Termine

The class of so-called distal theories was isolated by Simon as a class of purely unstable NIP theories. Examples include all o-minimal theories and the theory of the p-adics. A rare natural example of an NIP theory that is neither stable nor distal is the theory of algebraically closed valued fields (ACVF), which interprets an o-minimal part (the value group) and a stable part (the residue field). We introduce the notion of relative distality and show that ACVF is distal relative to the residue field. As an application, we strengthen a result of Bays and Martin by proving that valued fields with finite residue field uniformly satisfy incidence bounds in the sense of the Szemerédi?Trotter Theorem. The argument relies on distal incidence bounds due to Chernikov, Galvin, and Starchenko. I will review the relevant model-theoretic and combinatorial background, and explain the main ideas of the proofs as time permits.

The Friedmann--Lemaitre--Robertson--Walker family of spacetimes are the standard homogenous isotropic cosmological models in general relativity. Each member of this family describes a torus, evolving from a big bang singularity and expanding indefinitely to the future, with expansion rate encoded by a suitable scale factor. I will discuss a mixing effect which occurs for the Vlasov equation on these spacetimes when the expansion rate is suitably slow. This is joint work with Renato Velozo Ruiz (Imperial College London).

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In this talk, which is meant to be an introduction to oo-categories, I will use Grothendieck's "Homotopy Hypothesis" as a narrative thread to motivate and give intuition for oo-categories. I will further try to briefly indicate how the practice of this theory actually works, and, in the end, explain how oo-categories provide a framework to revisit and refine the homotopy hypothesis. No prior knowledge of oo-categories will be assumed, though a vague idea of what they might be will be helpful.

Further information

In the previous talk, we introduced the notion of geometric modularity in the sense that $\rho$ arises from a mod $p$ Hilbert modular form, and algebraic modularity in the sense that $\rho$ arises in the mod $p$ cohomology of a Shimura curve. In this talk, I will explain the idea of the proof that geometric modularity implies algebraic modularity of the same weight. The main strategies are the liftability of mod p Hilbert cusp forms, using weight-shifting operators, and the description of the Goren-Oort stratification.