TBA

Zarankiewicz's problem for hypergraphs asks for upper bounds on the number of edges of a hypergraph that has no complete sub-hypergraphs of a given size. Let M be an o-minimal structure. Basit-Chernikov-Starchenko-Tao-Tran (2021) proved that the following are equivalent:

(1) "linear Zarankiewicz's bounds" hold for hypergraphs whose edge relation is induced by a fixed relation definable in M

(2) M does not define an infinite field.

We prove that the following are equivalent:

(1') linear Zarankiewicz bounds hold for sufficiently "distant" hypergraphs whose edge relation is induced by a fixed relation definable in M

(2') M does not define a full field (that is, one whose domain is the whole universe of M).

This is joint work (in progress) with Aris Papadopoulos.

The design and analysis of finite element methods for high Reynolds flow remains a challenging task, not least because of the difficulties associated with turbulence. In this talk we will first revisit some theoretical results on interior penalty methods using equal order interpolation for smooth solutions of the Navier-Stokes’ equations at high Reynolds number and show some recent computational results for turbulent flows.

Then we will focus on so called pressure robust methods, i.e. methods where the smoothness of the pressure does not affect the upper bound of error estimates for the velocity of the Stokes’ system. We will discuss how convection can be stabilized for such methods in the high Reynolds regime and, for the lowest order case, show an interesting connection to turbulence modelling.

We call affine logic the fragment of continuous logic in which the connectives are limited to linear combinations and the constants (but quantification is allowed, in the usual continuous form). This fragment has been introduced and studied by S.M. Bagheri, the first to observe that this is the appropriate framework to consider convex combinations of metric structures and, more generally, ultrameans, i.e., ultraproducts in which the ultrafilter is replaced by a finitely additive probability measure. Bagheri has shown that many fundamental results of continuous logic hold in affine logic in an appropriate form, including Łoś's theorem, the compactness theorem, and the Keisler--Shelah isomorphism theorem.

In affine logic, type spaces are compact convex sets. In this talk I will report on an ongoing work with I. Ben Yaacov and T. Tsankov, in which we initiate the study of extremal models in affine logic, i.e., those that only realize extreme types.

From a model theoretic point of view, local fields of positive characteristic, i.e. fields of Laurent series over finite fields, are much less well understood than their characteristic zero counterparts - the fields of real, complex and p-adic numbers. I will discuss different approaches to axiomatize and decide at least their existential theory in various languages and under various forms of resolution of singularities. This includes new joint work with Sylvy Anscombe and Philip Dittmann.

A theorem of Chatzidakis, van den Dries and Macintyre, stemming ultimately from the Lang-Weil estimates, asserts, roughly, that if $\phi(x,y)$ is a formula in the language of rings (where $x,y$ are tuples) then the size of the solution set of $\phi(x,a)$ in any finite field $F_q $(where $a$ is a parameter tuple from $F_q$) takes one of finitely many dimension-measure pairs as $F_q$ and $a$ vary: for a finite set $E$ of pairs $(\mu,d)$ ($\mu$ rational, $d$ integer) dependent on $\phi$, any set $\phi(F_q,a)$ has size roughly $\mu q^d$ for some $(\mu,d) \in E$.

This led in work of Elwes, Steinhorn and myself to the notion of 'asymptotic class’ of finite structures (a class satisfying essentially the conclusion of Chatzidakis-van den Dries-Macintyre). As an example, by a theorem of Ryten, any family of finite simple groups of fixed Lie type forms an asymptotic class. There is a corresponding notion for infinite structures of  'measurable structure’ (e.g. a pseudofinite field, by the Chatzidakis-van den Dries-Macintyre theorem, or certain pseudofinite difference fields).

I will discuss a body of work with Sylvy Anscombe, Charles Steinhorn and Daniel Wolf which generalises this, incorporating a richer range of examples with fewer model-theoretic constraints; for example, the corresponding infinite 'generalised measurable’ structures, for which the definable sets are assigned values in some ordered semiring, need no longer have simple theory. I will also discuss a variant in which sizes of definable sets in finite structures are given exactly rather than asymptotically.

The Skolem Problem asks to decide whether a linearly recurrent sequence (LRS) over the rationals has a zero term.  It is sometimes considered as the halting problem for linear loops.   In this talk we will give an overview of two current approaches to establishing decidability of this problem.  First, we observe that the Skolem Problem for LRS with simple characteristic roots is decidable subject to the $p$-adic Schanuel conjecture and the exponential-local-global principle.  Next, we define a set $S$ of positive integers such that (i) $S$ has positive lower density and (ii) The Skolem Problem is decidable relative to $S$, i.e., one can effectively determine the set of all zeros of a given LRS that lie in $S$.

The talk is based on joint work with Y. Bilu, F. Luca, J. Ouaknine, D. Pursar, and J. Nieuwveld.  

We study the definability of valuation rings in ordered fields (in the language of ordered rings). We show that any henselian valuation ring that is definable in the language of ordered rings is already definable in the language of rings. However, this does not hold when we drop the assumption of henselianity.

This is joint work with Philip Dittmann, Sebastian Krapp and Salma Kuhlmann.

We consider the problem of finding the best memoryless stochastic policy for an infinite-horizon partially observable Markov decision process (POMDP) with finite state and action spaces with respect to either the discounted or mean reward criterion. We show that the (discounted) state-action frequencies and the expected cumulative reward are rational functions of the policy, whereby the degree is determined by the degree of partial observability. We then describe the optimization problem as a linear optimization problem in the space of feasible state-action frequencies subject to polynomial constraints that we characterize explicitly. This allows us to address the combinatorial and geometric complexity of the optimization problem using tools from polynomial optimization. In particular, we estimate the number of critical points and use the polynomial programming description of reward maximization to solve a navigation problem in a grid world. The talk is based on recent work with Johannes Müller.