Logic group

Partial combinatory algebras (pcas) are abstract models of computation. They are one of the earliest type of model that emerged in the 1930's when mathematicians were trying to define what it means to be computable. Notions such as recursive functions (used by Gödel in the proof of his incompleteness theorems), Turing machines, combinatory algebra and lambda calculus each turned out to be useful in the development of the theory of computation in their own way. Pcas are also used in the study of constructive mathematics. They play a key role in the connections between constructive mathematics, proof theory, and computability, and also serve as a basis for models of constructive mathematics.

In this talk we will give a quick review of the basics of combinatory algebra, and discuss some of the key examples, such as Kleene's models $\mathcal{K}_1$ (which describes the classical setting for computation on the natural numbers), $\mathcal{K}_2$ (which does the same for the real numbers), and Scott's graph model. We then discuss recent results about embeddings and completions of pcas, and in particular the complexity of various embeddings. Depending on time, we will also discus the complexity of the isomorphism problem, extensionality, and ordinal analysis of pcas.

Everyone loves a good decomposition. How can we break down a mathematical object — a graph, a group, or a function — efficiently into well-behaved (or regular) parts? And what conditions can we place on these objects to guarantee a higher degree of regularity, such as homogeneity? It turns out an excellent source of such conditions is model-theoretic dividing lines, that is, tameness properties of (first-order) structures. This is not a coincidence. In this talk, I will dive into the deep theory of these dividing lines in search of the source of homogeneity.

A graph has a countable chromatic number if the vertices can be labeled by natural numbers so that no edge has the same label on both ends. Verifying this property for a given graph can be somewhat difficult; consider e.g. the graph of points in the Euclidean space $ℝ^n$ connected with an edge iff their distance is a rational number. The countable chromatic number of this graph is an easy observation for $n=1$, but a nontrivial result for $n>1$ (Komjath, Schmerl). I will introduce an infinite determined game which provides a simple criterion for a definable (analytic on a Polish space) graph to be countably chromatic; it is sufficient to verify that the second player has a winning strategy. Using this criterion we can e.g. easily resolve the example of $ℝ^n$, as well as deduce interesting consequences on the structure of uncountably chromatic definable graphs.

The talk is based on the paper Chodounský, Zapletal: Two graph games (2024).

Given a topology on the natural numbers, how complicated is it to describe? To answer this question with tools from computability theory, we will restrict to the context of countable second-countable (CSC) topological spaces. One approach is to assign an index to each computable CSC space and determine the arithmetic complexity of the set of CSC spaces with some property. Another approach comes from computable structure theory; for example, given two computable copies of a CSC space, does there exist a computable homeomorphism between them? In this talk, we will explore these approaches and apply them in three running examples: the indiscrete, discrete, and initial segment topologies.

NOTES: the speaker will be online.
I will give a talk on several recent results regarding ultrahomogeneous graphs. A relational structure R is ultrahomogeneous if every isomorphism of finite induced substructures of R extends to an automorphism of R. We classify the finite vertex-colored oriented ultrahomogeneous graphs and the finite vertex-colored undirected graphs. The classifications comprise new general methods which govern how graphs can be combined or extended to create new ultrahomogeneous graphs. Further, we extend a classic theorem of Cameron (every 5-tuple regular graph is already ultrahomogeneous) to the setting of colored graphs.

This is joint work with Sofia Brenner, Eda Kaja, Thomas Schneider, and Pascal Schweitzer.