Abstract: In this talk, we introduce oriented abelian groups and present some tameness properties of these structures and their pairs. We show that, in certain theories of oriented abelian groups, the VC-density of formulas is bounded by the size of parameter variable. We further show that, for a specific pair structure, this bound becomes twice the size of parameter variable, and that these bounds are optimal. This is joint work with Ebru Nayir.
The automorphism group $\mathrm{Aut}(A)$ and the monoid of elementary embeddings $\mathrm{EEmb}(A)$ of a first-order structure $A$ are both endowed with a natural topology of pointwise convergence. When $A$ is $\omega$-categorical, these spaces of symmetries (together with their topologies) can be used to reconstruct the original structure up to bi-interpretability. This raises the question of when, given $\mathrm{Aut}(A)$ as a pure group, or $\mathrm{EEmb}(A)$ as a pure monoid, one can reconstruct its topology of pointwise convergence. Whilst the automorphism group version of this problem has been intensively studied over the last 40 years, its analogue for monoids has only recently received attention. In this talk, I will discuss various topological reconstruction problems for monoids of elementary embeddings of $\omega$-categorical structures. We prove that for a countable saturated structure $A$, if $\mathrm{Aut}(A) $ has automatic homeomorphicity with respect to closed subgroups of $S_\omega$ then $\mathrm{EEmb}(A)$ has automatic homeomorphicity with respect to closed submonoids of $\mathbb{N}^{\mathbb{N}}$. This result builds on previous work of Pech and Pech (2018), Behrisch, Truss, and Vargas-García (2017), and Bodirsky, Pinsker, and Pongrácz (2017), who proved special cases of it. We will also discuss when the topology of pointwise convergence ends up being minimal amongst Hausdorff semigroup topologies on $\mathrm{EEmb}(A)$. Interestingly, this seems to happen more easily than for $\mathrm{Aut}(A)$. This talk is based on an upcoming survey paper with Michael Pinsker, and on ongoing work with Javi de la Nuez Gonzalez, Zaniar Ghadernezhad, and Michael Pinsker.
Abstract: After a brief historical survey I will describe how some axioms of the theory of Hardy fields can be generalized so as to cover the case of differential fields of germs at a non-principal cut in an o-minimal ordered field. I will sketch how this can be used to prove that Tressl's signature alternative holds in a large class of exponential o-minimal theories.
There are many “paradoxical sets” of reals that can be obtained using a well-ordering of the reals or using a non-principal ultrafilter on ℕ, both consequences of the Axiom of Choice. In ZF, can we recover the well-ordering of the reals or the ultrafilter on ℕ from the existence of a given paradoxical set? Under certain amalgamation conditions, we give some negative answers to this question.
NOTES: unusual room, 2 hours seminar.
Abstract: F-V Kuhlmann's theory of tame (and separably tame) valued fields is one of the most general settings in which we have AKE principles. Such principles come in many flavours; in particular, we may constrain our attention to certain "subfragments" of the languages of valued fields/rings/ordered abelian groups. I will explain some of the underlying algebra, and show some recent work on such principles in expansions by sections of the residue map. This will touch on (separate) projects with Boissonneau and Fehm.
NOTES: unusual room and time.
Given an action of a group $G$ on a relational Fraïssé structure $M$, we call this action sharply $k$-homogeneous if, for each isomorphism $f : A \to B$ of substructures of $M$ of size $k$, there is exactly one element of $G$ whose action extends $f$. This generalises the well-known notion of a sharply $k$-transitive action on a set, and was previously investigated by Cameron, Macpherson and Cherlin. I will discuss recent results with J. de la Nuez González which show that a wide variety of Fraïssé structures admit sharply $k$-homogeneous actions for $k \leq 3$ by finitely generated virtually free groups. Our results also specialise to the case of sets, giving the first examples of finitely presented non-split infinite groups with sharply 2-transitive/sharply 3-transitive actions.
Abstract: Arising from extremal combinatorics, the global Zarankiewicz problem seeks an upper bound on the number of edges of a finite $r$-hypergraph where the edge relation is induced by some fixed hypergraph that does not contain the compete $r$-hypergraph $K_{k,\dots,k}$ as a subgraph for some $k$. Special cases of this problem, where the edge relation comes from a definable set in a particular structure, have been of interest to model theorists. This talk will give an introduction to the problem as well as some recent results.
The game of cops and robbers is played on a fixed graph, with the cop choosing a vertex to start at, then the robber chooses his, and then they take turns in moving to adjacent vertices. The game ends if the cop captures the robber (lands on its vertex). What graphs allow the cop to have a winning strategy, and how long does the game typically last, assuming optimal play? For finite graphs, the situation is very well understood — the cop-win graphs are precisely constructible graphs (constructed from a single vertex by repeatedly adding dominated vertices), and the capture time can be any finite ordinal (attained for example by finite paths).
In the infinite case, not much is known. In particular, there is no structural characterisation of cop-win graphs. What about the capture time? Is there an ordinal such that for any cop-win graph the sequence of moves of an optimal game is never that ordinal?
In this talk we will explore this question by showing that the answer is surprisingly 'no'.
NOTES: Unusual location.
Abstract: A pregeometric theory of fields is a theory T expanding ACF such that the model-theoretic algebraic closure acl satisfies the exchange property. Several important theories of fields satisfy this property, like ACF, RCF, ACVF, etc. Certain pregeometric fields have the additional property that we can expand the language with a generic derivation. We will discuss the structural implications of pregeometricness, and what we mean by a 'generic derivation'. Joint work with Antongiulio Fornasiero and Elliot Kaplan.
