, and long-range via diffusive signals. In addition, another class of cell-cell communication is by long, thin cellular protrusions that are ~100 microns in length and ~100 nanometers in width. These so-called non-canonical protrusions include cytonemes, nanotubes, and airinemes. But, before establishing communication, they must find their target cell. Here we demonstrate airinemes in zebrafish are consistent with a finite persistent random walk model. We study this model by stochastic simulation, and by numerically solving the survival probability equation using Strang splitting. The probability of contacting the target cell is maximized for a balance between ballistic search (straight) and diffusive (highly curved, random) search. We find that the curvature of airinemes in zebrafish, extracted from live cell microscopy, is approximately the same value as the optimum in the simple persistent random walk model. We also explore the ability of the target cell to infer direction of the airineme’s source, finding the experimentally observed parameters to be at a Pareto optimum balancing directional sensing with contact initiation.

Abstract:

Any countable group G gives rise to a von Neumann algebra L(G). The classification of these group von Neumann algebras is a central theme in operator algebras. I will survey recent rigidity results which provide instances when various algebraic properties of groups, such as the presence or absence of a direct product decomposition, are remembered by their von Neumann algebras. I will also explain the strongest such rigidity results, where L(G) completely remembers G, and discuss some of the open problems in the area.

Abstract:

A measured geodesic lamination on a hyperbolic surface encodes the
horizontal trajectory structure of certain quadratic differentials.
Thurston’s earthquake flow along such a lamination induces a dynamical
system on the moduli space of hyperbolic surfaces sharing many properties
with the classical Teichmüller horocycle flow. Mirzakhani gave a dynamical
correspondence between the earthquake and horocycle flows, defined
Lebesgue-almost everywhere. In this talk, we extend Mirzakhani’s conjugacy
and define an extension of the earthquake flow to an action of the upper
triangular group P in PSL(2,R) mapping certain flow lines to Teichmüller
geodesics. We classify the P-invariant ergodic probability measures as
those coming from affine invariant measures on quadratic differentials and
show that our map is a measurable isomorphism between P actions with
respect to these measures. This is joint work with Aaron Calderon.

Abstract:

To initiate movement, cells need to form a well-defined "front" and "rear" through the process of cellular polarization. Polarization is a crucial process involved in embryonic development and cell motility and it is not yet well understood. Mathematical models that have been developed to study the onset of polarization have explored either biochemical or mechanical pathways, yet few have proposed a combined mechano-chemical mechanism. However, experimental evidence suggests that most motile cells rely on both biochemical and mechanical components to break symmetry. I will describe a mechano-chemical mathematical model for emergent organization driven by both cytoskeletal dynamics and biochemical reactions. We have identified one of the simplest quantitative frameworks for a possible mechanism for spontaneous symmetry breaking for initiation of cell movement. The framework relies on local, linear coupling between minimal biochemical stochastic and mechanical deterministic systems; this coupling between mechanics and biochemistry has been speculated biologically, yet through our model, we demonstrate it is a necessary and sufficient condition for a cell to achieve a polarized state.

Abstract:

We consider a stochastic bistable two-species generalized Lotka-Volterra model of the microbiome and use it as a testbed to analytically and numerically explore the role of direct (e.g., fecal microbiota transplantation) and indirect (e.g., changes in diet) bacteriotherapies. Two types of noise are included in this model, representing the immigration of bacteria into and within the gut (additive noise) and variations in growth rate associated with the spatially inhomogeneous distribution of resources (multiplicative noise). The efficacy of a bacteriotherapy is determined by comparing the mean first-passage times (the average time required for the system to transition from one basin of attraction to the other) with and without the intervention. Concepts from transition path theory are used to investigate how the role of noise affects these bacteriotherapies.

Abstract:

Nonparametric density estimation is a challenging problem in theoretical statistics -- in general a maximum likelihood estimate (MLE) does not even exist! Introducing shape constraints allows a path forward.

In this talk I will first discuss non-parametric density estimation under total positivity (i.e. log-supermodularity) and log-concavity. Although they possess very special structure, totally positive random variables are quite common in real world data and have appealing mathematical properties. Given i.i.d. samples from a totally positive and log-concave distribution, we prove that the MLE exists with probability one assuming there are at least 3 samples. We characterize the domain of the MLE and if the observations are 2-dimensional, we show that the logarithm of the MLE is a tent function (i.e. a piecewise linear function) with "poles" at the observations, and we show that a certain convex program can find it.

I will finish by discussing density estimation for log-concave graphical models. As before, we show that the MLE exists and is unique with probability 1. We also characterize the domain of the MLE, and show how to find it if the graphical model corresponds to a chordal graph. I will conclude by discussing some future directions.

Speaker Biography

Dr. Robeva is an Assistant Professor with the Department of Mathematics at the University of British Columbia. From 2016 – 2019, Dr. Robeva was a Statistics Instructor and an NSF Postdoctoral Fellow in the Department of Mathematics and the Institute for Data, Systems, and Society, at the Massachusetts Institute of Technology. Dr. Robeva completed her PhD in 2016 from UC Berkeley, and won the Bernard Friedman Memorial Prize in Applied Mathematics, for her thesis.

