# Video Content by Date

Sep, 22: Transcendence of period integrals over function fields
Speaker: Jacob Tsimerman
Abstract:

Speaker: Jacob Tsimerman (University of Toronto)

Title: Transcendence of period integrals over function fields

Abstract: Periods are integrals of differential forms, and their study spans many branches of mathematics, including diophantine geometry, differential algebra, and algebraic geometry. If one restricts their attention to periods arising over Q, then the Grothendieck Period Conjecture is a precise way of saying that these are as transcendental as is allowed by the underlying geometry. While this is a remarkably general statement (and very open), it does not include another major (also open!) conjecture in transcendence theory - the Schanuel conjecture. In particular, e is not a period, even though it can be described through periods via the relation that the integral from 1 to e of dx/x is 1. We shall present a generalization due to André which unifies the two conjectures in a satisfactory manner.

In the (complex) function field case, a lot more is known. The Grothendieck Period Conjecture has been formulated and proved by Ayoub and Nori. We shall explain the geometric analogue of the André - Grothendieck Period Conjecture and present its proof. It turns out that this conjecture is (almost) equivalent to a functional-transcendence statement of extreme generality known as the Ax-Schanuel conjecture, which has been the subject of a lot of study over the past decade in connection with unlikely intersection problems. The version relevant to us is a comparison between the algebraic and flat coordinates of geometric local systems. We will explain the ideas behind the proofs of this Ax-Schanuel conjecture and explain how it implies the relevant period conjecture.

Abstract:

Euler’s famous formula tells us that (with appropriate caveats), a map on the sphere with f countries (faces), e borders (edges), and v border-ends (vertices) will satisfy v-e+f=2. And more generally, for a map on a surface with g holes, v-e+f=2-2g. Thus we can figure out the genus of a surface by cutting it into pieces (faces, edges, vertices), and just counting the pieces appropriately. This is an example of the topological maxim “think globally, act locally”. A starting point for modern algebraic geometry can be understood as the realization that when geometric objects are actually algebraic, then cutting and pasting tells you far more than it does in “usual” geometry. I will describe some easy-to-understand statements (with hard-to-understand proofs), as well as easy-to-understand conjectures (some with very clever counterexamples, by M. Larsen, V. Lunts, L. Borisov, and others). I may also discuss some joint work with Melanie Matchett Wood.

Speaker biography:

Ravi Vakil is a Professor of Mathematics and the Robert K. Packard University Fellow at Stanford University, and was the David Huntington Faculty Scholar. He received the Dean's Award for Distinguished Teaching, an American Mathematical Society Centennial Fellowship, a Frederick E. Terman fellowship, an Alfred P. Sloan Research Fellowship, a National Science Foundation CAREER grant, the presidential award PECASE, and the Brown Faculty Fellowship. Vakil also received the Coxeter-James Prize from the Canadian Mathematical Society, and the André-Aisenstadt Prize from the CRM in Montréal. He was the 2009 Earle Raymond Hedrick Lecturer at Mathfest, and a Mathematical Association of America's Pólya Lecturer 2012-2014. The article based on this lecture has won the Lester R. Ford Award in 2012 and the Chauvenet Prize in 2014. In 2013, he was a Simons Fellow in Mathematics.

Sep, 20: Turing Patterns on Growing Domains
Speaker: Irving Epstein
Abstract:

Turing patterns have been suggested as an explanation for morphogenesis in a variety of organisms. Despite the fact that morphogenesis occurs during growth, most studies of Turing patterns have been conducted on static domains. We present experimental and computational studies of Turing patterns in a chemical reaction-diffusion system on growing two-dimensional domains. We also investigate the effect of inert obstacles on pattern evolution. We find that the rate of domain growth significantly affects both how the patterns are laid down and their ultimate morphology.

Abstract:

Gabriel Verret (University of Auckland, New Zealand)

Many results in algebraic graph theory can be viewed as upper bounds on the size of the automorphism group of graphs satisfying various hypotheses. These kinds of results have many applications. For example, Tutte's classical theorem on 3-valent arc-transitive graphs led to many other important results about these graphs, including enumeration, both of small order and in the asymptotical sense. This naturally leads to trying to understand barriers to this type of results, namely graphs with large automorphism groups. We will discuss this, especially in the context of vertex-transitive graphs of fixed valency. We will highlight the apparent dichotomy between graphs with automorphism group of polynomial (with respect to the order of the graph) size, versus ones with exponential size.

Abstract:

We analyze the existence, linear stability, and slow dynamics of localized 1D spike patterns for a Keller-Segel model of chemotaxis that includes the effect of logistic growth of the cellular population. Our analysis of localized patterns for this two-component reaction-diffusion (RD) model is based, not on the usual limit of a large chemotactic drift coefficient, but instead on the singular limit of an asymptotically small diffusivity of the chemoattractant concentration field. In the limit, steady-state and quasi-equilibrium 1D multi-spike patterns are constructed asymptotically. To determine the linear stability of steady-state N-spike patterns, we analyze the spectral properties associated with both the “large” O(1) and the “small” o(1) eigenvalues associated with the linearization of the Keller-Segel model. By analyzing a nonlocal eigenvalue problem characterizing the large eigenvalues, it is shown that N-spike equilibria can be destabilized by a zero-eigenvalue crossing leading to a competition instability if the cellular diffusion rate exceeds a threshold, or from a Hopf bifurcation if a relaxation time constant is too large. In addition, a matrix eigenvalue problem that governs the stability properties of an N-spike steady-state with respect to the small eigenvalues is derived. From an analysis of this matrix problem, an explicit range of cellular diffusion rate where the N-spike steady-state is stable to the small eigenvalues is identified. Finally, for quasi-equilibrium spike patterns that are stable on an O(1) time-scale, we derive a differential algebraic system (DAE) governing the slow dynamics of a collection of localized spikes. Unexpectedly, our analysis of the KS model with logistic growth in the small chemical diffusion rate regime is rather closely related to the analysis of spike patterns for the Gierer-Meinhardt RD system.
This paper is a joint work with Professor Michael J. Ward and Juncheng Wei.

Abstract:

In this talk, we introduce some recent regularity results of free boundary in
optimal transportation. Particularly for higher order regularity, when
densities are Hölder continuous and domains are $C^2$, uniformly convex, we obtain the free boundary is $C^{2,\alpha}$ smooth. We also consider another mode
case that the target consists of two disjoint convex sets, in which
singularities of optimal transport mapping arise. Under similar assumptions,
we show that the singular set of the optimal mapping is an $(n-1)$-dimensional
$C^{2,\alpha}$ regular sub-manifold of $\mathbb{R}^n$. These are based on a
series of joint work with Shibing Chen and Xu-Jia Wang.

Abstract:

While Einstein’s theory of gravity is formulated in a smooth setting, the celebrated singularity theorems of Hawking and Penrose describe many physical situations in which this smoothness must eventually breakdown. In positive-definite signature, there is a highly successful theory of metric and metric-measure geometry which includes Riemannian manifolds as a special case, but permits the extraction of nonsmooth limits under curvature and dimension bounds analogous to the energy conditions in relativity: here sectional curvature is reformulated through triangle comparison, while Ricci curvature is reformulated using entropic convexity along geodesics of probability measures.

This lecture explores recent progress in the development of an analogous theory in Lorentzian signature, whose ultimate goal is to provide a nonsmooth theory of gravity. We highlight how the null energy condition of Penrose admits a nonsmooth formulation as a variable lower bound on timelike Ricci curvature.

