C5

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.

In this talk, I will discuss an approach to free boundary minimal surfaces which comes out of recent work by Struwe on a non-local energy, called the half-energy. I will introduce the gradient flow of this functional and its theory in the already studied case of disc type domains, covering existence, uniqueness, regularity and singularity analysis and highlighting the striking parallels with the theory of the classical harmonic map flow. Then I will go on to present new work, joint with Melanie Rupflin and Michael Struwe, which extends this theory to all compact surfaces with boundary. This relies upon combining the above ideas with those of the Teichmüller harmonic map flow introduced by Rupflin and Topping.

This talk will focus on various definitions of orientability for non-smooth spaces with Ricci curvature bounded from below. The stability of orientability and non-orientability will be discussed. As an application, we will prove the orientability of 4-manifolds with non-negative Ricci curvature and Euclidean volume growth. This work is based on a collaboration with E. Bruè and A. Pigati.

I introduce modal group theory, where one investigates the class of all groups using embeddability as a modal operator. By employing HNN extensions, I demonstrate that the modal language of groups is more expressive than the first-order language of groups. Furthermore, I establish that the theory of true arithmetic, viewed as sets of Gödel numbers, is computably isomorphic to the modal theory of finitely presented groups. Finally, I resolve an open question posed by Sören Berger, Alexander Block, and Benedikt Löwe by proving that the propositional modal validities of groups constitute precisely the modal logic S4.2.

Building on Leo’s talk last week, I will present the full Galois characterisation of henselianity and introduce some of the ‘explicit’ ingredients he referred to during his presentation. In particular, I will describe a Galois cohomology-inspired criterion for distinguishing between different characteristics. I will then outline the full proof of the Galois characterisation of p-adically closed fields, indicating how each of the ingredients enters the argument.

The absolute Galois group of ℚₚ determines its field structure: a field K is p-adically closed if and only if its absolute Galois group is isomorphic to that of ℚₚ. This Galois-theoretic characterisation was proved by Koenigsmann in 1995, building on previous work by Arason, Elman, Jacob, Ware, and Pop. Similar results were obtained by Efrat and further developed in his 2006 book.

Our project aims to provide an optimal proof of this characterisation, incorporating improvements and new developments. These include a revised proof strategy; Efrat's construction of valuations via multiplicative stratification; the Galois characterisation of henselianity; systematic use of the standard decomposition; and the function field analogy of Krasner-Kazhdan-Deligne type. Moreover, we replace arguments that use Galois cohomology with elementary ones.

In this talk, I will focus on two key components of the proof: the construction of valuations from rigid elements, and the role of the function field analogy as developed via the non-standard methods of Jahnke-Kartas.

This is joint work with Jochen Koenigsmann and Benedikt Stock.