About the Prize

The UBC-PIMS Mathematical Sciences Young Faculty Award prize was created by two founding donors, Anton Kuipers and Darrell Duffie, to recognize UBC researchers for their leading edge work in mathematics or its applications in the sciences. Dr Elina Robeva is the 2020 winner and will give her colloquium on Thursday April 21, 2021.

Abstract:

Cell division is a vital mechanism for cell proliferation, but it often breaks its symmetry during animal development. Symmetry-breaking of cell division, such as the orientation of the cell division axis and asymmetry of daughter cell sizes, regulates morphogenesis and cell fate decision during embryogenesis, organogenesis, and stem cell division in a range of organisms. Despite its significance in development and disease, the mechanisms of symmetry-breaking of cell division remain unclear. Previous studies heavily focused on the mechanism of symmetry-breaking at metaphase of mitosis, wherein a localized microtubule-motor protein activity pulls the mitotic spindle. Recent studies found that cortical flow, the collective migration of the cell surface actin-myosin network, plays an independent role in the symmetry-breaking of cell division after anaphase. Using nematode C. elegans embryos, we identified extrinsic and intrinsic cues that pattern cortical flow during early embryogenesis. Each cue specifies distinct cellular arrangements and is involved in a critical developmental event such as the establishment of the left-right body axis, the dorsal-ventral body axis, and the formation of endoderm. Our research started to uncover the regulatory mechanisms underlying the cortical flow patterning during early embryogenesis.

Abstract:

What does mathematics, materials science, biology, and quantum
information science have in common? It turns out, there are many
connections worth exploring. I this talk, I will focus on graphs and random
walks, starting from the classical mathematical constructs and moving on to
quantum descriptions and applications. We will see how the notions of graph
entropy and KL divergence appear in the context of characterizing
polycrystalline material microstructures and predicting their performance
under mechanical deformation, while also allowing to measure adaptation in
cancer networks and entanglement of quantum states. We will discover
unified conditions under which master equations for classical random walks
exhibit nonlocal and non-diffusive behavior and see how quantum walks allow
to realize the coveted exponential speedup in quantum Hamiltonian
simulations. Recent classical and quantum breakthroughs and open questions
will be discussed.

For other events in this series see the quanTA events website.

Abstract:

Cellular polarization plays a critical during cellular differentiation, development, and cellular migration through the establishment of a long-lived cell-front and cell-rear. Although mechanisms of polarization vary across cells types, some common biochemical players have emerged, namely the RhoGTPases Rac and Rho. The low diffusion coefficient of the active form of these molecules combined with their mutual inhibitory interaction dynamics have led to a prototypical pattern-formation system that can polarizes cell through a non-Turing pattern formation mechanism termed wave-pinning. We investigate the effects of paxillin, a master regulator of adhesion dynamics, on the Rac-Rho system through a positive feedback loop that amplifies Rac activation. We find that paxillin feedback onto the Rac-Rho system produces cells that (i) self-polarize in the absence of any input signal (i.e., paxllin feedback causes a Turing instability) and (ii) become arrested due to the development of multiple protrusive regions. The former effect is a positive finding that can be related to certain cell-types, while the latter outcome is likely an artefact of the model. In order to minimize the effects of this artefact and produce cells that can both self-polarize as well as migrate for extended periods of time, we revisit some of model's parameter values and use lessons from previous models of polarization. This approach allows us to draw conclusions about the biophysical properties and spatiotemporal dynamics of molecular systems required for autonomous decision making during cellular migration.

Abstract:

In The Hitchhiker’s Guide to the Galaxy, by Douglas Adams, the number 42 was revealed to be the “Answer to the Ultimate Question of Life, the Universe, and Everything”. But he didn’t say what the question was! I will reveal that here. In fact it is a simple geometry question, which then turns out to be related to the mathematics underlying string theory.

Speaker Biography

John Baez is a leader in the area of mathematical physics at the interface between quantum field theory and category theory, and has broad interests in mathematics, and science more generally. He created one of the earliest blogs "This week's finds in Mathematical Physics" (before the term blog existed!)

Baez did his PhD at MIT, and was a Gibbs Instructor at Yale before moving to University of California, Riverside in 1988.

About the series

Starting in 2021, PIMS has inaugurated a high-level network-wide colloquium series. Distinguished speakers will give talks across the full PIMS network with one talk per month during the academic term. The 2021 speaker series is part of the PIMS 25th Anniversary showcase.