Jul, 28: The Canadian regional climate model
Speaker: Carsten Abraham
Abstract:

High resolution climate simulations (horizontal resolutions of 25km or smaller) are a desired product for policy makers, the public, and renewable energy applications. Since running climate models at high resolution is not feasible, dynamical and statistical downscaling methods are applied. The latest version of the Canadian regional climate model (CanRCM5) is specifically designed to dynamically downscale future climate projections of its parent Canadian Earth System Model (CanESM5). The close relationship between these two Canadian models allows for improved RCM driving relative to independent RCM modelling centres, as all required prognostic variables are available on CanRCM5's lateral boundaries from its parent global model. Coupling different scale size models is challenging as regional climate models can easily develop their own climate. To keep the features of CanRCM5 consistent with CanESM5, the large-scale dynamics of CanRCM5 are typically nudged towards its parent model. We present a framework that identifies appreciable differences between the regional and global model and apply it to near-surface wind and precipitation fields showing that particularly the influence of better resolved topography yields substantial differences in the climate projections of CanRCM and CanESM. Finally, we discuss latest developments on research on bias correcting CanESM and CanRCM which allows for more accurate representations of the climate state. (Joint work with John Scinocca and Slava Kharin)

Abstract:

Boundary-layer wind and turbulence-profile theories as described in most textbooks apply to flat prairies, not to the rugged terrain of British Columbia (BC). For the mountainous terrain of BC, different turbines at the same wind farm experience different winds and turbulence associated with their locations relative to small-scale (unresolved) terrain features. Convolution of the resulting wind-speed distribution with the wind-turbine power curve for an individual turbine yields a wind-farm power curve that differs from the theoretical power curve. For several wind farms in BC, the farm-average power curve does not achieve the cube of wind speed even between the cut-in and rated-power speeds.

To partially compensate for these wind variations, we run an ensemble of up to 51 NWP model runs each day, with fewer ensemble members covering the more distant wind farms. These runs are based on a variety of initial/boundary conditions (from gov’t centers in Canada, USA, France, Germany), a variety of model cores (WRF-ARW, WRF-NMM, MM5, MPAS), a variety of horizontal grid spacings, and a variety of physics parameterizations. Each forecast is individually bias corrected based on recent-past observations at any wind farm, and then the separate runs are combined to yield ensemble-average and calibrated-probabilistic forecasts.

UBC has been making operational limited-area NWP forecasts of hub-height winds for all the active wind farms in BC for the past decade. BC Hydro uses our ensemble forecasts to better manage the integration of wind power with their much-greater hydro-power generation. BC Hydro also uses our forecasts of Bonneville Power Administration (Columbia River region) wind-farm hub-height winds to optimize their energy-trading to the USA.

Based on our experience, we created a course ATSC 313 “Renewable Energy Meteorology”, which covers meteorology for hydro, wind, and solar power.

Abstract:

Growing evidence is demonstrating the startling reach of wind farm wakes, which can often exceed 100 kilometers under frequently occurring stable atmospheric conditions. These "long wakes" can significantly affect the annual energy yield of neighboring wind farms. Unfortunately, this impact is not accounted for in industry-standard wind resource assessment methods, leading to consistent overestimations of energy production and the suboptimal placement of new wind farms. As the global wind farm fleet continues to rapidly grow, the development of tools to assess these impacts is crucial prior to making any major investments in new project sites.

In this presentation, we will elucidate how numerical weather prediction (NWP) modeling is emerging as the preferred tool for simulating long-wake propagation. We will initially discuss the physical mechanisms underlying long-wake propagation, and provide observational evidence substantiating their occurrence. Subsequently, we will delve into the suitability of the Weather Research and Forecasting (WRF) NWP model for long-wake modeling, specifically its ability to accurately simulate diurnal and seasonal atmospheric stability fluctuations and its capacity to model wake propagation through its wind farm parameterization option. In conclusion, we will present encouraging validation results from five onshore wind farms in the U.S., reinforcing the potential and accuracy of the WRF model in tackling this growing issue.

Jul, 28: Turbulence, wakes and wind farm control
Speaker: Dennice Gayme
Abstract:

The dynamics of the atmospheric boundary layer (ABL) play a fundamental role in wind farm power production, governing the velocity field that enters the farm as well as the turbulent mixing that regenerates energy for extraction at downstream rows. Understanding the dynamic interactions between turbines, wind farms, and the ABL can therefore be beneficial in improving the efficiency of wind farm design and control approaches. This talk introduces a suite of models that exploit this knowledge to improve predictions of both static and dynamic conditions in the wind farm. We first introduce the area localized coupled (ALC) model, which couples the steady state solution of a dynamic wake model with a localized top-down model that focuses on the effect of the farm on the ABL. The ALC model improves the accuracy of power output and local velocity predictions over both conventional wake models and top-down models, while extending the applicability of this type of coupled model to arbitrary wind farm layouts. In the second part of the talk, we focus on using attributes of the turbulent ABL to provide improved models for power production and wake behavior under turbine yawing, which has been shown to increase turbine power output potential. Finally, we demonstrate how these ideas can be extended to a control setting.

Jul, 27: PIMS/FACTS Panel: Tackling Climate Change and the Just Transition to Renewable Energy
Speaker: Réne Aïd, Gaël Giraud, Seth Klein, Judith Sayers, Andrew Weaver
Abstract:

This panel was organized as part of the PIMS Workshop on Forecasting and Mathematical Modeling for Renewable Energy and Public Panel Discussion on Climate Change. Ahead of the panel, the panelists were invited to give a short talk on their areas of expertise.

Abstract:

Andrew Weaver is a professor of climate science at the University of Victoria. He is also a lead author for the IPCC and a former BC MLA and leader of BC Green Party. This presentation was given ahead of his participation in a panel discussion on Tackling Climate Change and the Just Transition to Renewable Energy.

Abstract:

Judith Sayers is the President of the Nuu-chah-nulth Tribal Council, a lawyer, renewable energy leader, and chancellor of Vancouver Island University. This presentation was given ahead of a panel discussion on Tackling Climate Change and the Just Transition to Renewable Energy.

Jul, 27: Mobilizing Canada for the Climate Emergency
Speaker: Seth Klein
Abstract:

Seth Klein is a public policy researcher, author and team lead with the Climate Emergency Unit. This presentation as given ahead of his participation in a panel discussion on Tackling Climate Change and the Just Transition to Renewable Energy.

Jul, 27: Macroeconomics and Climate
Speaker: Gaël Giraud
Abstract:

Gaël Giraud is the Founding director of the Georgetown University Environmental Justice Program. He is also a professor at Georgetown and the former chief economist of the French Development agency CNRS. This presentation was given ahead of a panel discussion on Tackling Climate Change and the Just Transition to Renewable Energy in which Dr. Giraud also participated.

Abstract:

Réne Aïd is a professor of Economics at Université Paris-Dauphine and former Deputy-Director of EDF Research Energy Finance. This presentation was given ahead of the PIMS/FACTS panel discussion on Tackling Climate Change and the Just Transition to Renewable Energy in which the speaker also participated.

Abstract:

We formulate an equilibrium model of intraday trading in electricity markets. Agents face balancing constraints between their customers consumption plus intraday sales and their production plus intraday purchases. They have continuously updated forecast of their customers consumption at maturity. Forecasts are prone to idiosyncratic noise as well as common noise (weather). Agents production capacities are subject to independent random outages, which are each modeled by a Markov chain. The equilibrium price is defined as the price that minimizes trading cost plus imbalance cost of each agent and satisfies the usual market clearing condition. Existence and uniqueness of the equilibrium are proved, and we show that the equilibrium price and the optimal trading strategies are martingales. The main economic insights are the following. (i) When there is no uncertainty on generation, it is shown that the market price is a convex combination of forecasted marginal cost of each agent, with deterministic weights. Furthermore, the equilibrium market price is consistent with Almgren and Chriss's model, and we identify the fundamental part as well as the permanent market impact. It turns out that heterogeneity across agents is a necessary condition for Samuelson's effect to hold. We show that when heterogeneity lies only on costs, Samuelson's effect holds true. A similar result stands when heterogeneity lies only on market access quality. (ii) When there is production uncertainty only, we provide an approximation of the equilibrium for large number of players. The resulting price exhibits increasing volatility with time. (Joint work with Andrea Cosso and René Aïd)

Abstract:

We analyze the optimal regulatory incentives to foster the development of non-emissive electricity generation when the demand for power is served either by a one firm (monopoly) or by two interacting firms (competition). The regulator wishes to encourage green investments to limit carbon emissions, while simultaneously reducing intermittency of the total energy production. We find that the regulation of competing interacting firms is more efficient than the regulation of the monopoly situation as measured with the certainty equivalent of the principal’s value function. This higher efficiency is achieved thanks to a higher degree of freedom of the incentive mechanisms which involves cross-subsidies between firms. Joint work with Annika Kemper (Bielefeld University) and Nizar Touzi (Ecole Polytechnique).

