Mathematics

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.

Individual and collective cell polarity has fascinated mathematical modelers for a long time. Recently, a more subtle type of symmetry breaking started to attract attention of experimentalists and theorists alike - emergence of chirality in single cells and in cell groups. I will describe a joint project with Bershadsky/Tee lab to understand collective cell chirality on adhesive islands. From the initial microscopy data, two potential models emerged: in one, cells elongate and slowly rotate, and neighboring cells align with each other. When the collective rotation is stopped by the island boundaries, chirality emerges. In an alternative model, cells become chiral due to stress fibers turns inside the cells on the boundary, and then the polarity pattern propagates inward into the cellular groups. We used agent-based modeling to simulate these two hypotheses. The models make many predictions, and I will show how we discriminated between the models by comparing the data to these predictions.

Even though Euler products of L-functions are generally valid only to the right of the critical strip, there is a strong sense in which they should persist even inside the critical strip. Indeed, the behaviour of Euler products inside the critical strip is very closely related to several major problems in number theory including the Riemann Hypothesis and the Birch and Swinnerton-Dyer conjecture. In this talk, we will give an introduction to this topic and then discuss recent work on establishing asymptotics for partial Euler products of L-functions in the critical strip. We will end by giving applications of these results to questions related to Chebyshev's bias.

In this talk, we will discuss the question of establishing CLTs for empirical entropic optimal transport when choosing the regularisation parameter as a decreasing function of the sample size. Importantly, decreasing the regularisation parameter enables estimating the population unregularized quantities of interest. Furthermore, we will show an application to score function estimation, a central quantity in diffusion models, and will discuss parallels with recent work on the estimation of transport maps based on the linearization of the Monge—Ampère equation.

We show that the reduction of a projective endomorphism modulo a discrete valuation naturally takes the form of a set-theoretic correspondence. This raises the possibility of classifying "reduction types" of such dynamical systems, reminiscent of the additive/multiplicative dichotomy for elliptic curves. These correspondences facilitate the exact evaluation of certain integrals of dynamical Green's functions, which arise as local factors in the context of counting rational points ordered by the Call-Silverman canonical height. No prior knowledge of arithmetic dynamics will be assumed.