Tag - Groups and logic

Laura Ciobanu: Group equations, constraints and decidability

8th March, 2024

In this talk I will discuss group equations with non-rational constraints, a topic inspired by the long line of work on word equations with length constraints. Deciding algorithmically whether a word equation has solutions satisfying linear length constraints is a major open question, with deep theoretical and practical implications. I will introduce equations in groups and several kinds of constraints, and show that equations with length, abelian or context-free constraints are decidable in virtually abelian groups (joint with Alex Evetts and Alex Levine). This contrasts the fact that solving equations with abelian constraints is undecidable for non-abelian right-angled Artin groups and hyperbolic groups with ‘large’ abelianisation (joint work with Albert Garreta).

Andrey Nikolaev: Non-standard polynomials and non-standard free group

20th October, 2023

Interpretation and bi-interpretation offer a novel approach to studying all structures elementarily equivalent to a given one. We use this approach to describe and study non-standard models of the ring of polynomials, Laurent polynomials, and the group ring of a free group.

In the presence of interpretation but not bi-interpretation, this approach produces a family of structures elementarily equivalent to a given one. We exploit this to introduce non-standard models of a free group. As time permits, we discuss their main properties.

Slawomir Solecki: The dynamics and structure of transformation groups

17th January, 2023

This is a 24-lecture course, with each lecture being 75 minutes, given by Slawomir Solecki. Note that the 2nd lecture was not recorded. The other lectures might still be of significant interest, but this needs to be known.

This course focuses on the interaction between set theory, geometry, group theory, and dynamics. It will present parts of Rosendal’s Coarse Geometry of Topological Groups, Kechris-Pestov-Todorcevic’s Fraïssé Limits, Ramsey Theory, and Topological Dynamics of Automorphism Groups, as well as theory of Borel and measurable combinatorics.

Spencer Unger and Assaf Rinot: Set theory, algebra and analysis

9th January, 2023

This is a 23-lecture course, with each lecture being 75 minutes, given by Spencer Unger and Assaf Rinot.

This course will present a rigorous study of advanced set-theoretic methods including forcing, large cardinals, and methods of infinite combinatorics and Ramsey theory. An emphasis will be placed on their applications in algebra, topology, and real and functional analysis.

Alexei Miasnikov: Rich groups and weak second-order logic

21st April, 2022

"What can one describe by first-order formulas in a given group A?" - is an old and interesting question. Of course, this depends on the group A. For example, in a free group only cyclic subgroups (and the group itself) are definable in the first-order logic, but in a free monoid of finite rank any finitely generated submonoid is definable. A group A is called rich if the first-order logic in A is equivalent to the weak second-order logic. Surprisingly, there are a lot of interesting groups, rings, semigroups, etc., which are rich. I will describe various algebraic, geometric, and algorithmic properties that are first-order definable in rich groups and apply these to some open problems. Weak second-order logic can be introduced into algebraic structures in different ways: via HF-logic, or list superstructures over A, or computably enumerable infinite disjunctions and conjunctions, or via finite binary predicates, etc. I will describe a particular form of this logic which is especially convenient to use in algebra and show how to effectively translate such weak second order formulas into the equivalent first-order ones in the case of a rich group A.

James Parkinson: Automata for Coxeter groups

9th August, 2021

In 1993 Brink and Howlett proved that finitely generated Coxeter groups are automatic. In particular, they constructed a finite state automaton recognizing the language of reduced words in the Coxeter group. This automaton is constructed in terms of the remarkable set of "elementary roots" in the associated root system.

In this talk we outline the construction of Brink and Howlett. We also describe the minimal automaton recognizing the language of reduced words, and prove necessary and sufficient conditions for the Brink-Howlett automaton to coincide with this minimal automaton. This resolves a conjecture of Hohlweg, Nadeau, and Williams, and is joint work with Yeeka Yau.

Alex Lubotzky: First-order rigidity of high-rank arithmetic groups

16th July, 2021

The family of high-rank arithmetic groups is a class of groups playing an important role in various areas of mathematics. It includes SLn(ℤ), for n ≥ 3, SLn(ℤ[1/p]) for n ≥ 2, their finite-index subgroups and many more. A number of remarkable results about them have been proven including: Weil local rigidity, Mostow strong rigidity, Margulis superrigidity and the Schwartz-Eskin-Farb Quasi-isometric rigidity. We will add a new type of rigidity: 'first order rigidity'. Namely if D is such a non-uniform characteristic zero arithmetic group and L a finitely generated group which is elementary equivalent to D then L is isomorphic to D. This stands in contrast with Zlil Sela's remarkable work which implies that the free groups, surface groups and hyperbolic groups (many of which are low-rank arithmetic groups) have many non-isomorphic finitely generated groups which are elementarily equivalent to them.