Tag - Logic
Recently, Nies, Segal and Tent started an investigation of finite axiomatizability in the realm of profinite groups. Among the classes of profinite groups under their consideration is the class of p-adic analytic pro-p groups. In joint work with Benjamin Klopsch, we consider two key invariants of these groups, namely rank and dimension, and show that they can be characterized by a single first-order sentence. Before discussing these results I will introduce the relevant background. If time permits, I will also present some natural generalisations.
Erdős-style geometry is concerned with combinatorial questions about simple geometric objects, such as counting incidences between finite sets of points, lines, etc. These questions can be typically viewed as asking for the possible number of intersections of a given (semi-)algebraic variety with large finite grids of points. An influential theorem of Elekes and Szabó indicates that such intersections have maximal size only for varieties that are closely connected to algebraic groups. Techniques from model theory - Hrushovski's group configuration and its variants - are very useful in recognizing these groups, and allow to obtain higher arity and dimension generalizations of the Elekes-Szabó theorem. In fact, all of this is not just about polynomials and works in the larger setting of definable sets in o-minimal structures.
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.
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.
We discuss some recent progress on the model-theoretic problem of classifying the reducts of the complex field (with named parameters and up to interdefinability). The tools we use include Castle’s recent solution of the Restricted Trichotomy Conjecture in characteristic 0 and a generalized sumproduct result from additive combinatorics.
I will discuss some aspects of the first-order theory of homeomorphism groups of connected manifolds. The main result is as follows. Let M be a compact, connected manifold. There is a sentence S(M) in the language of groups such that if N is an arbitrary manifold and the homeomorphism group of N models S(M) then N is homeomorphic to M. This resolves a conjecture of Rubin from the 1980s. I will illustrate some of the ingredients of the proof, including an interpretation of second order arithmetic in the theory of homeomorphism groups of manifolds.
"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.
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