C5

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.

We establish that a criterion based on ring-theoretic amenability is both necessary and sufficient for the abelian version of the Elekes-Szabó theorem to be sharp in the case of positive characteristic. Moreover, the criterion is always sufficient. We provide illustrative examples in the theories ACF_p and DCF_0.

We shall explain how to represent a couple of basic notions in model theory by standard simplicial diagrams from homotopy theory. Namely, we shall see that the notions of a {definable/invariant type}, {convergence}, and {contractibility} are defined by the same simplicial formula, and so are that of a {complete E-M type} and an {idempotent of an oo-category}.  The first reformulation makes precise Hrushovski's point of view that a definable/invariant type is an operation on types rather than a property of a type depending on the choice of a model, and suggests a notion of a type over a {space} of parameters. The second involves the nerve of the category with a single idempotent non-identity morphism, and leads to a reformulation of {non-dividing} somewhat similar to that of lifting idempotents in an oo-category. If time permits, I shall also present simplicial reformulations of distality, NIP, and simplicity.

We do so by associating with a theory the simplicial set of its n-types, n>0. This simplicial set, or rather its symmetrisation, appeared earlier in model theory under the names of {type structure}  (M.Morley. Applications of topology to Lw1w. 1974), {type category} (R.Knight, Topological Spaces and Scattered Theories. 2007), {type space functors} (Haykazyan. Spaces of Types in Positive Model Theory. 2019; M.Kamsma. Type space functors and interpretations in positive logic. 2022).