# Video Content by Date

**Apr, 13**: Regularity of the cohomological equation for circle rotations

Speaker:

*Carlos Ospina (Utah)*

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.

**Apr, 5**: The second moment of symmetric square L-functions over Gaussian integers

Speaker:

*Olga Balkanova*

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*

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/

**Mar, 29**: Opinion Dynamics and Spreading Processes on Networks

Speaker:

*Mason Porter*

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.

**Mar, 27**: p-torsion of Jacobians for unramified Z/pZ-covers of curves

Speaker:

*Douglas Ulmer (University of Arizona)*

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*

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*

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.

**Mar, 20**: Forgotten conjectures of Andrews for Nahm-type sums

Speaker:

*Joshua Males, University of Manitoba.*

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−−−√]

.

**Mar, 15**: Sources, sinks, and sea lice: determining patch contribution and transient dynamics in marine metapopulations

Speaker:

*Peter Harrington, UBC*

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*

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.

**Mar, 13**: Orienteering on Supersingular Isogeny Volcanoes Using One Endomorphism

Speaker:

*Renate Scheidler, UCalgary*

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.

**Mar, 9**: Ratner/Masur equidistribution by orbit matching

Speaker:

*Jon Chaika*

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*

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*

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*

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.

**Mar, 1**: Localized Patterns in Population Models with the Large Biased Movement and Strong Allee Effect

Speaker:

*Fanze Kong*

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.

**Mar, 1**: A new explicit bound for the Riemann zeta function

Speaker:

*Ghaith Hiary*

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*

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.

**Feb, 27**: Fluctuations in the distribution of Frobenius automorphisms in number field extensions

Speaker:

*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*

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)*

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

**Feb, 16**: Understanding form and function in vascular tumours

Speaker:

*Helen Byrne*

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*

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*

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.

**Feb, 9**: An invitation to the algebraic geometry over idempotent semirings - lecture 2

Speaker:

*Cristhian Garay*

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

This lecture is the second part of a mini-course, please see also

**Feb, 8**: A logarithmic improvement in the Bombieri-Vinogradov theorem

Speaker:

*Alisa Sedunova*

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

**Feb, 6**: An invitation to the algebraic geometry over idempotent semirings - Lecture 1

Speaker:

*Cristhian Garay*

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

This lecture is the first part of a mini-course, please see also

**Feb, 6**: Generalized valuations and idempotization of schemes

Speaker:

*Cristhian Garay*

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.

**Feb, 1**: Filtrations, Mild groups and Arithmetic in an Equivariant context

Speaker:

*Oussama R. Hamza*

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.

**Feb, 1**: Optimal transport in statistics and Pitman efficient multivariate distribution-free testing

Speaker:

*Nabarun Deb, UBC*

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).

**Jan, 26**: Bregman divergence regularization of optimal transport problems on a finite set

Speaker:

*Asuka Takatsu*

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).

**Jan, 25**: An explicit error term in the prime number theorem for large x

Speaker:

*Daniel Johnston*

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.

**Jan, 25**: Dynamics and Wakes of a Fixed and Freely Moving Angular Particle in an Inertial Flow

Speaker:

*Guodong Gai*

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*

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*

TBA

**Jan, 11**: The Emergence of Spatial Patterns for Diffusion-Coupled Compartments with Activator-Inhibitor Kinetics in 1-D and 2-D

Speaker:

*Merlin Pelz*

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.

**Jan, 11**: Zeros of linear combinations of L-functions near the critical line

Speaker:

*Youness Lamzouri*

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*

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.