Abstract:

Electricity markets balance an increasingly intermittent supply with price-inelastic demand, while climate change and electrification of mobility are contributing to transforming diurnal and seasonal demand patterns. Electricity systems face an increasing level of stochasticity, and market participants need to inform their dispatch decisions based on 24-hour price forecasts for participation in Day-Ahead Markets, which in turn depend on supply and demand forecasts. The arrival of grid-scale electricity storage is also creating new scope for prices forecasts, while smaller scale storage systems act as price takers. In the long term, large-scale deployment of grid-scale electricity storage has the potential of significantly reducing price variation through arbitrage, which could shift the “value added” of forecasting from short-term (intra-day) to long-term predictions, and from supporting operational (dispatch) decisions to supporting capacity (investment) decisions.

Abstract:

This talk introduces a methodology for improving short-term wind speed forecasting in Alberta. Regime-switching spatio-temporal covariance models are applied using two datasets: (1) large-scale reanalysis dataset containing large scale atmospheric information for atmospheric clustering using k-means and hidden Markov models; (2) wind speed data from 131 weather stations across Alberta are used to train and test the covariance models. The predictive performance is assessed for different models and clustering methods.

Abstract:

The rising U.S. offshore wind sector holds great promise, both environmentally and economically, to unlock vast supplies of clean, domestic, and renewable energy. To harness this valuable resource, Gigawatt (GW)-scale offshore wind projects are already under way at several locations off of the U.S. coastline. This promising future, however, is still clouded with uncertainties on how to optimally manage those ultra-scale offshore wind assets, which would be operating under harsh environmental and operational conditions, in relatively under-explored territories, and at unprecedented scales. I will present some of our progress in formulating tailored forecasting and optimization models aimed at minimizing some of those uncertainties. Our models and analyses are largely tailored and tested using data from the U.S. Mid-Atlantic—where several GW-scale wind projects are currently under development.

Jul, 26: A Tribute to Bill Aiello
Speaker: Brian Marcus
Abstract:

A tribute to Bill Aiello

Abstract:

How much wind power will a turbine generate over its lifetime? To answer such questions, we can consider climate model output to generate very long-term wind power forecasts on the scale of years to decades. One major limitation of the data projected by climate models is their coarse temporal resolution that is usually not finer than three hours and can be as coarse as one month. However, wind speed distributions of low temporal resolution might not be able to account for high frequency variability which can lead to distributional shifts in the projected wind speeds. Even if these changes are small this can have a huge impact due to the highly non-linear relationship between wind and wind power and the long forecast horizons we consider. In my talk, I will discuss how the resolution of wind speed data from climate projections affects wind power forecasts.

Abstract:

Forecasts of renewable power production and electricity demand for multiple time periods and/or spatial expanses are required to operate modern power systems. Furthermore, probabilistic forecasts are necessary to facilitate economic decision-making and risk management. This gives rise to the challenge of producing forecasts that capture dependency between variables, over time, and between multiple locations. The Gaussian Copula has been widely used for multivariate energy forecasting, including for wind power, and is readily scalable given that the entire dependency structure is described by a single covariance matrix; however, estimating this covariance matrix in high dimensional problems remains a research challenge. Furthermore, it has been found empirically that this covariance matrix is often non-stationary and evolves over time. Two methods are presented for parameterising covariance matrices to enable conditioning on explanatory variables and as a step towards more robust estimation.

We consider two approaches, one based on modelling the parameters of covariance functions using additive models, and the second modelling individual elements of the modified Cholesky decomposition, again using additive models. We show how this gives rise to a wide range of possible parametric structures and discuss model selection and estimation strategies. Finally, we demonstrate though two case studies the improvement in forecast quality that these methods yield, and the importance and value of capturing the dynamics of dependency structures in wind power forecasting and net-load forecasting in the presence of embedded renewables.

Abstract:

The ability to forecast solar irradiance plays an indispensable role in solar power forecasting, which constitutes an essential step in planning and operating power systems under high penetration of solar power generation. Since solar radiation is an atmospheric process, solar irradiance forecasting, and thus solar power forecasting, can benefit from the participation of atmospheric scientists. In this talk, the two fields, namely, atmospheric science and power system engineering are jointly discussed with respect to how solar forecasting plays a part. Firstly, the state of affairs in solar forecasting is elaborated; some common misconceptions are clarified; and salient features of solar irradiance are explained. Next, five technical aspects of solar forecasting: (1) base forecasting methods, (2) post-processing, (3) irradiance-to-power conversion, (4) verification, and (5) grid-side implications, are reviewed. Following that, ten research topics moving into the future are enumerated; they are related to (1) data and tools, (2) numerical weather prediction, (3) forecast downscaling, (4) large eddy simulation, (5) dimming and brightening, (6) aerosols, (7) spatial forecast verification, (8) multivariate probabilistic forecast verification, (9) predictability, and (10) extreme weather events. Last but not least, a pathway towards ultra-high PV penetration is laid out, based on a recently proposed concept of firm generation and forecasting.

Abstract:

Topological insulators are materials that exhibit unique physical properties due to their non-trivial topological order. One of the most notable consequences of this order is the presence of protected edge states as well as closure of bulk spectral gaps, which is known as the bulk-edge correspondence.

In this talk, I will discuss the mathematical description of topological insulators and their related spectral properties. The presentation will begin with an overview of Floquet theory, Bloch bundles, and the Chern number. We will then examine the bulk-edge
correspondence in topological insulators before delving into our research on closure of bulk spectral gaps for topological insulators with general edges. This talk is based on a joint work with Alexis Drouot.

May, 31: Quantum symmetries of finite dimensional algebras
Speaker: Amrei Oswald
Abstract:

The classical notion of symmetry can be formalized by actions of groups. Quantum symmetry is a generalization of the notion of symmetry to the quantum setting, where symmetries can no longer be completely described by the actions of groups. In this setting, quantum symmetries are given by Hopf actions of quantum groups on algebras. I will start with background on quantum groups and Hopf actions and then give examples of quantum symmetries of quiver path algebras. Path algebras can be described in terms of directed graphs and play an important role in the representation theory of finite-dimensional algebras. While quantum symmetries are not straightforward to visualize, path algebras give us a nice tool for doing so. Then, I will discuss a tensor categorical perspective for understanding quantum symmetry and how this perspective can be applied to quantum symmetries of path algebras and finite-dimensional algebras.

Abstract:

The locally symmetric diffusions, also known as Brownian motions, on generalized Sierpinski carpets were constructed by Barlow and Bass in 1989. On a fixed carpet, by the uniqueness theorem (Barlow-Bass-Kumagai-Teplyaev, 2010), the reflected Brownians motion on level $n$ approximation Euclidean domain, running at speed $\lambda_n\asymp \eta^n$ with $\eta$ being a constant depending on the fractal, converges weakly to the Brownian motion on the Sierpinski carpet as $n$ tends to infinity. In this talk, we show the convergence of $\lambda_n/\eta^n$. We also give a positive answer to a closely related open question of Barlow-Bass (1990) about the convergence of the renormalized effective resistances between two opposite faces of approximation domains. This talk is based on a joint work with Zhen-Qing Chen.

May, 19: Paths and Pathways
Abstract:

We talk about how some simple sounding problems about straight line paths on surfaces require many different kinds of mathematical thinking to solve, focusing on the example of understanding straight line paths on Platonic solids. We'll use this to start a discussion of how we can emphasize teaching different ways of thinking, and why geometry is an important resource for students. There will be lots and lots of fun pictures and hopefully interesting and provocative ideas!

The following resources are referenced during this talk:

May, 17: Essential normality of Bergman modules on egg domains
Abstract:

During 2005-2006, Arveson and Douglas formulated a challenging conjecture in multivariable operator theory regarding the essential normality of compressed shifts in the usual Hilbert spaces of analytic functions, say, Bergman spaces on strongly pseudoconvex domains. (Essential normality means normality modulo compact operators.) In this talk, after stating this conjecture, I will report on a joint work with Xiang Tang about the essential normality of Bergman spaces on several classes of egg domains. These egg domains are generalizations of the unit ball and are weakly pseudoconvex in general. If time permits, I will say a few words about a resulting K-homology index theorem and discuss p-essential normality (that is normality modulo p-summable operators).

