World Logic Day Workshop 2025

Title: Model theoretic dynamics: the Newelski conjecture and beyond. 

Abstract: Around 15 years ago, Newelski first observed a fundamental link between the model theory and dynamics. In particular, there are deep connections between certain model theoretic invariants of definable groups (e.g. connected components) and dynamical invariants of groups acting on their spaces of types (e.g. Ellis groups). Since that time, this area has expanded and grown to touch serval other areas of model theory, including Keisler measures and convolution dynamics. The purpose of this talk is to give both a historical overview of this field as well as some recent advances in the subject.

Date and time: 17:00-17:50(Astana time), 14.01.2025

Title: Approaches to the countable spectrum of theories with a linear order

Abstract: One of the most intriguing open problems in model theory is Vaught's conjecture regarding the existence of a first-order theory with uncountably many, but fewer than the continuum, countable non-isomorphic models. In this talk, we explore various approaches to this question. We show that for each countable theory, there exists a countable linearly ordered theory with the same number of countable models. We discuss concepts such as convex closures of formulas and types, orthogonality of types, convex-to-the-right and convex-to-the-left formulas, as well as n-ary types and theories, and their relationship to Vaught's conjecture.

Date and time: 18:00-18:50(Astana time), 14.01.2025

Title: A survey of cohesive powers

Abstract: We survey recent results concerning cohesive products and powers, which are effective analogs of classical ultraproducts and ultrapowers where the structures are computable and cohesive sets play the role of ultrafilters.  We discuss general properties of cohesive products and powers, such as the extent to which the Łoś theorem, saturation theorems, and the Keisler--Shelah theorem hold in this context.  We then investigate the cohesive powers of computable linear orders and give several examples of classically-but-not-computably isomorphic structures whose cohesive powers are not elementarily equivalent.  These examples witness limitations to the Łoś theorem and the Keisler--Shelah theorem for cohesive powers.  Finally, we discuss the complexity of presenting cohesive products and powers.  Every cohesive product over a ∆_2 cohesive set has a ∆_3 presentation, and there are examples of computable structures where this complexity is tight.  There is also a computable sequence of linear orders whose cohesive products do not have computable presentations.

This survey is based on joint work with R. Dimitrov, V. Harizanov, A. Morozov, A. Soskova, and S. Vatev; and on joint work with D. Gonzalez.

Date and time: 19:00-19:50(Astana time), 14.01.2025