The global Zarankiewicz's problem for hypergraphs asks for an upper bound on the number of edges of a hypergraph, whose edge relation is induced by a fixed hypergraph $E$ that has no sub-hypergraphs of a given size. Basit-Chernikov-Starchenko-Tao-Tran (2021) obtained linear Zarankiewicz bounds in the case of a semilinear $E$, namely $E$ definable in a linear o-minimal structure. We extend this theorem to a broader range of "linear-like" structures, in o-minimal, Presburger arithmetic and stability theoretic settings. Some of the methods involved include (a) a reduction of the problem to the case of arbitrary subgroups $E$ of powers of groups, and (b) an abstract version of Zarankiewicz's problem in the saturated setting.
Joint work with Aris Papadopoulos.
In the context of a 3-dimensional real analytic vector field at a singular point, Cano, Moussu and Sanz introduced and studied the notion of integral pencils of trajectories at that point in order to obtain informations on the possible dynamical behaviours. We extend this approach on the formal side, taking advantage of the computability of transseries (in particular they are grid-based in the sense of Ecalle - van der Hoeven) when solving differential equations.
More precisely, for a real formal planar vector field at 0, we introduce a notion of transserial trajectories and provide an explicit description of all the possible transserial pencils. This is meant to be a first step toward the same sort of description in dimension 3.
As a motivation, we expect these transserial trajectories to reflect tameness properties of actual solutions, in a way similar to that of differentially algebraic transseries for germs in some Hardy fields: cf the recent results of Aschenbrenner-van den Dries-van der Hoeven.
Joint work in progress with Daniel Panazzolo and Fernando Sanz.
Abstract: In this talk I will discuss homotopy groups definable in the o-minimal setting. After giving a brief overview of the classical theory of homotopy groups, I will talk about previous results which have been proven in the o-minimal field case, before discussing my own work in the o-minimal linear case.
Thatcher and Wright showed that a property of trees of bounded degree is MSO-definable if and only if it is recognizable by a tree-automaton. In this talk we explore the question of when MSO-definability of a property of graphs is equivalent to the existence of a tree automata which, given a suitable expression encoding the input graph, recognizes the property. In this talk, I will survey the state of the art of the "definability equals recognizability" problem. For proving "definability equals recognizability" results the key step is to transduce a suitable tree-like decomposition of the input graph. I will present a new MSO-transduction which forms the core for transducing a particular type of graph decompositions.
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.