May, 12: Twelve on the twelfth
Speaker: Kristine Bauer
Abstract:

The under representation of women, especially women of color, has been persistently well documented (see for example the data dashboard on www.womendomath.org). One reason that this is a problem is that it can be difficult for women to identify role models - this can in turn make it harder for women to envision their own success. During my career, I found it very helpful to learn the stories of women in STEM and to draw on aspects of their success to try to invent my own path. In this talk, I will retell twelve stories of women in STEM that influenced me. I can’t promise that the stories will be historically accurate, but I will try to say what I learned from the stories as I heard them and what lessons I hope others might take from them as well.

Abstract:

In this talk I will be discussing the history of the Prime Number Theorem following the works of Legendre, Gauss, Riemann, Hadamard, and de la Vallée Poussin, followed by a survey on explicit bounds for $\psi(x)$ beginning with the work of Rosser in 1941. I will go over various improvements over the years including the works of Rosser and Schoenfeld, Dusart, Faber-Kadiri, and Büthe. I will finally briefly discuss my work on the survey of a paper by Büthe and its significance.

May, 12: On the quality of the ABC-solutions
Abstract:

Let the triple $(a, b, c)$ of integers be such that $\gcd{\left(a, b, c\right)} = 1$ and $a + b = c$. We call such a triple an ABC-solution. The quality of an ABC-solution $(a, b, c)$ is defined as
$$q(a, b, c) = \frac{\max\left\{\log |a|, \log |b|, \log |c|\right\}}{\log \operatorname{rad}\left({|abc|}\right)}$$
where $\operatorname{rad} (|abc|)$ is the product of distinct prime factors of $|abc|$. The ABC-conjecture states that given $\epsilon >0$ the number of the ABC-solutions $(a, b, c)$ with $q(a, b, c)\geq 1 + \epsilon$ is finite. In the first part of this talk, under the ABC-conjecture, we explore the quality of certain families of the ABC-solutions formed by terms in Lucas and associated Lucas sequences. We also introduce, unconditionally, a new family of ABC-solutions that has quality $> 1$. In the remainder of the talk, we provide a result on a conjecture of Erdõs on the solutions of the Brocard-Ramanujan equation
$$n! + 1 = m^2$$
by assuming an explicit version of the ABC-conjecture proposed by Baker.

May, 12: Mathematics for Humanity
Speaker: Laleh Behjat
Abstract:

The world is going through major changes with the climate crisis and the dual revolutions of artificial intelligence and bio-technology. These changes require us to rethink our systems and build new ones. In this talk, I will discuss why mathematics as the common language for sciences is the most important factor in building this new world and how diverse perspectives can help us solve these problems better.

Abstract:

Borsuk’s number b(n) is the smallest integer such that any set of diameter 1 in the n-dimensional space can be covered by b(n) sets of a smaller diameter. Exponential upper bounds on b(n) were first obtained by Shramm (1988) and later by Bourgain and Lindenstrauss (1989).

To obtain an upper bound on b(n), Bourgain and Lindenstrauss provided exponential bounds (both upper and lower) in Grünbaum's problem – the problem of determining the minimal number of open balls of diameter 1 needed to cover a set of diameter 1. On the other hand, Schramm provided an exponential upper bound on the illumination number of n-dimensional bodies of constant width. In 2015 Kalai asked if there exist n-dimensional convex bodies of constant width with illumination number exponentially large in n.

In this talk I will answer Kalai’s question in the affirmative and provide a new lower bound in the Grünbaum’s problem. This talk is based on a joint work with Andriy Bondarenko and Andriy Prymak.

May, 4: Linearised Optimal Transport Distances
Speaker: Matthew Thorpe
Abstract:

Optimal transport is a powerful tool for measuring the distances between signals and images. A common choice is to use the Wasserstein distance where one is required to treat the signal as a probability measure. This places restrictive conditions on the signals and although ad-hoc renormalisation can be applied to sets of unnormalised measures this can often dampen features of the signal. The second disadvantage is that despite recent advances, computing optimal transport distances for large sets is still difficult. In this talk I will extend the linearisation of optimal transport distances to the Hellinger–Kantorovich distance, which can be applied between any pair of non-negative measures, and the TLp distance, a version of optimal transport applicable to functions. Linearisation provides an embedding into a Euclidean space where the Euclidean distance in the embedded space is approximately the optimal transport distance in the original space. This method, in particular, allows for the application of off-the-shelf data analysis tools such as principal component analysis as well as reducing the number of optimal transport calculations from $O(n^2)$ to $O(n)$ in a data set of size n.

May, 3: Isotropy of quadratic forms in characteristic 2
Speaker: Kristýna Zemková
Abstract:

It is well-known that quadratic forms can be diagonalized over fields and that they are in a one-to-one correspondence with bilinear forms; the algebraic theory of quadratic forms is build on these two properties. But there is a catch -- they require division by two. Over a field of characteristic 2, neither of them is true, and the whole quadratic form theory needs to be rebuilt from scratch.
In the talk, we will give a brief introduction to the theory of quadratic forms in characteristic 2. Then we will focus on isotropy -- that is, whether we can find elements of the field on which the quadratic form in question gains the value zero. One of the classical problems is to describe, for any given quadratic form, "how much" it is isotropic over any field extension. We will see that there is basically only one type of field extensions that are relevant for this problem.

Abstract:

Biological populations are responding to climate-driven habitat shifts by either adapting in place or moving in space to follow their suitable temperature regime. The shifting speeds of temperature isoclines fluctuate in time and empirical evidence suggests that they may accelerate over time. We present a mathematical tool to study both transient behaviour of population dynamics and persistence within such moving habitats to discern between populations at high and low risk of extinction. We introduce a system of reaction–diffusion equations to study the impact of varying shifting speeds on the persistence and distribution of a single species. Our model includes habitat-dependent movement behaviour and habitat preference of individuals. These assumptions result in a jump in density across habitat types. We build and validate a numerical finite difference scheme to solve the resulting equations. Our numerical scheme uses a coordinate system where the location of the moving, suitable habitat is fixed in space and a modification of a finite difference scheme to capture the jump in density. We apply this numerical scheme to accelerating and periodically fluctuating speeds of climate change and contribute insights into the mechanisms that support population persistence in transient times and long term.

Apr, 19: Quaternion algebras for surgeries on knots
Speaker: Nicholas Rouse
Abstract:

Work of Thurston and Perelman implies that every compact 3-manifold decomposes into pieces each of which supports one of eight possible geometric structures. Among these eight geometries, the hyperbolic geometry leads to the richest and least well understood class of manifolds. Moreover, Mostow-Prasad rigidity implies that any such hyperbolic structure is unique in stark contrast to the situation in dimension 2. This rigidity also gives rise to number-theoretic invariants of hyperbolic 3-manifolds, and my talk will focus on these. In particular, associated to any finite volume hyperbolic 3-manifold is a number field called the trace field and a quaternion algebra over that trace field. For knot complements, this quaternion algebra is trivial in the sense that it is always a matrix algebra. However, for closed orbifolds such as those obtained by hyperbolic Dehn surgery on a hyperbolic knot complement, the algebra is often nontrivial. A conjecture of Chinburg, Reid, and Stover relates the algebras one can obtain by surgery to the Alexander polynomial of the knot. This problem involves the character variety of the knot and a generalization of quaternion algebras called Azumaya algebras. I will discuss the interplay of these objects as well as some work on the conjecture.

