The group configuration in stable theories
Dienstag, 3.5.22, 14:30-15:30, Raum 404, Ernst-Zermelo-Str. 1
A group configuration is a geometric incidence configuration consisting of 6 points which in stable theories is related to the existence of a type definable group. In the first part of the talk, we will introduce the above concepts and point out why a group gives rise to a group configuration. Moreover, by a result from Hrushovski, any group configuration in a stable theory yields the existence of a type definable group. We will discuss the basic ideas of this proof. The second part of the talk will present an application. Hrushovski and Pillay showed that any definable group in a real closed field F is locally isomorphic to the F-rational points of an algebraic group defined over F. This is achieved by considering a group configuration of the group in the algebraic closures of F.
Classes of graphs characterizable by finitely many homomorhism counts
Dienstag, 24.5.22, 14:30-15:30, Raum 404, Ernst-Zermelo-Str. 1
In 1967 Lovász showed that up to isomorphism every finite relational structure A is determined by the homomorphism counts hom(B,A), i.e, by the number of homomorphisms from B to A, where B ranges over all structures (of the same vocabulary as A).\nMoreover, it suffices that B ranges over the structures with at most as many elements as A.\n\nIn the talk, we deal with classes C of graphs characterizable by finitely many homomorphism counts, i.e., classes for which there are finitely many graphs F1,...,Fk such that for every graph G already hom(F1,G),...,hom(Fk,G) determines whether G is in C. Among others, we show which prefix classes of first-order logic have the property that each class of graphs definable by a sentence of this prefix class is characterizable by finitely many homomorphism counts.\n\n
The space of types with a spectral topology
Dienstag, 31.5.22, 14:30-15:30, Raum 404, Ernst-Zermelo-Str. 1
Influenced by results in real algebraic geometry, Pillay pointed out in 1988 that the space of types of an o-minimal expansion of a real closed field admits a spectral topology. With this topology, this space is quasi-compact and T_0, yet not Hausdorff. Nonetheless, the subspace of all closed points turns out to be quasi-compact and Hausdorff. \n\nIn this talk, I will relate the space of closed points with other topological spaces, such as the space of mu-types considered first by Peterzil and Starchenko. In addition, I will explain how to characterize coheir types of an o-minimal expansion of a real closed field within invariant types, using the spectral topology.\n\n