Abstract:

Cultures have their own identity; cultures interact. The medieval period contains within it widely varying cultures in Europe, India, and the middle East. The subject that eventually became modern mathematics did not live in a geographical cocoon during this period; it owes a great deal to several cultures. The journey of mathematics through Islam, for almost a millennium, changed it utterly. The shaping of algebra, the number system, arithmetic, geometry, optics, and mathematical astronomy had a major, yet unseen impact on how we think today. Yet, to understand the accomplishments of the medieval Islamic scientists, we must approach them on their terms. We shall explore some of the roots of modern mathematics, but also try to view the mathematical sciences in medieval Islam with eyes open to their vision --- not ours.​

Mar, 26: Khovanov homology and 4-manifolds
Speaker: Ciprian Manolescu
Abstract:

Over the last forty years, most progress in four-dimensional topology came from gauge theory and related invariants. Khovanov homology is an invariant of knots in of a different kind: its construction is combinatorial, and connected to ideas from representation theory. There is hope that it can tell us more about smooth 4-manifolds; for example, Freedman, Gompf, Morrison and Walker suggested a strategy to disprove the 4D Poincare conjecture using Rasmussen's invariant from Khovanov homology. It is yet unclear whether their strategy can work, and I will explain some of its challenges, as well as a new attempt to pursue it (joint work with Lisa Piccirillo). I will also review other topological applications of Khovanov homology, with regard to smoothly embedded surfaces in 4-manifolds.

Mar, 25: Reconsidering the History of Mathematics in India
Speaker: Clemency Montelle
Abstract:

Mathematics on the Indian subcontinent has been flourishing for over two and a half millennia, and this culture of inquiry has produced insights and techniques that are central to many of our mathematical practices today, such as the base ten decimal place value system and trigonometry. Indeed, many of their technical procedures, such as infinite series expansions for various mathematical relations predated those that were developed with the advent of the Calculus in Europe, but notably in contrasting intellectual circumstances with distinctly different epistemic priorities. However, while many histories of mathematics have centered on the so-called “western miracle” in their analysis of the ignition and flourishing of modern science, they have done so at the expense of other non-European traditions. This talk will highlight some of the significant mathematical achievements of India, and explore the work that remains to be done integrating them into more standard histories of mathematics.​

Mar, 25: Searching for the most likely evolution
Speaker: Giovanni Conforti
Abstract:

The theory of large deviations provides with a way to compute asymptotically the probability that an interacting particle system moves from a given configuration to another one over a fixed time interval. The problem of finding the most likely evolution realising the desired transition can be seen as a prototype of stochastic optimal transport problem, whose specific formulation depends on the choice of interaction mechanism. The first goal of this talk is to present some notable examples of this family of transport problems such as the Schrödinger problem and its mean field and kinetic counterparts. The second goal of the talk is to discuss some (possibly open) questions on the ergodic behaviour of optimal solutions and how their answer relies upon a combination of tools coming from Riemannian geometry, functional inequalities and stochastic control.

Abstract:

A celebrated theorem of Jorgens-Calabi-Pogorelov says that global convex solutions to the Monge-Ampere equation det(D^2u) = 1 are quadratic polynomials. On the other hand, an example of Pogorelov shows that local solutions can have line singularities. It is natural to ask what kinds of singular structures can appear in functions that solve the Monge-Ampere equation outside of a small set. We will discuss examples of functions that solve the equation away from finitely many points but exhibit polyhedral and Y-shaped singularities. Along the way we will discuss geometric and applied motivations for constructing such examples, as well as their connection to a certain obstacle problem for the Monge-Ampere equation.

Abstract:

One of the most important techniques provided by modern logic is the use of models to show the consistency of theories. The technique burst onto the scene in the late 19th century, and had its most important early instance in demonstrating the consistency of non-Euclidean geometries. This talk investigates the development of that technique as it transitions from a geometric tool to an all-purpose tool of logic. I’ll argue that the standard narrative, according to which our modern technique provides answers to centuries-old questions, is mistaken. Once we understand how modern models work, I’ll argue, we see important differences between the kinds of consistencyclaims that would have made sense e.g. to Kant and the kinds of consistency-claims that we can demonstrate today. We’ll also see some philosophically-interesting shifts, over this time period, in the kinds of things that we take proofs to demonstrate.

Speaker

Patricia Blanchette is Professor of Philosophy and Glynn Family Honors Collegiate Chair in the Department of Philosophy at the University of Notre Dame. Prior to coming to Notre Dame, Blanchette taught in the Department of Philosophy at Yale University. Blanchette works in the history and philosophy of logic, philosophy of mathematics, history of analytic philosophy, and philosophy of language. She is an editor of the Bulletin of Symbolic Logic, and serves on the editorial boards of the Notre Dame Journal of Formal Logic and of Philosophia Mathematica. She is the author of Frege’s Conception of Logic (Oxford University Press 2012).