Abstract:

The sign test (Arbuthnott, 1710) and the Wilcoxon signed-rank test (Wilcoxon, 1945) are among the first examples of a nonparametric test. These procedures — based on signs, (absolute) ranks and signed-ranks — yield distribution-free tests for symmetry in one-dimension. However, multivariate distribution-free generalizations of these tests are not known in the literature. In this talk we propose a novel framework — based on the theory of optimal transport — which leads to distribution-free generalized multivariate signs, ranks and signed-ranks, and, as a consequence to analogues of the sign and Wilcoxon signed-rank tests that share many of the appealing properties of their one-dimensional counterparts. In particular, the proposed tests are exactly distribution-free in finite samples, with an asymptotic normal distribution, and adapt to various notions of multivariate symmetry such as central symmetry, sign symmetry, and spherical symmetry. We study the consistency of the proposed tests and their behaviors under local alternatives, and show that the proposed generalized Wilcoxon signed-rank test is particularly powerful against location shift alternatives. We show that in a large class of models, our generalized Wilcoxon signed-rank test suffers from no loss in (asymptotic) efficiency, when compared to the Hotelling’s T^2 test, despite being nonparametric and exactly distribution-free. These ideas can also be used to construct distribution-free confidence sets for the location parameter for multivariate distributions.

This is joint work with Zhen Huang at Columbia University.

Abstract:

Given an smooth function h, this talk will focus on solving the equation \psi(Rz)-\psi(z) = h(z) for circle rotations. We will see how the Diophantine condition on the rotation implies smooth solutions.

Abstract:

Vertex operator algebras are the symmetry algebras of two dimensional conformal field theory. In a famous series of papers, Kazhdan and Lusztig proved an equivalence between particular semisimple categories of modules over affine Lie algebras and quantum groups, the former of which can also be realized as modules over a corresponding vertex operator algebra. Such equivalences between representation categories of vertex operator algebras and quantum groups are now broadly referred to as the Kazhdan-Lusztig correspondence. There has been substantial research interest over the last two decades in understanding the Kazhdan-Lusztig correspondence for vertex operator algebras with non-semisimple representation theory. In this talk I will present an overview of this research area and discuss recent results and future directions.

Apr, 12: Equivalences of Categories of Modules Over Quantum Groups and Vertex Algebras
Speaker: Matthew Rupert, University of Saskatchewan
Abstract:

Vertex operator algebras are the symmetry algebras of two dimensional conformal field theory. In a famous series of papers, Kazhdan and Lusztig proved an equivalence between particular semisimple categories of modules over affine Lie algebras and quantum groups, the former of which can also be realized as modules over a corresponding vertex operator algebra. Such equivalences between representation categories of vertex operator algebras and quantum groups are now broadly referred to as the Kazhdan-Lusztig correspondence. There has been substantial research interest over the last two decades in understanding the Kazhdan-Lusztig correspondence for vertex operator algebras with non-semisimple representation theory. In this talk I will present an overview of this research area and discuss recent results and future directions.

Abstract:

Stochastic parameterizations (SMCM) are continuously providing promising simulations of unresolved atmospheric processes for global climate models (GCMs). One of the features of earlier SMCM is to mimic the life cycle of the three most common cloud types (congestus, deep, and stratiform) in tropical convective systems. In this present study, a new cloud type, namely shallow cloud, is included along with the existing three cloud types to make the model more realistic. Further, the cloud population statistics of four cloud types (shallow, congestus, deep, and stratiform) are taken from Indian (Mandhardev) radar observations. A Bayesian inference technique is used here to generate key time scale parameters required for the SMCM as SMCM is most sensitive to these time scale parameters as reported in many earlier studies. An attempt has been made here for better representing organized convection in GCMs, the SMCM parameterization is adopted in one of the state-of-art GCMs namely the Climate Forecast System version 2 (CFSv2) in lieu of the pre-existing simplified Arakawa–Schubert (default) cumulus scheme and has shown important improvements in key large-scale features of tropical convection such as intra-seasonal wave disturbances, cloud statistics, and rainfall variability. This study also shows the need for further calibration the SMCM with rigorous observations for the betterment of the model's performance in short term weather and climate scale predictions.

Abstract:

We prove an explicit formula for the first moment of Maass form symmetric square L-functions defined over Gaussian integers. As a consequence, we derive a new upper bound for the second moment. This is joint work with Dmitry Frolenkov.

Apr, 3: Expansion, divisibility and parity
Speaker: Harald Andrés Helfgott
Abstract:

Harald Andrés Helfgott University of Göttingen, Germany, and Institut de Mathématiques de Jussieu, France)

We will discuss a graph that encodes the divisibility properties of integers by primes. We prove that this graph has a strong local expander property almost everywhere. We then obtain several consequences in number theory, beyond the traditional parity barrier, by combining our result with Matomaki-Radziwill. For instance: for $\lambda$ the Liouville function (that is, the completely multiplicative function with $\lambda(p) = -1$ for every prime), $(1/\log x) \sum_{n\leq x} \lambda(n) \lambda(n+1)/n = O(1/\sqrt(\log \log x))$, which is stronger than well-known results by Tao and Tao-Teravainen. We also manage to prove, for example, that $\lambda(n+1)$ averages to $0$ at almost all scales when n restricted to have a specific number of prime divisors $\Omega(n)=k$, for any "popular" value of $k$ (that is, $k = \log \log N + O(\sqrt(\log \log N))$ for $n \leq N$).

For the Full abstract, please see: https://www.cs.uleth.ca/~nathanng/ntcoseminar/

Abstract:

People interact with each other in social and communication networks, which affect the processes that occur on them. In this talk, I will give an introduction to dynamical proceses on networks. I will focus my discussion on opinion dynamics, and I will also discuss coupled opinion and disease dynamics on networks. Time-permitting, I may also briefly discuss a model of COVID-19 that centers on disabled people and their caregivers.

Abstract:

Douglas Ulmer (University of Arizona, USA)

It is a classical problem to understand the set of Jacobians of curves among all abelian varieties, i.e., the image of the map Mg→Ag which sends a curve X to its Jacobian JX. In characteristic p, Ag has interesting filtrations, and we can ask how the image of Mg
interacts with them. Concretely, which groups schemes arise as the p-torsion subgroup JX[p] of a Jacobian? We consider this problem in the context of unramified Z/pZ covers Y→X of curves, asking how JY[p] is related to JX[p]. Translating this into a problem about de Rham cohmology yields some results using classical ideas of Chevalley and Weil. This is joint work with Bryden Cais.

Mar, 23: Free boundary regularity for the obstacle problem
Speaker: Alessio Figalli
Abstract:

The classical obstacle problem consists of finding the equilibrium position of an elastic membrane whose boundary is held fixed and which is constrained to lie above a given obstacle. By classical results of Caffarelli, the free boundary is smooth outside a set of singular points. However, explicit examples show that the singular set could be, in general, as large as the regular set. This talk aims to introduce this beautiful problem and describe some classical and recent results on the regularity of the free boundary.

Speaker Biography: Alessio Figalli is a leading figure in the areas of Optimal Transport, partial differential equations and the calculus of variations. He received his Ph.D. from the Scuola Normale Superiore di Pisa and the Ecole Normale Superieur de Lyon and has held positions in Paris and Austin, Texas. He is currently a Professor at ETH Zurich. His work has been recognized with many awards including the Prize of the European Mathematical Society in 2012 and the Fields Medal in 2018.

Mar, 22: Exceptional Chebyshev's bias over finite fields
Speaker: Alexandre Bailleul
Abstract:

Chebyshev's bias is the surprising phenomenon that there is usually more primes of the form 4n+3 than of the form 4n+1 in initial intervals of the natural numbers. More generally, following work from Rubinstein and Sarnak, we know Chebyshev's bias favours primes that are not squares modulo a fixed integer q compared to primes which are squares modulo q. This phenomenon also appears over finite fields, where we look at irreducible polynomials modulo a fixed polynomial M. However, in the finite field case, there are a few known exceptions to this phenomenon, appearing as a result of multiplicative relations between zeroes of certain L-functions. In this work, we show, improving on earlier work by Kowalski, that those exceptions are rare. This is joint work with L. Devin, D. Keliher and W. Li.

Abstract:

Joshua Males (University of Manitoba, Canada)

In his famous '86 paper, Andrews made several conjectures on the function σ(q) of Ramanujan, including that it has coefficients (which count certain partition-theoretic objects) whose sup grows in absolute value, and that it has infinitely many Fourier coefficients that vanish. These conjectures were famously proved by Andrews-Dyson-Hickerson in their '88 Invent. paper, and the function σ has been related to the arithmetic of Z[6–√]by Cohen (and extensions by Zwegers), and is an important first example of quantum modular forms introduced by Zagier.

