C6

On this talk we will focus on the family of aggregation-diffusion equations

$$\frac{\partial \rho}{\partial t} = \mathrm{div}\left(\mathrm{m}(\rho)\nabla (U'(\rho) + V) \right).$$

Here, $\mathrm{m}(s)$ represents a continuous and compactly supported nonlinear mobility (saturation) not necessarily concave. $U$ corresponds to the diffusive potential and includes all the porous medium cases, i.e. $U(s) = \frac{1}{m-1} s^m$ for $m > 0$ or $U(s) = s \log (s)$ if $m = 1$. $V$ corresponds to the attractive potential and it is such that $V \geq 0$, $V \in W^{2, \infty}$.

Taking advantage of a family of approximating problems, we show the existence of $C_0$-semigroups of $L^1$ contractions. We study the $\omega$-limit of the problem, its most relevant properties, and the appearance of free boundaries in the long-time behaviour. Furthermore, since this problem has a formal gradient-flow structure, we discuss the local/global minimisers of the corresponding free energy in the natural topology related to the set of initial data for the $L^\infty$-constrained gradient flow of probability densities. Finally, we explore the properties of a corresponding implicit finite volume scheme introduced by Bailo, Carrillo and Hu.

The talk presents joint work with Prof. J.A. Carrillo and Prof. D.  Gómez-Castro.

In this talk, we discuss the critical threshold phenomena in a large class of one-dimensional pressureless Euler-Poisson (EP) equations with non-vanishing background states. First, we establish local-in-time well-posedness in appropriate regularity spaces, specifically involving negative Sobolev spaces, which are adapted to ensure the neutrality condition holds. We show that this negative homogeneous Sobolev regularity is necessary by proving an ill-posedness result in classical Sobolev spaces when this condition is absent. Next, we examine the critical threshold phenomena in pressureless EP systems that satisfy the neutrality condition. We show that, in the case of attractive forcing, the neutrality condition further restricts the sub-critical region, reducing it to a single line in the phase plane. Finally, we provide an analysis of the critical thresholds for repulsive EP systems with variable backgrounds. As an application, we analyze the critical thresholds for the damped EP system in the context of cold plasma ion dynamics, where the electron density is governed by the Maxwell-Boltzmann relation. This talk is based on joint work with Dong-ha Kim, Dowan Koo, and Eitan Tadmor.

Hoheisel's theorem states that there is some $\delta> 0$ and some $x_0>0$ such that for all $x > x_0$ the interval $[x,x+x^{1-\delta}]$ contains prime numbers. Classically this is proved using the Riemann zeta function and results about its zeros such as the zero-free region and zero density estimates. In this talk I will describe a new elementary proof of Hoheisel's theorem. This is joint work with Kaisa Matomäki (Turku) and Joni Teräväinen (Cambridge). Instead of the zeta function, our approach is based on sieve methods and ideas coming from additive combinatorics, in particular, the transference principle. The method also gives an L-function free proof of Linnik's theorem on the least prime in arithmetic progressions.

In algebraic geometry, the technique of dévissage reduces many questions to the case of curves. In difference and differential algebra, this is not the case, but the obstructions can be closely analysed. In difference algebra, they are difference varieties defined by equations of the form $\si(x)=g x$, determined by an action of an algebraic group and an element g of this group. This is joint work with Zoé Chatzidakis.

In the vanishing viscosity limit from the Navier-Stokes to Euler equations on domains with boundaries, a main difficulty comes from the mismatch of boundary conditions and, consequently, the possible formation of a boundary layer. Within a purely interior framework, Constantin and Vicol showed that the two-dimensional viscosity limit is justified for any arbitrary but finite time under the assumption that on each compactly contained subset of the domain, the enstrophies are bounded uniformly along the viscosity sequence. Within this framework, we upgrade to local strong convergence of the vorticities under a similar assumption on the p-enstrophies, p > 2. The talk is based on a recent publication with Christian Seis and Emil Wiedemann.

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.