C6

TBC

The Keating-Snaith conjecture for quadratic twists of elliptic curves predicts the central values should have a log-normal distribution. I present recent progress towards establishing this in the range of large deviations of order of the variance. This extends Selberg’s Central Limit Theorem from ranges of order of the standard deviation to ranges of order of the variance in a variety of contexts, inspired by random walk theory. It is inspired by recent work on large deviations of the zeta function and central values of L-functions.
 

The large sieve inequality for Dirichlet characters is a central result in analytic number theory, which encodes a strong orthogonality property between primitive characters of varying conductors. This can be viewed as a statement about $\mathrm{GL}_1$ automorphic representations, and it is a key open problem to prove similar results in the higher $\mathrm{GL}_m$ setting; for $m \ge 2$, our best bounds are far from optimal. We'll outline two approaches to such results (sketching them first in the elementary case of Dirichlet characters), and discuss work-in-progress of Thorner and the author on an improved $\mathrm{GL}_m$ large sieve. No prior knowledge of automorphic representations will be assumed.

The Martinet flat structure is one of the simplest sub-Riemannian manifolds that has many non-Riemannian features: it is not equiregular, it has abnormal geodesics, and the Carnot-Carathéodory sphere is not sub-analytic. I will review how the geometry of the Martinet flat structure is tied to the equations of the pendulum. Surprisingly, the Measure Contraction Property (a weak synthetic formulation of Ricci curvature bounds in non-smooth spaces) fails, and we will try to understand why. If time permits, I will also discuss how this can be generalised to some Carnot groups that have abnormal extremals. This is a joint work in progress with Luca Rizzi.

We investigate an interacting particle model to simulate a foraging colony of ants, where each ant is represented as a so-called active Brownian particle. Interactions among ants are mediated through chemotaxis, aligning their orientations with the upward gradient of the pheromone field. We show how the empirical measure of the interacting particle system converges to a solution of a mean-field limit (MFL) PDE for some subset of the model parameters. We situate the MFL PDE as a non-gradient flow continuity equation with some other recent examples. We then demonstrate that the MFL PDE for the ant model has two distinctive behaviors: the well-known Keller--Segel aggregation into spots and the formation of lanes along which the ants travel. Using linear and nonlinear analysis and numerical methods we provide the foundations for understanding these particle behaviors at the mean-field level. We conclude with long-time estimates that imply that there is no infinite time blow-up for the MFL PDE.

The Brunn-Minkowski inequality gives a lower bound on the volume of the set of midpoints of line segments joining two sets. On a Riemannian manifold, line segments are replaced by geodesic segments, and the Brunn-Minkowski inequality characterizes manifolds with nonnegative Ricci curvature. I will present a generalization of the Riemannian Brunn-Minkowski inequality where geodesics are replaced by magnetic geodesics, which are minimizers of a functional given by length minus the integral of a fixed one-form on the manifold. The Brunn-Minkowski inequality is then equivalent to nonnegativity of a suitably defined magnetic Ricci curvature. More generally, I will present a magnetic version of the Borell-Brascamp-Lieb inequality of Cordero-Erausquin, McCann and Schmuckenschläger. The proof uses the needle decomposition technique.

In recent years, the theme of asymptotically flat spacetimes has come back to the fore, fueled by the discovery of gravitational waves and the growing interest in what flat holography could be. In this quest, the standard tools pertaining to asymptotically anti-de Sitter spacetimes have been insufficiently exploited. I will show how Ricci-flat spacetimes are generally reached as a limit of Einstein geometries and how they are in fact constructed by means of data defined on the conformal Carrollian boundary that is null infinity. These data, infinite in number, are obtained as the coefficients of the Laurent expansion of the energy-momentum tensor in powers of the cosmological constant. This approach puts this tensor back at the heart of the analysis, and at the same time reveals the versatile role of the boundary Cotton tensor. Both appear in the infinite hierarchy of flux-balance equations governing the gravitational dynamics.  

For a complete theory T, Lascar associated with it a Galois group which we call the Lacsar group. We will talk about some of my work on recovering the Lascar group as the fundamental group of Mod(T) and some recent progress in understanding the higher homotopy groups.