A closer inspection of Andrews' '86 paper reveals several more functions that have been a little left in the shadow of their sibling σ , but which also exhibit extraordinary behaviour. In an ongoing project with Folsom, Rolen, and Storzer, we study the function v1(q) which is given by a Nahm-type sum and whose coefficients count certain differences of partition-theoretic objects. We give explanations of four conjectures made by Andrews on v1, which require a blend of novel and well-known techniques, and reveal that v1 should be intimately linked to the arithmetic of the imaginary quadratic field Q[−3−−−√]
.

Abstract:

Sea lice are a threat to the health of both wild and farmed salmon and an economic burden for salmon farms. Open-net salmon farms act as reservoirs for sea lice in near coastal areas, which can lead to elevated sea louse levels on wild salmon. With a free living larval stage, sea lice can disperse tens of kilometers in the ocean, both from salmon farms onto wild salmon and between salmon farms. This larval dispersal connects local sea louse populations on salmon farms and thus modelling the collection of salmon farms as a metapopulation can lead to a better understanding of which salmon farms are driving the overall growth of sea lice in a salmon farming region. In this talk I will discuss using metapopulation models to specifically study sea lice on salmon farms in the Broughton Archipelago, BC, and more broadly to better understand the transient and asymptotic dynamics of marine metapopulations.

Mar, 15: The Bootstrap Learning Algorithm
Speaker: Jyoti Bhadana, University of Alberta
Abstract:

Constructing and training the neural network depends on various types of Stochastic Gradient Descent (SGD) methods, with adaptations that help with convergence by boosting the speed of the gradient search. Convergence for existing algorithms requires a large number of observations to achieve high accuracy with certain classes of functions. We work with a different, non-curve-tracking technique with the potential of achieving better speeds of convergence. In this talk, the new idea of 'decoupling' hidden layers by bootstrapping and using linear stochastic approximation is introduced. By utilizing resampled observations, the convergence of this process is quick and requires a lower number of data points. This proposed bootstrap learning algorithm can deliver quick and accurate estimates. This boost in speed allows the approximation of classes of functions within a fraction of the observations required with traditional neural network training methods.

Abstract:

Renate Scheidler (University of Calgary, Canada)

Elliptic curve isogeny path finding has many applications in number theory and cryptography. For supersingular curves, this problem is known to be easy when one small endomorphism or the entire endomorphism ring are known. Unfortunately, computing the endomorphism ring, or even just finding one small endomorphism, is hard. How difficult is path finding in the presence of one (not necessarily small) endomorphism? We use the volcano structure of the oriented supersingular isogeny graph to answer this question. We give a classical algorithm for path finding that is subexponential in the degree of the endomorphism and linear in a certain class number, and a quantum algorithm for finding a smooth isogeny (and hence also a path) that is subexponential in the discriminant of the endomorphism. A crucial tool for navigating supersingular oriented isogeny volcanoes is a certain class group action on oriented elliptic curves which generalizes the well-known class group action in the setting of ordinary elliptic curves.

Abstract:

In a 1992 article where she surveyed her recent breakthrough on unipotent flows on homogeneous spaces, Ratner presented an argument for the equidistribution of horospherical orbits in the context of horocycle flow on SL(2,R)/Lattice. This idea is separate from the ideas in her celebrated work on unipotent flows and I will present her argument for horospherical equidistribution in the simplest situation I can think of: proving the ergodicity of a particular directional flow on the flat two torus. Ratner's argument has similarities to Masur's criterion for unique ergodicity of translation flows, proven around the same time. Time permitting I will comment on Masur's criterion as well.

Mar, 9: Kantorovich operators and their ergodic properties
Speaker: Nassif Ghoussoub
Abstract:

Our introduction of the notion of a non-linear Kantorovich operator was motivated by the celebrated duality in the mass transport problem, hence the name. In retrospect, we realized that they -and their iterates- were omnipresent in several branches of analysis, even those that are focused on linear Markov operators and their semi-groups such as classical ergodic theory, potential theory, and probability theory. The Kantorovich operators that appear in these cases, though non-linear, are all positively 1-homogenous rendering most classical operations on measures and functions conducted in these theories “cost-free”: From “filling schemes” in ergodic theory, to “balayage of measures” in potential theory, to dynamic programming of "gambling houses" in probability theory. General Kantorovich operators arise when one assigns “a cost” to such operations.

Kantorovich operators are also Choquet capacities and are the “least non-linear” extensions of Markov operators, which make them a relatively “manageable” subclass of non-linear maps, where they play the same role that convex envelopes play for numerical functions. Motivated by the stochastic counterpart of Aubry-Mather theory for Lagrangian systems and Fathi-Mather weak KAM theory, as well as ergodic optimization of dynamical systems, we study the asymptotic properties of general Kantorovich operators.

Mar, 8: Euler's divergent series and primes in arithmetic progressions
Speaker: Anne-Maria Ernvall-Hytönen
Abstract:

Euler's divergent series $\sum_{n>0} n!z^n$ which converges only for $z = 0$ becomes an interesting object when evaluated with respect to a p-adic norm (which will be introduced in the talk). Very little is known about the values of the series. For example, it is an open question whether the value at one is irrational (or even non-zero). As individual values are difficult to reach, it makes sense to try to say something about collections of values over sufficiently large sets of primes. This leads to looking at primes in arithmetic progressions, which is in turn raises a need for an explicit bound for the number of primes in an arithmetic progression under the generalized Riemann hypothesis.
During the talk, I will speak about both sides of the story: why we needed good explicit bounds for the number of primes in arithmetic progressions while working with questions about irrationality, and how we then proved such a bound.

The talk is joint work with Tapani Matala-aho, Neea Palojärvi and Louna Seppälä. (Questions about irrationality with T. M. and L. S. and primes in arithmetic progressions with N. P.)

Mar, 8: Central Limit Theorems in Analytic Number Theory
Speaker: Fatma Çiçek, University of Northern British Columbia
Abstract:

Central limit theorem is a significant result in probability. It states that under some assumptions, the behavior of the average of identically distributed independent random variables tends towards that of the standard Gaussian random variable as the number of variables tends to infinity. In number theory, Erdős-Kac theorem is an example of this which is about the distribution of an arithmetic function while Selberg's central limit theorem is about the distribution of the Riemann zeta-function. In this talk, we aim to provide some explanations toward the proofs of these results and mention some versions of Selberg's theorem.

Abstract:

The strong Allee effect plays an important role on the evolution of population in ecological systems. One important concept is the Allee threshold that determines the persistence or extinction of the population in a long time. In general, a small initial population size is harmful to the survival of a species since when the initial data is below the Allee threshold, the population tends to extinction, rather than persistence. Another interesting feature of population evolution is that a species whose movement strategy follows a conditional dispersal strategy is more likely to persist. To study the interaction between Allee effect and the biased movement strategy, we mainly consider the pattern formation and local dynamics for a class of single species population models that is subject to the strong Allee effect. We first rigorously show the existence of multiple localized solutions when the directed movement is strong enough. Next, the spectrum analysis of the associated linear eigenvalue problem is established and used to investigate the stability properties of these interior spikes. This analysis proves that there exist not only unstable but also linear stable steady states. Finally, we extend results of the single equation to coupled systems for two interacting species, each with different advective terms, and competing for the same resources. We also construct several non-constant steady states and analyze their stability.

This is a work in progress talk by a local graduate student.

Abstract:

I give a new explicit bound for the Riemann zeta function on the critical line. This is joint work with Dhir Patel and Andrew Yang. The context of this work highlights the importance of reliability and reproducibility of explicit bounds in analytic number theory.