Abstract:

The Arab mathematician al-Khwarizmi is usually said to be the ‘father of algebra’, or otherwise that ‘the Arabs invented algebra’. There is probably nothing in the previous sentence that is true (except the ‘usually’). It turns out that the traditional story is just intellectually, mathematically, and culturally lazy. A little bit of thinking about the original texts, the mathematics, and a little bit of historical context leads to a much more problematic, culturally rich, and technically subtle story. We still don’t know the whole story – there is lots of room for further research, if you have the languages – and a lot of room for thinking about past mathematics (and by symmetry present day mathematics) as existing in a rich, complex social and intellectual matrix, and not just as a succession of correct theorems. The story might even involve the Sogdians, and you have never heard of them!​

Abstract:

yperbolic Lattices are tessellations of the hyperbolic plane
using, for instance, heptagons or octagons. They are relevant for quantum
error correcting codes and experimental simulations of quantum physics in
curved space. Underneath their perplexing beauty lies a hidden and,
perhaps, unexpected periodicity that allows us to identify the unit cell
and Bravais lattice for a given hyperbolic lattice. This paves the way for
applying powerful concepts from solid state physics and, potentially,
finding a generalization of Bloch's theorem to hyperbolic lattices. In my
talk, I will explain some of the mathematics underlying this hyperbolic
crystallography.

For other events in this series see the quanTA events website
.

Mar, 17: The geometry of the spinning string
Speaker: Peter Kristel
Abstract:

The development of quantum electrodynamics is one of the major achievements of theoretical physics and mathematics of the 20th century, called the "Jewel of physics" by Richard Feynman. This talk is not about that. Instead, I explain two of its basic ingredients - Feynman diagrams, and Spinor bundles - and then describe how these can be adapted to "electron-like" strings. This will lead us naturally to the Spinor bundle on loop space, which I will describe in some detail. An element of loop space, i.e. a smooth function from the circle into some fixed manifold, is supposed to represent a string at a fixed moment in time. I will then explain the notion of a fusion product (on this bundle), and argue that this is a manifestation of the principle of locality, which is ubiquitous in physics. If time permits, I will discuss some ongoing work, in collaboration with Matthias Ludewig, Darvin Mertsch, and Konrad Waldorf, where we describe how this fusive spinor bundle on loop space fits beautifully in the higher categorical framework of 2-vector bundles.

Mar, 11: Ergodic theorems along trees
Speaker: Anush Tserunyan
Abstract:

In the classical pointwise ergodic theorem for a probability measure preserving (pmp) transformation $T$, one takes averages of a given integrable function over the intervals $\{x, T(x), T^2(x), \hdots, T^n(x)\}$ in the forward orbit of the point $x$. In joint work with Jenna Zomback, we prove a “backward” ergodic theorem for a countable-to-one pmp $T$, where the averages are taken over subtrees of the graph of $T$ that are rooted at $x$ and lie behind $x$ (in the direction of $T^{-1}$). Surprisingly, this theorem yields (forward) ergodic theorems for countable groups, in particular, one for pmp actions of free groups of finite rank where the averages are taken along subtrees of the standard Cayley graph rooted at the identity. For free group actions, this strengthens the best known result in this vein due to Bufetov (2000). After reviewing the subject history and discussing the statements of our theorems in the first half of the talk, we will highlight some ingredients of proofs in the second half.

Abstract:

he asymmetric exclusion process (ASEP) is a model of particles hopping on a one-dimensional lattice. While it was initially introduced by Macdonald-Gibbs-Pipkin to provide a model for translation in protein synthesis, the stationary distribution of the ASEP and its variants has surprising connections to combinatorics. I will explain how the study of the ASEP on a ring leads to new formulas for Macdonald polynomials, a remarkable family of multivariate polynomials which generalize Schur polynomials. In a different direction, the inhomogeneous ASEP on a ring is closely connected to Schubert polynomials, which represent classes of Schubert varieties in the flag variety. This talk is based on joint work with Corteel-Mandelshtam, and joint work with Donghyun Kim.

Abstract:

The $s$-colour size-Ramsey number of a hypergraph $H$ is the minimum number of edges in a hypergraph $G$ whose every $s$-edge-colouring contains a monochromatic copy of $H$. We show that the $s$-colour size-Ramsey number of the $t$-power of the $r$-uniform tight path on $n$ vertices is linear in $n$, for every fixed $r, s, t$, thus answering a question of Dudek, La Fleur, Mubayi, and R\"odl (2017). In fact, we prove a stronger result that allows us to deduce that powers of bounded degree hypergraph trees and of `long subdivisions' of bounded degree hypergraphs have size-Ramsey numbers that are linear in the number of vertices. This extends recent results about the linearity of size-Ramsey numbers of powers of bounded degree trees and of long subdivisions of bounded degree graphs.

This is joint work with Shoham Letzter and Alexey Pokrovskiy.

Mar, 4: Mathematics, Colonization and Empire
Speaker: Tom Archibald, SFU
Abstract:

Mathematics has been an important tool in various colonizing enterprises; and in the last 2 centuries the colonizing enterprise has often involved teaching mathematics to the new subjects of the imperial or colonial regime. In this rather informal discussion we will look at mathematics and mathematicians as instruments in this process, using examples from various time periods and places.

