# Mathematics

## On the average least negative Hecke eigenvalue

The least quadratic non-residue has been a central problem in number theory for centuries. The average least quadratic non-residue was explored by Erdős in the 1960s, and many extensions of this problem such as to the average least character non-residue (Martin, Pollack) have been explored. In this talk, we look in to the average first sign change of Fourier coefficients of newforms (equivalently Hecke eigenvalues). We discuss the distribution of Hecke eigenvalues through the so-called 'horizontal' and 'vertical' Sato-Tate distributions, and we also discuss large sieve inequalities for cusp forms that are uniform in both the weight and the level.

## Skeletal muscle: modeling and computation

Skeletal muscle is composed of cells collectively referred to as fibers, which themselves contain contractile proteins arranged longtitudinally into sarcomeres. These latter respond to signals from the nervous system, and contract; unlike cardiac muscle, skeletal muscles can respond to voluntary control. Muscles react to mechanical forces - they contain connective tissue and fluid, and are linked via tendons to the skeletal system - but they also are capable of activation via stimulation (and hence, contraction) of the sacromeres. The restorative along-fibre force introduce strong mechanical anisotropy, and depend on departures from a characteristic length of the sarcomeres; diseases such as cerebral palsy cause this characteristic length to change, thereby impacting muscle force. In the 1910s, A.V. Hill [1] posited a mathematical description of skeletal muscles which approximated muscle as a 1-dimensional nonlinear and massless spring. This has been a remarkably successful model, and remains in wide use. Yet skeletal muscle is three dimensional, has mass, and a fairly complicated structure. Are these features important? What insights are gained if we include some of this complexity in our models? Many mathematical questions of interest in skeletal muscle mechanics arise: how to model this system, how to discretize it, and what theoretical properties does it have? In this talk, we survey recent work on the modeling, parameter estimation, simulation and validation of a fully 3-D continuum elasticity approach for skeletal muscle dynamics. This is joint work based on a long-standing collaboration with James Wakeling (Dept. of Biomedical Physiology and Kinesiology, SFU).

## Computational Modeling for Medical Devices

The Food and Drug Administration (FDA) is responsible for ensuring the safety and effectiveness of medical devices marketed in the US. For several decades, in a handful of niche applications, medical device industry has used computational modeling to provide evidence for safety or effectiveness, complementing bench, animal, or clinical testing. In recent years, the use of computational modeling in medical device regulatory submissions has grown significantly. FDA’s medical device Center is now tasked with evaluating a wide range of computational models of medical devices, as well as computational models implemented in medical device software (for example, patient-specific model-based software devices, closely related to the concept of a digital twin), and in silico clinical trials. This talk will discuss how computational models are relevant to medical devices, and then delve in model credibility assessment. We will discuss key activities involved in evaluating computational models for medical devices, overview recent FDA-led Standards and Guidances, and summarize recent work expanding these methods to the new frontiers of patient-specific models and in silico clinical trials.

## Environmental feedback maintains cooperation in phage $\Phi_6$

The evolution and maintenance of cooperation is a fundamental problem in evolutionary biology. Because cooperative behaviors impose a cost, Cooperators are vulnerable to exploitation by Defectors that do not pay the cost to cooperate but still benefit from the cooperation of others. The bacteriophage $\Phi_6$ exhibits cooperative and defective phenotypes in infection: during replication, phages produce essential proteins in the host cell cytoplasm. Coinfection between multiple phages is possible. A given phage cannot guarantee exclusive access to its own proteins, so Cooperators contribute to the common pool of proteins while Defectors contribute less and instead appropriate proteins from Cooperators. Previous experimental work found that $\Phi_6$ was trapped in a prisoner's dilemma, predicting that the cooperative phenotype should disappear. Here we propose that environmental feedback, or interplay between phage and host densities, can maintain cooperation in $\Phi_6$ populations by modulating the rate of co-infection and shifting the advantages of cooperation vs. defection. We build and analyze an ODE model and find that for a wide range of parameter values, environmental feedback allows Cooperation to survive.

## Torsion of Rational Elliptic Curves over the Cyclotomic Extensions of ℚ

Let E be an elliptic curve defined over ℚ. Let p > 3 be a prime such that p - 1 is not divisible by 3, 4, 5, 7, 11. In this article, we classify the groups that can arise as E(ℚ(ζp))tors up to isomorphism. The method illustrates techniques for eliminating possible structures that can appear as a subgroup of E(ℚab)tors.

## Subconvexity for GL(2) L-functions and Shifted MDS

Subconvexity problems have maintained extreme interest in analytic number theory for decades. Critical barriers such as the convexity, Burgess, and Weyl bounds hold particular interest because one usually needs to drastically adjust the analytic techniques involved in order to break through them. It has recently come to light that shifted Dirichlet series can be used to obtain subconvexity results. While these Dirichlet series do not admit Euler products, they are amenable to study via spectral methods. In this talk, we construct a shifted multiple Dirichlet series (MDS) and leverage its analytic continuation via spectral decompositions to obtain the Weyl bound in the conductor-aspect for the L-function of a holomorphic cusp form twisted by an arbitrary Dirichlet character. This improves upon the corresponding bound for quadratic characters obtained by Iwaniec-Conrey in 2000. This work is joint with Jeff Hoffstein, Nikos Diamantis, and Min Lee.

## Mathematical modeling applied to women's health

Several years ago, Dr Kathryn Isaac, a UBC professor and clinical surgeon contacted me with an intriguing problem. In her work on cosmetic reconstructive for post-breast-cancer-surgery patients, she encounters cases of failure that result (weeks or months later) in "capsular contraction" (CC). This painful condition arises when the healing tissue (the "capsule") that forms surrounding a breast implant undergoes pathological contraction and deformation - necessitating a new rounds of surgery. In this talk, I describe work in our team to help understand the root causes of CC, the risk factors, and possible preventative treatments. We use mathematical modeling to depict and investigate hypotheses for cell-level mechanisms involved in initiating CC. I will describe two rounds of modeling, the earliest joint with Cheryl Dyck MacDonald, and the latest joint with Ms Yuqi Xiao (UBC MSc student) and Prof. Alain Goriely (Oxford).

## Orienteering with One Endomorphism

Given two elliptic curves, the path finding problem asks to find an isogeny (i.e. a group homomorphism) between them, subject to certain degree restrictions. Path finding has uses in number theory as well as applications to 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.

## Parametrization of rings of finite rank - a geometric approach and their use in counting number fields

We describe parametrizations of rings that generalize the notions of monogenic rings and binary rings. We use these parametrizations to give better lower bounds on the number of number fields of degree n and bounded discriminant.

## Moments of real Dirichlet L-functions and multiple Dirichlet series

There are two ways to compute moments in families of L-functions: one uses the approximation by Dirichlet polynomials, and the other is based on multiple Dirichlet series. We will introduce the two methods to study the family of real Dirichlet L-functions, compare them and show that they lead to the same results. We will then focus on obtaining the meromorphic continuation of the associated multiple Dirichlet series.