Philipp Hieronymi and I have recently established that if E is a boolean combination of open subsets of real euclidean n-space, and the expansion of the real field by E does not define the set of all integers, then the Lebesgue covering dimension of E is equal to its Assouad dimension (hence also to its Hausdorff and packing dimensions, and also to its upper Minkowski dimension if E is bounded). The proof is too technical to attempt in a seminar talk, but the result is surprisingly easy to prove for the case n=1. Indeed, I will prove the

(possibly) stronger result that all reasonable (in a way that I will make precise) Lipschitz invariant metric dimensions then coincide on E.

3:45 I. Martin Isaacs: Orbit sizes and an analog of the Alperin weight conjecture.

Let G be a finite group acting on a finite vector space V. Then G also acts on the dual space of V, and by general principles, the numbers of orbits in these two actions are equal. Although the sizes of the orbits in these actions generally do not agree, there are, nevertheless, some subtle relationships among these orbit sizes. The proof of the relevant theorem is not hard, but it involves a formula that is formally identical to the still unproved Alperin weight conjecture, which will be explained.

The notion of PAC fields has been generalized by Basarab and by Prestel to ordered fields. Prestel defines a field M to be Pseudo Real Closed field (PRC) if M is existentially closed (in the language of rings) in every regular extension L to which all orderings of M extend. Equivalently, if every absolutely irreducible variety defined over M that has a rational point in every real closure of M, has an M-rational point.

In the first part of the talk I will present a short summary of the required preliminaries on pseudo real closed fields.The main theorem is a positive answer to the conjecture by A. Chernikov, I. Kaplan and P. Simon: If M is a PRC field, then Th(M) is NTP2 if and only if M is bounded.

In the second part of the talk I will give a sketch of the proof.

We give a basic introduction to the projective class group of a group ring and the Whitehead group of a group and discuss applications to topology such as Wall's finiteness obstruction, the s-Cobordism Theorem and topological rigidity.

In simple theory, n-amalgamation is the property that any coherent, independent system of types indexed by proper subsets of {1, ..., n} has a consistent extension. For stable theories, there is a three-way connection (first discovered by Hrushovski) between 4-amalgamation of types, definable groupoids, and the splitting of certain finite covers.

The present talk reports on joint work with Byunghan Kim and Alexei Kolesnikov which generalizes the equivalence of 4-amalgamation and eliminability of definable groupoids: in a stable theory, if n is minimal such that n-amalgamation fails, then in a mild expansion (adding a predicate for a Morley sequence) the theory interprets a homogeneous, locally finite structure which we call an (n-2)-ary polygroupoid that witnesses the failure of amalgamation. A 2-ary polygroupoid is an ordinary groupoid, and a k-ary polygroupoid has a k-ary "composition" operation which satisfies a generalized associativity law.

We give an introduction to the K- and L-theoretic Farrell-Jones Conjecture and discuss its status. e.g, recently it has been proved for all lattices in almost connected Lie groups. We give a panorama of its large variety of applications, for instance to the Novikov Conjecvture about the homotopy invariance of higher signatures, the Borel Conjecture about the
topological rigidity of aspherical manifolds and to hyperbolic groups with spheres as boundary. Finally we dicsuss some connections to equivariant homotopy and homology for proper actions of infinite groups.

The Goodwillie derivatives of a functor from based spaces to spectra possess additional structure that allows the Taylor tower of the functor to be reconstructed. I will describe this structure as a 'module' over the 'pro-operad' formed by the Koszul duals of the little disc operads. For certain functors this structure arises from an actual module over the little L-discs operad for some L. In particular, this is the case for functors that are left Kan extensions from a category of 'pointed framed L-dimensional manifolds' (which are examples of the zero-pointed manifolds of Ayala and Francis). As an application I will describe where Waldhausen's algebraic K-theory of spaces fits into this picture. This is joint work with Greg Arone (and, additionally, with Andrew Blumberg for the application to K-theory).

Given a "non-special" section of a semi-abelian scheme over a curve, the relative Manin-Mumford conjecture (RMM) asserts that its image W meets only finitely many torsion curves. I will explain how a relative version of the points constructed (in a Kummer theoretical setting) by the organizer of this seminar  provide ``very special" examples of infinite intersection. However, the corresponding curves W recover a "normally special" status, when viewed in the setting of Pink's conjecture on mixed Shimura varieties. Furthermore, these sections form the only counterexample to the standard version of RMM for semi-abelian surfaces. These are joint results with B. Edixhoven, and with D. Masser, A. Pillay and U. Zannier.

Motivic integration was invented by Kontsevich about 20 years ago for proving that Hodge numbers of Calabi-Yau are birational invariants and has developed at a fast pace since. I will start by presenting informally a device for defining and computing motivic integrals due to Cluckers and myself. I will then focus on  some recent applications. In particular I plan to explain how motivic integration provides tools for proving uniformity results for $p$-adic integrals occuring in the Langlands program and  how one may use "motivic harmonic analysis" to study certain generating series counting curves in enumerative algebraic geometry.

The talk will be built around  examples of how model theory informs our understanding in areas of mathematics such as algebra, number theory, algebraic geometry and analysis. The model theory behind these applications includes concepts such as compactness, definability, stability and o-minimality. However, we assume  no special expertise in model theory, but rather aim  to illustrate the kinds  of  mathematical questions for which a model theoretical perspective has proven to be useful.