Abstract:

The optimal transport problem provides a fundamental and quantitative way to measure the distance between probability distributions. Recently, it has been successfully used to analyze the evolutionary dynamics in physics and biology. Motivated by questions of pricing in financial mathematics and control of distributed agents, stochastic variants of optimal transport have been developed. Over the past few years, my postdoc supervisors at the University of British Columbia (Nassif Ghoussoub and Young-Heon Kim) and I have developed a robust method to analyze these problems using convex duality, stochastic optimal control theory, and partial differential equation analysis.

This talk will focus on these variants of optimal transport, their applications, and our methods of analysis. Particular attention will be paid to the connections with mean field games and to a new direction of research that incorporates the practical limitation of partial information.

Feb, 24: Fusion rings and their categorifications
Speaker: Andrew Schopieray
Abstract:

Fusion rings are a special class of associative unital rings with nonnegative integer structure constants and a notion of duality. For example, the group ring of a finite group is a fusion ring. We study fusion rings mainly because they arise as Grothendieck rings of categories associated to Hopf algebras, semisimple Lie algebras, vertex operator algebras, etc. In turn, these categories have application to topological quantum field theory, invariants of knots and links, and quantum computation, to name a few. In this talk we will discuss the brief history of the classification of categorifiable fusion rings and how number-theoretic properties of fusion rings dictate the existence of, or properties of, their categorifications.

Abstract:

Any translation surface can be presented as a collection of polygons in the plane with sides identified. By acting linearly on the polygons, we obtain an action of GL(2,R) on moduli spaces of translation surfaces. Recent work of Eskin, Mirzakhani, and Mohammadi showed that GL(2,R) orbit closures are locally described by linear equations on the edges of the polygons. However, which linear manifolds arise this way is mysterious.
In this lecture series, we will describe new joint work that shows that when an orbit closure is sufficiently large it must be a whole moduli space, called a stratum in this context, or a locus defined by rotation by π symmetry.
We define "sufficiently large" in terms of rank, which is the most important numerical invariant of an orbit closure, and is an integer between 1 and the genus g. Our result applies when the rank is at least 1+g/2, and so handles roughly half of the possible values of rank.
The five lectures will introduce novel and broadly applicable techniques, organized as follows:
An introduction to orbit closures, their rank, their boundary in the WYSIWYG partial compactification, and cylinder deformations.
Reconstructing orbit closures from their boundaries (this talk will explicate a preprint of the same name).
Recognizing loci of covers using cylinders (this talk will follow a preprint titled “Generalizations of the Eierlegende-Wollmilchsau”).
An overview of the proof of the main theorem; marked points (following the preprint “Marked Points on Translation Surfaces”); and a dichotomy for cylinder degenerations.
Completion of the proof of the main theorem.

Abstract:

Any translation surface can be presented as a collection of polygons in the plane with sides identified. By acting linearly on the polygons, we obtain an action of GL(2,R) on moduli spaces of translation surfaces. Recent work of Eskin, Mirzakhani, and Mohammadi showed that GL(2,R) orbit closures are locally described by linear equations on the edges of the polygons. However, which linear manifolds arise this way is mysterious.
In this lecture series, we will describe new joint work that shows that when an orbit closure is sufficiently large it must be a whole moduli space, called a stratum in this context, or a locus defined by rotation by π symmetry.
We define "sufficiently large" in terms of rank, which is the most important numerical invariant of an orbit closure, and is an integer between 1 and the genus g. Our result applies when the rank is at least 1+g/2, and so handles roughly half of the possible values of rank.
The five lectures will introduce novel and broadly applicable techniques, organized as follows:
An introduction to orbit closures, their rank, their boundary in the WYSIWYG partial compactification, and cylinder deformations.
Reconstructing orbit closures from their boundaries (this talk will explicate a preprint of the same name).
Recognizing loci of covers using cylinders (this talk will follow a preprint titled “Generalizations of the Eierlegende-Wollmilchsau”).
An overview of the proof of the main theorem; marked points (following the preprint “Marked Points on Translation Surfaces”); and a dichotomy for cylinder degenerations.
Completion of the proof of the main theorem.

Feb, 11: New lower bounds for van der Waerden numbers
Speaker: Ben Green
Abstract:

Colour ${1,\ldots,N}$ red and blue, in such a manner that no 3 of the blue elements are in arithmetic progression. How long an arithmetic progression of red elements must there be? It had been speculated based on numerical evidence that there must always be a red progression of length about $\sqrt{N}$. I will describe a construction which shows that this is not the case - in fact, there is a colouring with no red progression of length more than about $\exp{\left(\left(\log{N}\right)^{3/4}\right)}$, and in particular less than any fixed power of $N$.

I will give a general overview of this kind of problem (which can be formulated in terms of finding lower bounds for so-called van der Waerden numbers), and an overview of the construction and some of the ingredients which enter into the proof. The collection of techniques brought to bear on the problem is quite extensive and includes tools from diophantine approximation, additive number theory and, at one point, random matrix theory

Abstract:

Establishing inequalities among graph densities is a central pursuit in extremal graph theory. One way to certify the nonnegativity of a graph density expression is to write it as a sum of squares or as a rational sum of squares. In this talk, we will explore how one does so and we will then identify simple conditions under which a graph density expression cannot be a sum of squares or a rational sum of squares. Tropicalization will play a key role for the latter, and will turn out to be an interesting object in itself. This is joint work with Greg Blekherman, Mohit Singh, and Rekha Thomas.

