Scientific

For over a century, scientists have studied striking spatiotemporal patterns during the continual tooth replacement of reptiles. Aside from the compelling aesthetics of this phenomenon, it is thought that understanding the underlying mechanisms may provide the insight required to trigger adult tooth replacement in humans. Theoretical frameworks have long been proposed to understand the rules behind the observed spatiotemporal order, but have only been analyzed mathematically more recently. Starting from Edmund's observations in crocodiles and proposed theory of replacement waves, we show how a simple model consisting of a row of non-interacting phase oscillators predicts several experimental observations. Next, inspired by the hypothesis put forth by Osborn, we consider a variation of the phase model with ODEs that account for mutual inhibition between tooth sites, and use continuation methods to thoroughly search parameter space for experimentally validated solutions. We then extend the model to a PDE that explicitly accounts for the diffusion of inhibitory signals between teeth, yielding some novel solution types. Using continuation methods once again, we delineate parameter regimes with solutions that closely resemble experimental observations in leopard geckos.

Given an elliptic curve $E/\mathbb{Q}$, we can consider its trace of Frobenius, denoted as $a_p(E)$, where $p$ is a prime. We will discuss the race problem arising from these ap values and the general strategy in attacking these problems.

It has been known since the 80s, thanks to Conrey and Ghosh, that the average of the square of the Riemann zeta function, summed over the extreme points of zeta up to a height $T$, is $\frac{1}{2}(e^2 −5)\log T$ as $T\rightarrow \infty$. This problem and its generalisations are closely linked to evaluating asymptotics of joint moments of the zeta function and its derivatives, and for a time was one of the few cases in which Number Theory could do what Random Matrix Theory could not. RMT then managed to retake the lead in calculating these sorts of problems, but we may now tell the story of how Number Theory is fighting back, and in doing so, describe how to find a full asymptotic expansion for this problem, the first of its kind for any nontrivial joint moment of the Riemann zeta function. This is joint work with Chris Hughes and Solomon Lugmayer

The Farey sequence $\mathcal{F}_Q$ of order $Q$ is the ascending sequence of fractions $\frac{a}{b}$ in the unit interval $(0, 1]$ with $gcd(a, b) = 1$ and $0 < a \leq b \leq Q$. The study of the Farey fractions is of major interest because of their role in problems related to Diophantine approximation. Also, there is a connection between the distribution of Farey fractions and the Riemann hypothesis, which further motivates their study. In this talk, we will discuss the distribution of Farey fractions with some divisibility constraints on denominators by studying their pair-correlation measure. This is based on joint work with Sneha Chaubey.

Recently Martin, Mossinghoff, and Trudgian investigated comparative number theoretic results for a family of arithmetic functions called “fake $\mu$’s”. In their paper, they focused on the bias and oscillation of the summatory function of a fake $\mu$ at the $\sqrt{x}$ scale, while acknowledging that a function with no bias at this scale could well see one at a smaller scale. In this spirit, I will discuss some new oscillation results for the summatory functions of general fake $\mu$’s. This is joint work with Greg Martin.

Please note, this recording is incomplete due to a problem with the room system.