Scientific

A fundamental aim of evolutionary biology is to describe and explain biodiversity patterns; this aim centers around questions of how many "species" exist, where they are most/least abundant, how this distribution is changing over time, and why. Practically speaking, deciphering biodiversity trends and understanding their underlying ecological and evolutionary drivers is important for monitoring and managing both the biodiversity crisis and emergent epidemics. In this seminar I will discuss 100 years of biodiversity mathematics, beginning with Yule's 1924 foundational work on the model that now bears his name. Despite the twists and turns of the intervening years, I will then introduce recent work in my group with direct connections to Yule's. Throughout, I will highlight the importance of using math and models to clarify biological thinking and will argue that a fully interdisciplinary approach that integrates math, biology, and statistics is necessary to understand biodiversity, be it at the macroevolutionary or epidemiological scale.

The Moran process models the evolutionary dynamics between two competing types in a population, traditionally assuming a fixed population size. We investigate an extension to this process which adds ecological aspects through variable population sizes. For the original Moran process, birth and death events are correlated to maintain a constant population size. Here we decouple the two events and derive the stochastic differential equation that represents the dynamics in a well-mixed population and captures its behaviour as the population size becomes arbitrarily large. Our analysis explores the impact of this decoupling on two key determinants of the evolutionary process: fixation probabilities and fixation times. In evolutionary graph theory, these statistics depend significantly on the population structure, such that structures have been identified that act as ‘amplifiers’ of selection while others are ‘suppressors’ of selection. However, these features are crucially dependent on the sequence of events, such as birth-death vs death-birth – a seemingly small change with significant consequences. In our extension of the Moran process this distinction is no longer necessary or possible. We determine the fixation probabilities and times for the well-mixed population, regular graphs as well as amplifiers and suppressors, and compare them to the original Moran process.

For a number field $K$, the P\'olya group of $K$, denoted by $Po(K)$, is the subgroup of the ideal class group of $K$ generated by the classes of the products of maximal ideals of $K$ with the same norm. In this talk, after reviewing some results concerning $Po(K)$, I will generalize this notion to the relative P\'olya group $Po(K/F)$, for $K/F$ a finite extension of number fields. Accordingly, I will generalize some results in the literature about P\'olya groups to the relative case. Then, due to some essential observations, I will explain why we need to modify the notion of the relative P\'olya group to the Ostrowski quotient $Ost(K/F)$ to get a more 'accurate' generalization of $Po(K)$. The talk is based on a joint work with Ali Rajaei (Tarbiat Modares University) and Ehsan Shahoseini (Institute For Research In Fundamental Sciences).

In 1973, assuming the Riemann hypothesis (RH), Montgomery studied the vertical distribution of zeta zeros, and conjectured that they behave like the eigenvalues of some random matrices. We will discuss some models for zeta zeros starting from the random matrix model but going beyond it and related questions, conjectures and results on statistical information on the zeros. In particular, assuming RH and a conjecture of Chan for how often gaps between zeros can be close to a fixed non-zero value, we will discuss our proof of a conjecture of Berry (1988) for the number variance of zeta zeros, in a regime where random matrix models alone do not accurately predict the actual behavior (based on joint work with Meghann Moriah Lugar and Micah B. Milinovich).

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).