Abstract:

Any translation surface can be presented as a collection of polygons in the plane with sides identified. By acting linearly on the polygons, we obtain an action of GL(2,R) on moduli spaces of translation surfaces. Recent work of Eskin, Mirzakhani, and Mohammadi showed that GL(2,R) orbit closures are locally described by linear equations on the edges of the polygons. However, which linear manifolds arise this way is mysterious.

In this lecture series, we will describe new joint work that shows that when an orbit closure is sufficiently large it must be a whole moduli space, called a stratum in this context, or a locus defined by rotation by π symmetry.

We define "sufficiently large" in terms of rank, which is the most important numerical invariant of an orbit closure, and is an integer between 1 and the genus g. Our result applies when the rank is at least 1+g/2, and so handles roughly half of the possible values of rank.

The five lectures will introduce novel and broadly applicable techniques, organized as follows:

An introduction to orbit closures, their rank, their boundary in the WYSIWYG partial compactification, and cylinder deformations.
Reconstructing orbit closures from their boundaries (this talk will explicate a preprint of the same name).
Recognizing loci of covers using cylinders (this talk will follow a preprint titled “Generalizations of the Eierlegende-Wollmilchsau”).
An overview of the proof of the main theorem; marked points (following the preprint “Marked Points on Translation Surfaces”); and a dichotomy for cylinder degenerations.
Completion of the proof of the main theorem.

Abstract:

A connected matching is a matching contained in a connected component. A well-known method due to Łuczak reduces problems about monochromatic paths and cycles in complete graphs to problems about monochromatic matchings in almost complete graphs. We show that these can be further reduced to problems about monochromatic connected matchings in complete graphs.

I will describe Łuczak's reduction, introduce the new reduction, and mention potential applications of the improved method.

Abstract:

Over the millennia, from Theano (born c. 546 B.C.), the wife of the Greek mathematician Pythagoras and herself a mathematician, to Maryam Mirzakhani (May 1977 – July 2017), who in 2014 became the first woman to win the Fields medal, the most prestigious award in mathematics, there have been many brilliant female mathematicians working in all areas of math. I will mention a few who were active in the late 19th and the first half of the 20th centuries, and discuss the work and impact of one of them in greater depth.​

Jan, 30: Vector Copulas and Vector Sklar Theorem
Speaker: Yanqin Fan
Abstract:

This talk introduces vector copulas and establishes a vector version of Sklar’s theorem. The latter provides a theoretical justification for the use of vector copulas to characterize nonlinear or rank dependence between a finite number of random vectors (robust to within vector dependence), and to construct multivariate distributions with any given non-overlapping multivariate marginals. We construct Elliptical, Archimedean, and Kendall families of vector copulas and present algorithms to generate data from them. We introduce a concordance ordering for two random vectors with given within-dependence structures and generalize Spearman’s rho to random vectors. Finally, we construct empirical vector copulas and show their consistency under mild conditions.

Abstract:

Suppose you want to open up 7 coffee shops so that people in the downtown area have to walk the least amount to get their morning coffee. That’s a classical problem in Optimal Transport, minimizing the Wasserstein distance between the sum of 7 Dirac measures and the (coffee-drinking) population density. But in reality things are trickier. If the 7 coffee shops go well, you want to open an 8th and a 9th and you want to remain optimal in this respect (and the first 7 are already fixed). We find optimal rates for this problem in ($W_2$) in all dimensions. Analytic Number Theory makes an appearance and, in fact, Optimal Transport can tell us something new about $\sqrt{2}$ . All of this is also related to the question of approximating an integral by sampling in a number of points and a conjectured extension of the Kantorovich-Rubinstein duality regarding the $W_1$ distance and testing of two measures against Lipschitz functions.

Jan, 30: Deep kernel-based distances between distributions
Speaker: Danica Sutherland
Abstract:

Optimal transport, while widespread and effective, is not the only game in town for comparing high-dimensional distributions. This talk will cover a set of related distances based on kernel methods, in particular the maximum mean discrepancy, and especially their use with learned kernels defined by deep networks. This set of distance metrics allows for effective use in a variety of applications; we will cover foundational properties and develop variants useful for distinguishing distributions, training generative models, and other machine learning applications.

