# Scientific

## Free boundary regularity for the obstacle problem

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.

## Opinion Dynamics and Spreading Processes on Networks

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.

## Expansion, divisibility and parity

**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/

## Forgotten conjectures of Andrews for Nahm-type sums

**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−−−√]

.

## The second moment of symmetric square L-functions over Gaussian integers

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.

## Exceptional Chebyshev's bias over finite fields

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.

## p-torsion of Jacobians for unramified Z/pZ-covers of curves

**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.

## Orienteering on Supersingular Isogeny Volcanoes Using One Endomorphism

**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.

## Kantorovich operators and their ergodic properties

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.

## Adversarial training through the lens of optimal transport

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.