# Scientific

## Understanding adversarial robustness via optimal transport.

Deep learning-based approaches have succeeded surprisingly in various fields of sciences. In particular, one of the first and most successful achievements of them is image classification. Now, deep learning-based algorithms perform even better than humans in classification problems. However, people in machine learning community observed that it is possible to deteriorate the performance of neural networks seriously by adding a well-designed small noise, which is called ‘adversarial attack’. Although humans still classify this new image correctly. the machine completely fails to classify this new image. Since this issue is serious in practice, for example security or self-driving car, practioners want to develop more robust machines against such adversarial attack, which motivates ‘adversarial training problem’. However, until very recently there has been no theoretical understanding of it. In this talk, I will present the recent progress of understanding adversarial training problem. The key idea of connecting the two areas originates from (Wasserstein) barycenter problem, one of the famous implications of optimal transport theory. I will introduce ‘generalized barycenter problem’, the extension of classical barycenter problem, and its multimarginal optimal transport formulations. Through the lens of those tools, one can understand the geometric structure of adversarial training problems. One crucial advantage of this result is that it allows to utilize many computational optimal transport tools. Lastly, if time is permitted, I will give the result of the existence of optimal robust classifiers which not only extends the binary setting case to the multiclass one but also provides a clean interpretation by duality.

## What conifer trees can show us about how organs are positioned in developing organisms

One of the central questions in developmental biology is how organs form in the correct positions in order to create a functional mature organism. Plant leaves offer an easily observable example of organ positioning, with species-specific motifs for leaf arrangement (phyllotaxis). These patterns arise through a combination of chemical pattern formation, mechanical stresses and growth. Mathematical modelling in each of these areas (and their combinations) contributes to quantitative understanding of developmental mechanisms and morphogenesis in general. Conifer trees are some of the most characteristic plants of BC. They also display a type of ring patterning of their embryonic leaves (cotyledons), which I believe offers a unique route to understanding plant phyllotaxis in general. I will discuss how early work at UBC on similar patterning in algae led to application of reaction-diffusion models in conifer development. This framework has guided experiments at BCIT and recently led to a model that accounts for the natural variability in conifer cotyledon number. The model involves the kinetics of a highly conserved gene regulation module and therefore sheds light on the chemical pattern formation control of phyllotaxis across plants. Conifer patterning also demonstrates scaling of position to organism size, an active area of research in animal development: the model provides some mechanistic insight into how this can occur via chemical kinetics.

## Sign changes of the error term in the Piltz divisor problem

For an integer k≥3; Δk (x) :=∑n≤xdk(n)-Ress=1 (ζk(s)xs/s), where dk(n) is the k-fold divisor function, and ζ(s) is the Riemann zeta-function. In the 1950's, Tong showed for all large enough X; Δk(x) changes sign at least once in the interval [X, X + CkX1-1/k] for some positive constant Ck. For a large parameter X, we show that if the Lindelöf hypothesis is true, then there exist many disjoint subintervals of [X, 2X], each of length X1-1/k-ε such that Δk (x) does not change sign in any of these subintervals. If the Riemann hypothesis is true, then we can improve the length of the subintervals to << X1-1/k (logX)-k^2-2. These results may be viewed as higher-degree analogues of a theorem of Heath-Brown and Tsang, who studied the case k = 2. This is joint work with Siegfred Baluyot.

## Transcendence of period integrals over function fields

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.

## Understanding arithmetic and geometry through cutting and pasting

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.

## Vertex-transitive graphs with large automorphism groups

**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.

## Turing Patterns on Growing Domains

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.

## Existence, Stability and Slow Dynamics of Spikes in a 1D Minimal Keller–Segel Model with Logistic Growth

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.

## PIMS/FACTS Panel: Tackling Climate Change and the Just Transition to Renewable Energy

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.

- Réne Aïd: More electricity demand response for less carbon emissions
- Gaël Giraud: Macroeconomics and Climate
- Seth Klein: Mobilizing Canada for the Climate Emergency
- Judith Sayers: First Nation Leadership in Clean Energy and Climate Action
- Andrew Weaver: Privilege, agency, and the climate scientist’s role in the global warming debate

## The Canadian regional climate model

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)