Abstract:

The past decade has seen a rapid development of data-driven plant breeding strategies based on the two significant technological developments. First, the use of high throughput DNA sequencing technology to identify millions of genetic markers on that characterize the available genetic diversity captured by the thousands of available accessions in each major crop species. Second, the development of high throughput imaging platforms for estimating quantitative traits associated with easily accessible above-ground structures such as shoots, leaves and flowers. These data-driven breeding strategies are widely viewed as the basis for rapid development of crops capable of providing stable yields in the face of global climate change. Roots and other below-ground structures are much more difficult to study yet play essential roles in adaptation to climate change including as uptake of water and nutrients. Estimation of quantitative traits from images remains a significant technical and scientific bottleneck for both above and below-ground structures. The focus of this talk, inspired by the analytical results of Kac, van den Berg and many others in the area of spectral geometry, is to describe a computational and statistical methodology that employs stochastic processes as quantitative measurement tools suitable for characterizing images of multi-scale dendritic structures such as plant root systems. The substrate for statistical analyses in Wasserstein space are hitting distributions obtained by simulation. The practical utility of this approach is demonstrated using 2D images of sorghum roots of different genetic backgrounds and grown in different environments.

Jan, 29: Bandit learning of Nash equilibria in monotone games
Speaker: Maryam Kamgarpour
Abstract:

Game theory is a powerful framework to address optimization and learning of multiple interacting agents referred to as players. In a multi-agent setting, the notion of Nash equilibrium captures a desirable solution as it exhibits stability, that is, no player has incentive to deviate from this solution. From the viewpoint of learning the question is whether players can learn their Nash equilibrium strategies with limited information about the game. In this talk, I address our work on designing distributed algorithms for players so that they can learn the Nash equilibrium based only on information regarding their experienced payoffs. I discuss the convergence of the algorithm and its applicability to a large class of monotone games.

Abstract:

Optimal transport (OT) has recently gained lot of interest in machine learning. It is a natural tool to compare in a geometrically faithful way probability distributions. It finds applications in both supervised learning (using geometric loss functions) and unsupervised learning (to perform generative model fitting). OT is however plagued by the curse of dimensionality, since it might require a number of samples which grows exponentially with the dimension. In this talk, I will explain how to leverage entropic regularization methods to define computationally efficient loss functions, approximating OT with a better sample complexity. More information and references can be found on the website of our book “Computational Optimal Transport” https://optimaltransport.github.io/

Abstract:

The initial value problem for many important PDEs (Burgers, Euler, Hamilton-Jacobi, Navier-Stokes equations, systems of conservation laws with convex entropy, etc…) can be often reduced to a convex minimization problem that can be seen as a generalized optimal transport problem involving matrix-valued density fields. The time boundary conditions enjoy a backward-forward structure of “ballistic” type, just as in mean-field game theory.

Abstract:

Any translation surface can be presented as a collection of polygons in the plane with sides identified. By acting linearly on the polygons, we obtain an action of GL(2,R) on moduli spaces of translation surfaces. Recent work of Eskin, Mirzakhani, and Mohammadi showed that GL(2,R) orbit closures are locally described by linear equations on the edges of the polygons. However, which linear manifolds arise this way is mysterious.

In this lecture series, we will describe new joint work that shows that when an orbit closure is sufficiently large it must be a whole moduli space, called a stratum in this context, or a locus defined by rotation by π symmetry.

We define "sufficiently large" in terms of rank, which is the most important numerical invariant of an orbit closure, and is an integer between 1 and the genus g. Our result applies when the rank is at least 1+g/2, and so handles roughly half of the possible values of rank.

The five lectures will introduce novel and broadly applicable techniques, organized as follows:

An introduction to orbit closures, their rank, their boundary in the WYSIWYG partial compactification, and cylinder deformations.
Reconstructing orbit closures from their boundaries (this talk will explicate a preprint of the same name).
Recognizing loci of covers using cylinders (this talk will follow a preprint titled “Generalizations of the Eierlegende-Wollmilchsau”).
An overview of the proof of the main theorem; marked points (following the preprint “Marked Points on Translation Surfaces”); and a dichotomy for cylinder degenerations.
Completion of the proof of the main theorem.

Abstract:

What is Khovanov homology, and when is it boring?

Khovanov homology, though relatively young, is difficult to survey in an hour. This talk will nevertheless attempt to do so, by focussing on the problem of characterizing thin links—those links with simplest-possible Khovanov homology. This is a story that is still unfolding; I will describe some progress that is part of a joint project with Artem Kotelskiy and Claudius Zibrowius.

Abstract:

I will give an overview of a few places where combinatorial structures have an interesting role to play in quantum field theory and which I have been involved in to varying degrees, from the Connes-Kreimer Hopf algebra and other renormalization Hopf algebras, to the combinatorics of Dyson-Schwinger equations and the graph theory of Feynman integrals.

For other events in this series see the quanTA events website.

Abstract:

Any translation surface can be presented as a collection of polygons in the plane with sides identified. By acting linearly on the polygons, we obtain an action of GL(2,R) on moduli spaces of translation surfaces. Recent work of Eskin, Mirzakhani, and Mohammadi showed that GL(2,R) orbit closures are locally described by linear equations on the edges of the polygons. However, which linear manifolds arise this way is mysterious.

In this lecture series, we will describe new joint work that shows that when an orbit closure is sufficiently large it must be a whole moduli space, called a stratum in this context, or a locus defined by rotation by π symmetry.

