# Scientific

## 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.

## The Bootstrap Learning Algorithm

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.

## Central Limit Theorems in Analytic Number Theory

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.

## L-Functions of Elliptic Curves Modulo Integers

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.

## Total Variation Flow on metric measure spaces

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

## Sources, sinks, and sea lice: determining patch contribution and transient dynamics in marine metapopulations

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.