C5

In this talk I will give a hopefully not too technical introduction to one of the techniques that allowed Taylor and Wiles to prove the modularity theorem that was the final step for proving Fermat's Last Theorem.
After explaining how the patching works, I will present some generalisations of the method to different contexts. If time permits, I will also briefly explain how patching was used to produce a candidate for the p-adic local Langlands correspondence.

We study two-dimensional characteristic free-boundary problems in ideal compressible magnetohydrodynamics. For current-vortex sheets, surface-wave effects yield derivative loss and only weak (neutral) stability; under a sufficient stability condition on the background state we obtain anisotropic weighted Sobolev energy estimates and prove local-in-time existence and nonlinear stability via a Nash-Moser scheme, confirming stabilization by strong magnetic fields against Kelvin-Helmholtz instability. For the plasma-vacuum interface, coupling hyperbolic MHD with elliptic pre-Maxwell dynamics, we establish local existence and uniqueness provided at least one magnetic field is nonzero along the initial interface.

In the talk, I'll first describe a more general context of Sárközy-type problems and interesting directions in which they can be pursued. Then, I'll focus on the specific case of bounding the size of sets A s. t. A - A + 1 contains no prime. After describing the progress on the problem for integers, I'll pass on to considering an analogous question for function fields and (after a general introduction to function fields) I'll speak about my recent result in this area.

We consider a one-dimensional compressible Navier-Stokes model for reacting gas mixtures with the same γ-law in dynamic combustion. The unknowns of the PDE system consist of the inverse density, velocity, temperature, and mass fraction of the reactant (Z). First, we show that the graph of Z cannot form cusps or corners near the points where the reactant in the combustion process is completely depleted at any time, based on a Bernis-type inequality by M. Winkler (2012) and the recent works by T. Cieślak et al (2023). In addition, we establish the global well-posedness theory of small BV weak solutions for initial data that are small perturbations around the constant equilibrium state (1, 0, 1, 0) in the L¹(R)∩BV(R)-norm, via an analysis of the Green's function of the linearised system. The large-time behaviour of the global BV weak solutions is also characterised. This is motivated by and extends the recent global well-posedness theory for BV weak solutions to the one-dimensional isentropic Navier-Stokes and Navier-Stokes-Fourier systems developed by T. Liu and S.-H. Yu (2022).

*Joint with Prof. Haitao Wang and Miss Jianing Yang (SJTU)

A longstanding folklore conjecture in combinatorial number theory is the following: given an additive set $S$ not containing the identity, $S$ can be ordered as $s_1, \ldots, s_k$ so that the partial sums $s_1+\cdots+s_j$ are distinct for each $j\in[k]$. We discuss a recent resolution of this conjecture in the finite field model (where the ambient group is $\mathbb{F}_2^n$, or more generally, any bounded exponent abelian group). This is joint work with B. Bedert, M. Bucic, N. Kravitz, and R. Montgomery.

Neurons interact via spikes, which is a pulse-like, discontinuous mechanism. Their mean-field PDE description gives Fokker-Planck equations with novel nonlinearities. From a probability point of view, these give rise to Mckean-Vlasov equations involving hitting times. Similar mechanisms also arise in models for systemic risk in mathematical finance, and the supercooled Stefan problem. In this talk, we will first present models for spiking neurons: both microscopic particle models and macroscopic PDE models, with an emphasis on the general mathematical structure. A central question for these equations is the finite-time blow-up of the firing rate, which scientifically corresponds to the synchronization of a neuronal network. We will discuss how to continue the solution physically after the blow-up, by introducing a new timescale. The new timescale also helps us to understand the long term behavior of the equation, as it reveals a hidden contraction structure in the hyperbolic case. Finally, we will present a recently developed numerical solver based on this framework. Numerical tests show that during the synchronization the standard microscopic solver suffers from a rather demanding time step requirement, while our macro-mesoscopic solver does not.