We define "sufficiently large" in terms of rank, which is the most important numerical invariant of an orbit closure, and is an integer between 1 and the genus g. Our result applies when the rank is at least 1+g/2, and so handles roughly half of the possible values of rank.

The five lectures will introduce novel and broadly applicable techniques, organized as follows:

  1. An introduction to orbit closures, their rank, their boundary in the WYSIWYG partial compactification, and cylinder deformations.
  2. Reconstructing orbit closures from their boundaries (this talk will explicate a preprint of the same name).
  3. Recognizing loci of covers using cylinders (this talk will follow a preprint titled “Generalizations of the Eierlegende-Wollmilchsau”).
  4. An overview of the proof of the main theorem; marked points (following the preprint “Marked Points on Translation Surfaces”); and a dichotomy for cylinder degenerations.
  5. Completion of the proof of the main theorem.
Jan, 14: The Erdos-Hajnal conjecture for the five-cycle
Speaker: Sophie Spirkl
Abstract:

The Erdos-Hajnal conjecture states that for every graph H there exists c > 0 such that every n-vertex graph G either contains H as an induced subgraph, or has a clique or stable set of size at least n^c. I will talk about a proof of this conjecture for the case H = C5 (a five-cycle), and related results. The proof is based on an extension of a lemma about bipartite graphs due to Pach and Tomon. This is joint work with Maria Chudnovsky, Alex Scott, and Paul Seymour.

Jan, 13: Quantum State Transfer on Graphs
Speaker: Christopher van Bommel
Abstract:

Quantum computing is believed to provide many advantages over traditional computing, particularly considering the speed at which computations can be performed. One of the challenges that needs to be resolved in order to construct a quantum computer is the transmission of information from one part of the computer to another. This transmission can be implemented by spin chains, which can be modeled as a graph, and analyzed using algebraic graph theory. The ideal situation is that of perfect state transfer, where there exists a time interval during which the information is perfectly moved from one location to another. As perfect state transfer is relatively rare, we also consider pretty good state transfer, where for any desired level of accuracy, there exists a time interval during which the information transfer achieves this accuracy. We will discuss determining whether graphs admit perfect or pretty good state transfer.

Jan, 3: Geometry of Numbers: Lecture 13 of 13
Speaker: Barak Weiss
Abstract:

This lecture is part of a course on the geometry of numbers.

For more information about these lectures, please see the course website (external).

Jan, 3: Geometry of Numbers: Lecture 12 of 13
Speaker: Barak Weiss
Abstract:

This lecture is part of a course on the geometry of numbers.

For more information about these lectures, please see the course website (external).

Jan, 3: Geometry of Numbers: Lecture 11 of 13
Speaker: Barak Weiss
Abstract:

This lecture is part of a course on the geometry of numbers.

For more information about these lectures, please see the course website (external).

Jan, 2: Geometry of Numbers: Lecture 10 of 13
Speaker: Barak Weiss
Abstract:

This lecture is part of a course on the geometry of numbers.

For more information about these lectures, please see the course website (external).

Jan, 2: Geometry of Numbers: Lecture 9 of 13
Speaker: Barak Weiss
Abstract:

This lecture is part of a course on the geometry of numbers.

For more information about these lectures, please see the course website (external).

Jan, 2: Geometry of Numbers: Lecture 8 of 13
Speaker: Barak Weiss
Abstract:

This lecture is part of a course on the geometry of numbers.

For more information about these lectures, please see the course website (external).

Jan, 2: Geometry of Numbers: Lecture 7 of 13
Speaker: Barak Weiss
Abstract:

This lecture is part of a course on the geometry of numbers.

For more information about these lectures, please see the course website (external).

Jan, 2: Geometry of Numbers: Lecture 6 of 13
Speaker: Barak Weiss
Abstract:

This lecture is part of a course on the geometry of numbers.

For more information about these lectures, please see the course website (external).

Jan, 2: Geometry of Numbers: Lecture 5 of 13
Speaker: Barak Weiss
Abstract:

This lecture is part of a course on the geometry of numbers.

For more information about these lectures, please see the course website (external).

Jan, 2: Geometry of Numbers: Lecture 4 of 13
Speaker: Barak Weiss
Abstract:

This lecture is part of a course on the geometry of numbers.

For more information about these lectures, please see the course website (external).

Jan, 1: Geometry of Numbers: Lecture 3 of 13
Speaker: Barak Weiss
Abstract:

This lecture is part of a course on the geometry of numbers.

For more information about these lectures, please see the course website (external).

Jan, 1: Geometry of Numbers: Lecture 2 of 13
Speaker: Barak Weiss
Abstract:

This lecture is part of a course on the geometry of numbers.

For more information about these lectures, please see the course website (external).

Jan, 1: Geometry of Numbers: Lecture 1 of 13
Speaker: Barak Weiss
Abstract:

This lecture is part of a course on the geometry of numbers.

For more information about these lectures, please see the course website (external).

2021