This event is part of the PIMS CRG Group on L-Functions in Analytic Number Theory. More details can be found on the webpage here: https://sites.google.com/view/crgl-functions/crg-weekly-seminar

Mar, 1: L-Functions of Elliptic Curves Modulo Integers
Speaker: Félix Baril Boudreau, University of Lethbridge
Abstract:

Elliptic curves are one of the major objects of study in number theory. Over finite fields, their zeta functions were proven to be rational by F. K. Schmidt in 1931. In 1985, R. Schoof devised an algorithm to compute zeta functions of elliptic curves over finite fields by directly computing the numerators of these rational functions modulo sufficiently many primes. Over function fields of positive characteristic p, we know from the work of A. Grothendieck, M. Artin, J.L. Verdier (1964/1965) and others, that their L-functions are rational. They are even polynomials with integer coefficients if we assume that their j-invariants are nonconstant rational functions, as shown by P. Deligne in 1980 using a result of J.-I. Igusa (1959).

Therefore, we can meaningfully study the reduction of the L-function of an elliptic curve E with nonconstant j-invariant modulo an integer N. In 2003, C. Hall gave a formula for that reduction modulo N, provided the elliptic curve had rational N-torsion.

In this talk, we first obtain, under the assumptions of C. Hall, a formula for the L-function of any of the infinitely many quadratic twists of E. Secondly, without any condition on the rational 2-torsion subgroup of E, we give a formula for the quotient modulo 2 of L-functions of any two quadratic twists of E. Thirdly, we illustrate that sometimes the reduced L-function is enough to determine important properties of the L-function itself. More precisely, we use the previous results to compute the global root numbers of an infinite family of quadratic twists of some elliptic curve and, under extra assumptions, find in most cases the exact analytic rank of each of these quadratic twists. Finally, we use our formulas to compute directly some degree 2 L-functions, in analogy with the algorithm of Schoof.

Abstract:

Florent Jouve (Institut de Mathématiques de Bordeaux, France)

Given a Galois extension of number fields L/K, the Chebotarev Density Theorem asserts that, away from ramified primes, Frobenius automorphisms equidistribute in the set of conjugacy classes of Gal(L/K). In this talk we report on joint work with D. Fiorilli in which we study the variations of the error term in Chebotarev's Theorem as L/K runs over certain families of extensions. We shall explain some consequences of this analysis: regarding first “Linnik type problems” on the least prime ideal in a given Frobenius set, and second, the existence of unconditional “Chebyshev biases” in the context of number fields. Time permitting we will mention joint work with R. de La Bretèche and D. Fiorilli in which we go one step further and study moments of the distribution of Frobenius automorphisms.

Feb, 23: Adversarial training through the lens of optimal transport
Speaker: Nicolas Garcia Trillos
Abstract:

Modern machine learning methods, in particular deep learning approaches, have enjoyed unparalleled success in a variety of challenging application fields like image recognition, medical image reconstruction, and natural language processing. While a vast majority of previous research in machine learning mainly focused on constructing and understanding models with high predictive power, consensus has emerged that other properties like stability and robustness of models are of equal importance and in many applications are essential. This has motivated researchers to investigate the problem of adversarial training —or how to make models robust to adversarial attacks— but despite the development of several computational strategies for adversarial training and some theoretical development in the broader distributionally robust optimization literature, there are still several theoretical questions about it that remain relatively unexplored. In this talk, I will take an analytical perspective on the adversarial robustness problem and explore two questions: 1) Can we use analytical tools to find lower bounds for adversarial robustness problems?, and 2) How do we use modern tools from analysis and geometry to solve adversarial robustness problems? In this talk I will showcase how ideas from optimal transport theory can provide answers to these questions.

This talk is based on joint works with Camilo Andrés García Trillos, Matt Jacobs, and Jakwang Kim.

Feb, 22: Total Variation Flow on metric measure spaces
Speaker: Cintia Pacchiano (UAlberta)
Abstract:

In this project, we discuss some fine properties and the existence of variational solutions to the Total Variation Flow. Instead of the classical Euclidean setting, we intend to work mostly in the general setting of metric measure spaces.

During the past two decades, a theory of Sobolev functions and BV functions has been developed in this abstract setting. A central motivation for developing such a theory has been the desire to unify the assumptions and methods employed in various specific spaces, such as weighted Euclidean spaces, Riemannian manifolds, Heisenberg groups, graphs, etc.

The Total Variation Flow can be understood as the process of diminishing the total variation using the gradient descent method. This idea can be reformulated using variational solutions, and it gives rise to a definition of parabolic minimizers. The approach’s advantages using a minimization formulation include much better convergence and stability properties. This is essential as the solutions naturally lie only in the space of BV functions.

More details on the abstract are available here: https://www.pims.math.ca/scientific-event/230222-tppfscp

Abstract:

The past twenty-five years have heralded an unparalleled increase in understanding of cancer. At the same time, mathematical modelling has emerged as a natural tool for unravelling the complex processes that contribute to the initiation and progression of tumours, for testing hypotheses about experimental and clinical observations, and assisting with the development of new approaches for improving its treatment. In this talk I will reflect on how increased access to experimental data is stimulating the application of new theoretical approaches for studying tumour growth. I will focus on two case studies which illustrate how mathematical approaches can be used to characterise and quantify tumour vascular networks, and to understand how microstructural features of these networks affect tumour blood flow.

Feb, 13: The principal Chebotarev density theorem
Speaker: Kelly O'Connor
Abstract:

Kelly O'Connor (Colorado State University, USA)

Let K/k be a finite Galois extension. We define a principal version of the Chebotarev density theorem which represents the density of prime ideals of k that factor into a product of principal prime ideals in K . We find explicit equations to express the principal density in terms of the invariants of K/k and give an effective bound which can be used to verify the non-splitting of the Hilbert exact sequence.

Feb, 10: A Reintroduction to Proofs
Speaker: Emily Riehl, John Hopkins University
Abstract:

In an introduction to proofs course, students learn to write proofs informally in the language of set theory and classical logic. In this talk, I'll explore the alternate possibility of teaching students to write proofs informally in the language of dependent type theory. I'll argue that the intuitions suggested by this formal system are closer to the intuitions mathematicians have about their praxis. Furthermore, dependent type theory is the formal system used by many computer proof assistants both "under the hood" to verify the correctness of proofs and in the vernacular language with which they interact with the user. Thus, students could practice writing proofs in this formal system by interacting with computer proof assistants such as Coq and Lean.

Abstract:

Idempotent semi-rings have been relevant in several branches of applied mathematics, like formal languages and combinatorial optimization.

They were brought recently to pure mathematics thanks to its link with tropical geometry, which is a relatively new branch of mathematics that has been useful in solving some problems and conjectures in classical algebraic geometry.

However, up to now we do not have a proper algebraic formalization of what could be called “Tropical Algebraic Geometry”, which is expected to be the geometry arising from idempotent semi-rings.

In this mini-course we aim to motivate the necessity for such theory, and we recast some old constructions in order theory in terms of commutative algebra of semi-rings and modules over them.

### Mini-Course

Abstract:

We improve the best known to date result of Dress-Iwaniec-Tenenbaum, getting ($\log x)^2$ instead of $\left(log x\right)^(5/2)$. We use a weighted form of Vaughan's identity, allowing a smooth truncation inside the procedure, and an estimate due to Barban-Vehov and Graham related to Selberg's sieve. We give effective and non-effective versions of the result.

This event is part of the PIMS CRG Group on L-Functions in Analytic Number Theory. More details can be found on the webpage here: https://sites.google.com/view/crgl-functions/crg-weekly-seminar?authuser=0

Abstract:

Idempotent semi-rings have been relevant in several branches of applied mathematics, like formal languages and combinatorial optimization.

They were brought recently to pure mathematics thanks to its link with tropical geometry, which is a relatively new branch of mathematics that has been useful in solving some problems and conjectures in classical algebraic geometry.

However, up to now we do not have a proper algebraic formalization of what could be called “Tropical Algebraic Geometry”, which is expected to be the geometry arising from idempotent semi-rings.

In this mini-course we aim to motivate the necessity for such theory, and we recast some old constructions in order theory in terms of commutative algebra of semi-rings and modules over them.

### Mini-Course

Feb, 6: Generalized valuations and idempotization of schemes
Speaker: Cristhian Garay
Abstract:

Cristhian Garay (CIMAT Guanajuato, Mexico)

Classical valuation theory has proved to be a valuable tool in number theory, algebraic geometry and singularity theory. For example, one can enrich spectra of rings with new points coming from valuations defined on them and taking values in totally ordered abelian groups.

