# Mathematics

## PIMS Summer Public Lecture: John Mighton

Math provides us with mental tools of incredible power. When we learn math we learn to see patterns, to think logically and systematically, to draw analogies, to perceive risk, to understand cause and effect--among many other critical skills.

Yet we tolerate and in fact expect a vast performance gap in math among students and live in a world where many adults aren't equipped with these crucial tools. This learning gap is unnecessary, dangerous and tragic, and it has led us to a problem of intellectual poverty which is apparent everywhere--in fake news, political turmoil, floundering economies, even in erroneous medical diagnoses.

The study of math is an ideal starting point to break down social inequality and empower individuals to build a smarter, kinder, more equitable world. In this talk Mighton will share his vision for a numerate society for all, not just a chosen few.

### Speaker Biography

Dr. John Mighton is a playwright turned mathematician and author who founded JUMP Math as a charity in 2001. His work in fostering numeracy and in building children's self-confidence through success in math has been widely recognized. He has been named a Schwab Foundation Social Entrepreneur of the Year, an Ernst & Young Social Entrepreneur of the Year for Canada, an Ashoka Fellow, an Officer of the Order of Canada, and has received five honorary doctorates. John is also the recipient of the 10th Annual Egerton Ryerson Award for Dedication to Public Education.

John developed JUMP Math to address both the tragedy of low expectations for students and that of math anxiety in teachers. John began tutoring children in math as a financially-struggling playwright, and his success in helping students achieve levels of success that teachers and parents had thought impossible fueled his belief that everyone has great untapped potential.

The experience of repeatedly witnessing the heart-breaking paradox of high potential and low achievement led him to conclude that the widely-held assumption that mathematical talent is a rare genetic gift has created a self-fulfilling prophecy of low achievement. A generally high level of math anxiety among many elementary school teachers, itself an outcome of that belief system, creates an additional challenge.

John had to overcome his own "massive math anxiety" before making the decision to earn a Ph.D. in Mathematics at the University of Toronto. He was later awarded an NSERC Fellowship for postdoctoral research in knot and graph theory. He is currently a Fellow of the Fields Institute for Research in Mathematical Sciences and has taught mathematics at the University of Toronto. He has also lectured in philosophy at McMaster University, where he received a master’s degree in philosophy.

His plays have been performed around the world and he is the recipient of several national awards for theatre, including two Governor General’s Awards. He played the role of Tom in the film Good Will Hunting.

## Mathematical modelling of the emergence and spread of antimalarial drug resistance

Malaria is a leading cause of death in many low-income countries despite being preventable, treatable and curable. One of the major roadblocks to malaria elimination is the emergence and spread of antimalarial drug resistance, which evolves when malaria parasites are exposed to a drug for prolonged periods. In this talk, I will introduce several statistical and mathematical models for monitoring the emergence and spread of antimalarial drug resistance. Results will be presented from a Bayesian geostatistical model that have generated spatio-temporal predictions of resistance based on prevalence data available only at discrete study locations and times. In this way, the model output provides insight into the spatiotemporal spread of resistance that the discrete data points alone cannot provide. I will discuss how the results of these models have been used to update public health policy.

## Micro-Pharmacology: Recognizing and Overcoming the Tissue Barriers to Drug Delivery

Systemic chemotherapy is one of the main anticancer treatments used for most kinds of clinically diagnosed tumors. However, the efficacy of these drugs can be hampered by the physical attributes of the tumor tissue, such as tortuous vasculature, dense and fibrous extracellular matrix, irregular cellular architecture, metabolic gradients, and non-uniform expression of the cell membrane receptors. This can impede the transport of therapeutic agents to tumor cells in quantities sufficient to exert the desired effect. In addition, tumor microenvironments undergo dynamic spatio-temporal changes during treatment, which can also obstruct the observed drug efficacy. To examine ways to improve drug delivery on a cell-to-tissue scale (single-cell pharmacology), we developed the microscale pharmacokinetics/pharmacodynamics modeling framework “microPKPD”. I will present how this framework can be used to design optimal schedules for various treatments and to investigate the development of drug-induced resistance.

