C6

The goal of the talk is to give an overview of the metric theory of currents by Ambrosio-Kirchheim, together with some recent progress in the setting of Banach spaces. Metric currents are a generalization to the metric setting of classical currents. Classical currents are the natural generalization of oriented submanifolds, as distributions play the same role for functions. We present a structure result for 1-metric currents as superposition of 1-rectifiable sets in Banach spaces, which generalizes a previous result by Schioppa. This is based on an approximation result of metric 1-currents with normal 1-currents. This is joint work with D. Bate, J. Takáč, P. Valentine, and P. Wald (Warwick).

Sard’s theorem asserts that the set of critical values of a smooth map from one Euclidean space to another one has measure zero. A version of this result for infinite-dimensional Banach manifolds was proven by Smale for maps with Fredholm differential. However, when the domain is infinite dimensional and the range is finite dimensional, the result is not true – even under the assumption that the map is “polynomial” – and a general theory is still lacking. In this seminar, I will provide sharp quantitative criteria for the validity of Sard’s theorem in this setting, obtained combining a functional analysis approach with new tools in semialgebraic geometry. As an application, I will present new results on the Sard conjecture in sub-Riemannian geometry. Based on a joint work with A. Lerario and L. Rizzi.

The Alfvén waves are fundamental wave phenomena in magnetized plasmas. Mathematically, the dynamics of Alfvén waves are governed by a system of nonlinear partial differential equations called the magnetohydrodynamics (MHD) equations. Let us introduce some recent results about inverse scattering of Alfvén waves in ideal MHD, which are intended to establish the relationship between Alfvén waves emanating from the plasma and their scattering fields at infinities.The proof is mainly based on the weighted energy estimates. Moreover, the null structure inherent in MHD equations is thoroughly examined, especially when we estimate the pressure term.

Several physical models are available to understand the dynamics of fluid mixtures, including the so-called Baer-Nunziato models. The partial differential equations associated with these models look like those of Navier-Stokes, with the addition of new relaxation terms. One strategy to obtain these models is homogenisation: starting from a mesoscopic mixture, where two pure fluids satisfying the compressible Navier-Stokes equations share the space between them, a change of scale is performed to obtain a macroscopic mixture, where the two fluids can coexist at any point in space.

This problem concerns the study of the Navier-Stokes equations with strongly oscillating initial data. We'll start by explaining some results in this framework, in one dimension of space and on the torus, for barotropic fluids. We will then detail the various steps involved in demonstrating homogenisation. Finally, we'll explain how to adapt this reasoning to homogenisation for perfect gases, with and without heat conduction.

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