Totally ordered groups are examples of idempotent semirings, and generalized valuations appear when we replace totally ordered abelian groups with more general idempotent semirings. An important example of idempotent semiring is the tropical semifield.

As an application of this set of ideas, we show how to associate an idempotent version of the structure sheaf of a scheme, which behaves particularly well with respect to idempotization of closed subschemes.

This is a joint work with Félix Baril Boudreau.

Abstract:

Oussama R. Hamza (University of Western Ontario, Canada)

Pro-p groups arise naturally in number theory as quotients of absolute Galois groups over number fields. These groups are quite mysterious. During the 60's, Koch gave a presentation of some of these quotients. Furthermore, around the same period, Jennings, Golod, Shafarevich and Lazard introduced two integer sequences (a_n) and (c_n), closely related to a special filtration of a finitely generated pro-p group G, called the Zassenhaus filtration. These sequences give the cardinality of G, and characterize its topology. For instance, we have the well-known Gocha's alternative (Golod and Shafarevich): There exists an integer n such that a_n=0 (or c_n has a polynomial growth) if and only if G is a Lie group over p-adic fields.

In 2016, Minac, Rogelstad and Tan inferred an explicit relation between a_n and c_n. Recently (2022), considering geometrical ideas of Filip and Stix, Hamza got more precise relations in an equivariant context: when the automorphism group of G admits a subgroup of order a prime q dividing p-1.

In this talk, we present equivariant relations inferred by Hamza (2022) and give explicit examples in an arithmetical context.

Abstract:

In recent years, the problem of optimal transport has received significant attention in statistics and machine learning due to its powerful geometric properties. In this talk, we introduce the optimal transport problem and present concrete applications of this theory in statistics. In particular, we will propose a general framework for distribution-free nonparametric testing in multi-dimensions, based on a notion of "multivariate ranks" defined using the theory of optimal transport. We demonstrate the applicability of this approach by constructing exactly distribution-free tests for testing the equality of two multivariate distributions. We investigate the consistency and asymptotic distributions of these tests, both under the null and local contiguous alternatives. We further study their local power and asymptotic (Pitman) efficiency, and show that a subclass of these tests achieve attractive efficiency lower bounds that mimic the classical efficiency results of Hodges and Lehmann (1956) and Chernoff and Savage (1958).

Abstract:

In optimal transport problems on a finite set, one successful approach to reducing its computational burden is the regularization by the Kullback-Leibler divergence. Then a natural question arises: Are other divergences not admissible for regularization? What kinds of properties are required for divergences? I introduce required properties for Bregman divergences and provide a non-asymptotic error estimate for the optimal transport problem regularized by such Bregman divergences. This convergence is possibly faster than exponential decay as the regularized parameter goes to zero.

This talk is based on joint work with Koya Sakakibara (Okayama U. of Science) and Keiichi Morikuni (U. of Tsukuba).

Abstract:

In 1896, the prime number theorem was established, showing that π(x) ∼ li(x). Perhaps the most widely used estimates in explicit analytic number theory are bounds on |π(x)-li(x)| or the related error term |θ(x)-x|. In this talk we discuss methods one can use to obtain good bounds on these error terms when x is large. Moreover, we will explore the many ways in which these bounds could be improved in the future.

Abstract:

We investigate the interaction between a Platonic solid and an unbounded inertial flow. For a fixed Platonic particle in the flow, we consider three different angular positions: face facing the flow, edge facing the flow, and corner facing the flow, to elucidate the effects of the particle angularity on the flow regime transitions. The impact of these angular positions, notably on drag and lift coefficients, is discussed. The particle cross-section area has a prominent influence on the drag coefficients for low Reynolds numbers, but for higher Reynolds numbers, the impacts of angular positions are more significant. As for the freely moving particle, the change in symmetry of the wake region and path instabilities are strongly related to the particle's angular position and the transverse forces. We analyze and determine the two well-known regimes transitions: the loss of symmetry of the wake and the loss of stationarity of the flow.

Jan, 23: Kummer Theory for Number Fields
Speaker: Antonella Perucca
Abstract:

Antonella Perucca (University of Luxembourg, Luxembourg)

Kummer theory is a classical theory about radical extensions of fields in the case where suitable roots of unity are present in the base field. Motivated by problems close to Artin's primitive root conjecture, we have investigated the degree of families of general Kummer extensions of number fields, providing parametric closed formulas. We present a series of papers that are in part joint work with Christophe Debry, Fritz Hörmann, Pietro Sgobba, and Sebastiano Tronto.

Jan, 18: Least quadratic non-residue and related problems
Speaker: Enrique Treviño
Abstract:

TBA

Abstract:

Since Alan Turing's pioneering publication on morphogenetic pattern formation obtained with reaction-diffusion (RD) systems, it has been the prevailing belief that two-component reaction diffusion systems have to include a fast diffusing inhibiting component (inhibitor) and a much slower diffusing activating component (activator) in order to break symmetry from a uniform steady-state. This time-scale separation is often unbiological for cell signal transduction pathways.
We modify the traditional RD paradigm by considering nonlinear reaction kinetics only inside compartments with reactive boundary conditions to the extra-compartmental space that provides a two-species diffusive coupling. The construction of a nonlinear algebraic system for all existing steady-states enables us to derive a globally coupled matrix eigenvalue problem for the growth rates of eigenperturbations from the symmetric steady-state, on finite domains in 1-D and 2-D and a periodically extended version in 1-D.

We show that the membrane reaction rate ratio of inhibitor rate to activator rate is a key bifurcation parameter leading to robust symmetry-breaking of the compartments. Illustrated with Gierer-Meinhardt, FitzHugh-Nagumo and Rauch-Millonas intra-compartmental reaction kinetics, our compartmental-reaction diffusion system does not require diffusion of inhibitor and activator on vastly different time scales.
Our results elucidate a possible mechanism of the ubiquitous biological cell specialization observed in nature.

Abstract:

In this talk, I will present a recent joint work with Yoonbok Lee, where we investigate the number of zeros of linear combinations of $L$-functions in the vicinity of the critical line. More precisely, we let $L_1, \dots, L_J$ be distinct primitive $L$-functions belonging to a large class (which conjecturally contains all $L$-functions arising from automorphic representations on $\text{GL}(n)$), and $b_1, \dots, b_J$ be real numbers. Our main result is an asymptotic formula for the number of zeros of $F(\sigma+it)=\sum_{j\leq J} b_j L_j(\sigma+it)$ in the region $\sigma\geq 1/2+1/G(T)$ and $t\in [T, 2T]$, uniformly in the range $\log \log T \leq G(T)\leq (\log T)^{\nu}$, where $\nu\asymp 1/J$. This establishes a general form of a conjecture of Hejhal in this range. The strategy of the proof relies on comparing the distribution of $F(\sigma+it)$ to that of an associated probabilistic random model.

This event is part of the PIMS CRG Group on L-Functions in Analytic Number Theory. More details can be found on the webpage here: https://sites.google.com/view/crgl-functions/crg-weekly-seminar?authuser=0

Jan, 9: A walk on Legendre paths
Speaker: Youness Lamzouri
Abstract:

The Legendre symbol is one of the most basic, mysterious and extensively studied objects in number theory. It is a multiplicative function that encodes information about whether an integer is a square modulo an odd prime p. The Legendre symbol was introduced by Adrien-Marie Legendre in 1798, and has since found countless applications in various areas of mathematics as well as in other fields including cryptography. In this talk, we shall explore what we call "Legendre paths", which encode information about the values of the Legendre symbol. The Legendre path modulo p is defined as the polygonal path in the plane formed by joining the partial sums of the Legendre symbol modulo p. In particular, we will attempt to answer the following questions as we vary over the primes p: how are these paths distributed? how do their maximums behave? and what proportion of the path is above the real axis? Among our results, we prove that these paths converge in law, in the space of continuous functions, to a certain random Fourier series constructed using Rademakher random multiplicative functions. Part of this work is joint with Ayesha Hussain.