Scientific

Let ord𝑝(𝑎)be the order of 𝑎in (ℤ/𝑝ℤ)∗. In 1927, Artin conjectured that the set of primes 𝑝for which an integer 𝑎≠−1,◻is a primitive root (i.e. ord𝑝(𝑎)=𝑝−1) has a positive asymptotic density among all primes. In 1967 Hooley proved this conjecture assuming the Generalized Riemann Hypothesis (GRH). In this talk, we will study the behaviour of ord𝑝(𝑎)as 𝑝varies over primes. In particular, we will show, under GRH, that the set of primes 𝑝for which ord𝑝(𝑎)is “𝑘prime factors away” from 𝑝−1− 1 has a positive asymptotic density among all primes, except for particular values of 𝑎and 𝑘. We will interpret being “𝑘prime factors away” in three different ways:
𝑘=𝜔(𝑝−1ord𝑝(𝑎)),𝑘=Ω(𝑝−1ord𝑝(𝑎)),𝑘=𝜔(𝑝−1)−𝜔(ord𝑝(𝑎)).

We will present conditional results analogous to Hooley’s in all three cases and for all integer 𝑘. From this, we will derive conditionally the expectation for these quantities.

Furthermore, we will provide partial unconditional answers to some of these questions.

This is joint work with Leo Goldmakher and Greg Martin.

Cell movement requires long-range coordination of the cytoskeletal machinery that organizes cell morphogenesis. We have found that reciprocal interactions between biochemical signals and physical forces enable this long-range signal integration. Through a combination of optogenetic inputs, mechanical measurements, and mathematical modeling, we resolve a recent controversy regarding the role of membrane tension propagation in this process and reveal the requirements for long-range transmission of tension in cells. Most cells don't move in isolation-- they collectively migrate by sharing information similar to the flocking of birds, the schooling of fish, and the swarming of ants. We reveal a novel active signal relay system that rapidly and robustly ensures the proper level of immune cell recruitment to sites of injury and infection.

This talk was given as a guest lecture for the PIMS Network Wide Course Analytic Number Theory II in the 2022-2023 academic year.

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.