Mathematik Studium Mathematik Kursmaterialien WiSe 2022/23 Oberseminar Modelltheorie

# Oberseminar Modelltheorie - Wintersemester 2022/23

Responsible: Prof. Immanuel Halupczok

Time: Tue, 14:30-16:00, Room 25.22.03.73

Prerequisites: Model theory

If you are interested (either in giving a talk or just in listening to talks), you can send me an e-mail so that I put you onto the mailing list.

### Planning

This term, we will have mixed topics, each speaker speaking either about (some aspect of) their research... or any other topic around model theory they deem interesting.

• 18.10.: David: Trees of Trees and their automorphism groups

Abstract: Some permutation groups are best represented/constructed as kinds-of-products or of limits of permutation groups. We discuss a kind of structure called a "limit of betweenness relations" that can be viewed as a kind of tree of trees. Making use of a generalisation of a construction called a Fraissé-limit (a tool from model theory that we will sketch in the talk) how we have constructed a (new) family of such structures. Further we plan to say how their automorphism groups fit into the landscape of infinite primitive Jordan permutation groups, and the structure theory of Jordan groups established by S. Adeleke, D. Macpherson, and P. M. Neumann. Joint work with John Truss (Leeds).

• (25.10., 1.11.: no seminar)
• 8.11.: Immi: Stratifications using non-standard models

Abstract: An important tool in singularity theory is the notion of stratifications. In this talk, I will explain what this is about and present a particular kind of stratifications which uses is defined using a non-standard model of the field one is interested in. This is joint work with David.

• 15.11.: Peter: Combining propositional logics via homotopy colimits

Abstract: Mathematicians mostly only work with one propositional logic: classical logic. In philosophy, computer science and linguistics, however, a multitude of non-classical propositional logics are in use. One can relate these logics to each other by translations, i.e. consequence preserving maps between their formal languages. Often one wishes to combine several logics into a bigger one, and this has been recognized as the formation of a colimit in the category of logics and translations. However, it only works for very simple translations. In this talk I will use Łoś and Tarski's definitions of "logic" and "translation", and show that the category of logics and translations carries the structure of a so-called cofibration category. This means that it is a setting in which one can mimic the classical homotopy theory of topological spaces. In particular one can form homotopy colimits (a notion I will explain) and with this has much ampler possibilities of combining and manipulating logics.

• 22.11.: Thor: Operations on Milnor-Witt K-theory

Abstract: The talk will consist of three parts:
(i) Trying to convince you that Milnor-Witt K-theory is an interesting object that arises naturally
(ii) What is Milnor-Witt K-theory? Basic definitions and properties.
(iii) Explaining the title - towards a full description of all operations on Milnor-Witt K-theory
Note for potentially interested Wuppertal people: Thor w ill give a similar talk in the Oberseminar in Wuppertal (11.01.23).

• 29.11.: Ana: Sheaves on Heyting Algebras and H-sets

Abstract: Intuitionistic propositional logic corresponds to complete Heyting algebras H because we can translate theorems of classical propositional calculus as equations of complete Heyting algebras and vice-versa. First-order logic is an extension of propositional logic. In this talk, we introduce Heyting-valued sets or H-sets, which are models for intuitionistic first-order logic, and sheaves on H. Recent researches try to replace H by a noncommutative generalization, called quantale, and establish the suitable notions of quantale-valued sets and sheaves on quantales. Furthermore, since the categories of H-sets and of sheaves on H are equivalent, an equivalence in the quantalic setting may favour the use of sheaves to model other non-classical logics.

• (6.12.: no seminar)
• 13.12.: Zeynep: Uniform Rationality for Compact $p$-adic Analytic Groups

Abstract: For a group $G$, it is natural to ask how many irreducible complex representations of a given dimension it has. $G$ is called \textit{representation rigid}, if, for each $n$, the number $r_n(G)$ of isomorphism classes of complex irreducible $n$-dimensional representations of $G$ is finite. The asymptotic behaviour of the sequence $\lbrace r_n(G) \rbrace_n$ indicates the representation growth of $G$; we encode $r_n(G)$ in a Dirichlet generating series called the \textit{representation zeta function} of $G$. In this talk, we will discuss the rationality of representation zeta functions of FAb (a property ensuring rigidity) compact $p$-adic analytic groups (e.g., $\operatorname{SL}_n(\mathbb{Z}_{p})$) and how the model theory of valued fields plays a role in studying them with a focus on $p$-dependence.

• 20.12.: Florian: tba

Abstract: tba

• (10.1.: no seminar)
• 17.1.: Alex: Mechanical Mathematicians

Abstract: Since the 1960s, some researchers have dreamed of computer programs that would find proofs of mathematical conjectures that no human was able to prove before. Despite some successes, today’s automatic theorem provers are still far from achieving this vision. Nonetheless, I would like to talk about some of the methods that come closest to it today. In my talk, I will give an introduction to the automated theorem proving methods resolution and superposition. These calculi operate on first-order logic and can be shown to be refutationally complete using model-theoretic arguments. I will present my PhD project, in which I extended these calculi to operate on higher-order logic, which allows us to express mathematics more elegantly and to offer more powerful tools for proof assistants.

• 24.1.: Hamed: tba

Abstract: tba

• 31.1.: Saba: tba

Abstract: tba