# Algebra Seminar talk

2024-04-19

Robert **Sullivan***Generic embeddings into Fraïssé structures*

Abstract:

This project, in the writing-up stage, is work with A. Codenotti (Münster), A. Panagiotopoulos (Vienna) and J.Winkel.

Let M be a Fraïssé structure (eg the random graph), and let A be a countably infinite structure which is embeddable in M. If M has free amalgamation, then there exists a Katetov embedding of A into M: an embedding such that each automorphism of A extends to an automorphism of M. Is this embedding "common" or "uncommon"?

To answer this, we investigate generic embeddings of A into M. An embedding of A into M is said to be generic if it lies in a comeagre set inside the Polish space Emb(A, M).

We will answer the following three questions:

- When are two embeddings of A into M generically isomorphic via an automorphism of M?

- When is A generically corigid (i.e. Aut(M/A) trivial)?

- Let g lie in Aut(A). When is g generically extensible to an automorphism of M?

We will also discuss a wide range of examples in the context of these three questions.