Program

Program:

10am - Coffee

10:30am - Athar Abdul-Quader

Abstract: The lattice problem for models of PA is to determine which lattices can be represented either as lattices of elementary substructures of a model of PA or, more generally, which can be represented as lattices of elementary substructures of a model N that contain a given elementary substructure M of N. In this talk, we will review some of what is known and discuss the main technique for realizing finite lattices as interstructure lattices, due to Schmerl in 1986. This technique involves representations of lattices as lattices of equivalence relations and the combinatorial properties of such representations. We will focus on some specific examples that we hope can motivate this technique and show the relevant combinatorial properties.

11:30am - Vince Guingona

Abstract: We will examine probably approximately correct (PAC) learning and differentially private PAC learning through the lens of model theory. In 1989, A. Blumer, A. Ehrenfeucht, D. Haussler, and M. Warmuth proved that PAC learnability is equivalent to having finite VC dimension (model theoretically, having NIP). In 2022, N. Alon, M. Bun, R. Livni, M. Malliaris, and S. Moran showed that differentially private PAC learnability is equivalent to having finite Littlestone dimension (model theoretically, being stable). In ongoing work, we are imposing additional model theoretic strength, beyond stability, to see if this improves the best-known bounds of the sample complexity of a differentially private PAC learning algorithm. In particular, we are currently examining equational formulas. This is joint work with Alexei Kolesnikov, Miriam Parnes, and Natalie Piltoyan.

12:30pm - Lunch

2:00pm - David Marker

Abstract: In joint work with Charles Steinhorn we construct a non-archimedean real closed field of transcendence degree two with no non-trivial automorphisms. This is the first countable example of a nonstandard rigid real closed field. In this talk I will discuss our original result plus outline an idea we have for extending this result and an interesting algebraic problem that is an obstacle.

3:00pm - Margaret Thomas

Abstract: The application of o-minimality to diophantine geometry is an active area of research that has become well established over the past couple of decades, stemming from the seminal Pila--Wilkie Theorem counting rational points (and, in later variants, algebraic and semi-rational points) on definable sets. One main focus for improving this theorem and its applications has been on seeking an effective version of this theorem (something not given by the original proof), and during the last few years there has been significant progress on this topic. This has led to a uniform, effective version of this theorem (and its variants) in certain settings, as well as effective diophantine applications, for example to an effective, uniform Manin--Mumford statement for products of elliptic curves with complex multiplication (joint work with Binyamini, Jones and Schmidt). But it has also led to new perspectives on how to approach the question of effective (applications of) o-minimality more generally, including the introduction of `sharply' and `effectively o-minimal structures' -- for example, it was by combining and developing various of these ideas that Binyamini, Novikov and Zak obtained improved counting bounds and Binyamini proved Khovanskii's conjecture. I will aim to survey these developments and this perspective, which has the potential to broaden the scope of effective applications of o-minimality, and try to give some ideas about what could be next on this agenda.

4:00pm - Coffee Break

4:30pm - Michał Szachniewicz

Abstract: I will recall the notion of Weil heights on algebraic numbers and present a proof that if heights are treated in discrete first-order logic, in the structure one can define totally real algebraic integers (which are known to be undecidable by Julia Robinson). On the other hand, while treated in unbounded continuous logic, I suspect that the theory of algebraic numbers with Weil heights is decidable. I will give some motivations and state problems of similar kind. This talk is based on observations from a joint paper with Itaï Ben Yaacov, Pablo Destic and Ehud Hrushovski, and on a work in progress with Nuno Hultberg.