## Something's wrong in the (cellular) neighborhood: Mechanisms of epithelial wound detection

The first response of epithelial cells to local wounds is a dramatic increase in cytosolic calcium. This increase occurs quickly – calcium floods into damaged cells within 0.1 s, moves into adjacent cells over ~20 s, and appears in a much larger set of surrounding cells via a delayed second expansion over 40-300 s – but calcium is nonetheless a reporter: cells must detect wounds even earlier. Using the calcium response as a proxy for wound detection, we have identified an upstream G-protein-coupled-receptor (GPCR) signaling pathway, including the receptor and its chemokine ligand. We present experimental and computational evidence that multiple proteases released during cell lysis/wounding serve as the instructive signal, proteolytically liberating active ligand to diffuse to GPCRs on surrounding epithelial cells. Epithelial wounds are thus detected by the activation of a protease bait. We will discuss the experimental evidence and a corresponding computational model developed to test the plausibility of these hypothesized mechanisms. The model includes calcium currents between each cell’s cytosol and its endoplasmic reticulum (ER), between cytosol and extracellular space, and between the cytosol of neighboring cells. These calcium currents are initiated in the model by cavitation-induced microtears in the plasma membranes of cells near the wound (initial influx), by flow through gap junctions into adjacent cells (first expansion), and by the activation of GPCRs via a proteolytically activated diffusible ligand (second expansion). We will discuss how the model matches experimental observations and makes experimentally testable predictions.

Supported by NIH Grant 1R01GM130130.

## Stationary measure and orbit closure classification for random walks on surfaces

We study the problem of classifying stationary measures and orbit closures for non-abelian action on surfaces. Using a result of Brown and Rodriguez Hertz, we show that under a certain average growth condition, the orbit closures are either finite or dense. Moreover, every infinite orbit equidistributes on the surface. This is analogous to the results of Benoist-Quint and Eskin-Lindenstrauss in the homogeneous setting, and the result of Eskin-Mirzakhani in the setting of moduli spaces of translation surfaces.

We then consider the problem of verifying this growth condition in concrete settings. In particular, we apply the theorem to two settings, namely discrete perturbations of the standard map and the \Out(F_2)-action on a certain character variety. We verify the growth condition analytically in the former setting, and verify numerically in the latter setting.

## Quantitative weak mixing for random substitution tilings

"Quantitative weak mixing" is the term used to bound the dimensions of spectral measures of a measure-preserving system. This type of study has gained popularity over the last decade, led by a series of results of Bufetov and Solomyak for a large class of flows which include general one-dimensional tiling spaces as well as translation flows on flat surfaces, as well as results on quantitative weak mixing by Forni. In this talk I will present results which extend the results for flows to higher rank parabolic actions, focusing on quantitative results for a broad class of tilings in any dimension. The talk won't assume familiarity with almost anything, so I will define all objects in consideration.

## Unique ergodicity of horocycle flows on compact quotients of SL(2,R) following Coudène

Furstenberg proved that the horocycle flow on any compact quotient of SL(2,R) is uniquely ergodic. This has been generalized by many people. I will present a proof due to Yves Coudène, which I find elegant and can prove some of the generalizations of Furstenberg's theorem too.

## The counting formula of Eskin and McMullen

Given a lattice acting on the hyperbolic plane, how many orbits of a point intersect the ball the radius of r as r gets big? Similarly, given a hyperbolic surface with a geodesic gamma, how many lifts of gamma to the hyperbolic plane intersect the ball of radius r? Using the mixing of geodesic flow on hyperbolic surfaces, Eskin and McMullen found a short beautiful argument to find the asymptotics for these counting questions (and more general ones on affine symmetric spaces). The key insight is to relate the counting problems to the equidistribution of circles under geodesic flow. In this talk I will discuss how to deduce circle equidistribution and counting problem asymptotics from mixing. The talk will involve many pictures and focus on the case of hyperbolic surfaces, however, the arguments presented will be general and their application to counting on general affine symmetric spaces will be explained at the end of the talk.

## Specification and the measure of maximal entropy

There are various proofs that a transitive uniformly hyperbolic dynamical system has a unique measure of maximal entropy. I will outline a proof due to Bowen that uses the specification and expansivity properties, focusing on the example of shift spaces. If time permits, I will describe how Bowen's proof works for equilibrium states associated to nonzero potential functions.

## Geometric Langlands for hypergeometric sheaves

Generalised hypergeometric sheaves are rigid local systems on the punctured projective line with remarkable properties. Their study originated in the seminal work of Riemann on the Euler--Gauss hypergeometric function and has blossomed into an active field with connections to many areas of mathematics. I will report on a joint work with Lingfei Yi, where we construct the Hecke eigensheaves whose eigenvalues are the irreducible hypergeometric local systems. This confirms a central conjecture of the geometric Langlands program for hypergeometrics. The key tool we use is the notion of rigid automorphic data due to Zhiwei Yun. This talk is based on the preprint arXiv:2006.10870.