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.
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.
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).
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.
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.
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).
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.
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.
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.
Abstract: In this talk, I will start by introducing the concept of o-minimal structures and their significance in the field of mathematics. O-minimal structures are a generalization of semialgebraic and subanalytic structures and have important applications in many areas of mathematics such as real algebraic geometry and Diophantine geometry. I will give a brief overview of the basic properties of o-minimal structures and some examples of them.After that, I will delve deeper into the Pila-Wilkie Theorem and its implications. Specifically, the Pila-Wilkie Theorem bounds the number of rational points of bounded height lying on the transcendental part of a definable set in an o-minimal structure. This theorem is a powerful tool for understanding the distribution of rational points on algebraic varieties and has been used to prove many important results in arithmetic geometry. I will provide an overview of the key ingredients and proof of this theorem.If time permits, I will also discuss some of the applications of the Pila-Wilkie Theorem. For example, it has been used to give a new proof of the Manin-Mumford Conjecture. I will also explain the Pila-Zannier strategy of proving the Manin-Mumford conjecture in a special case of elliptic curves. This strategy is based on the Pila-Wilkie Theorem and has led to many new results in Diophantine geometry. By the end of this talk, I hope to have provided a comprehensive overview of the Pila-Wilkie Theorem and its applications in mathematics, and to have sparked an interest in the study of o-minimal structures and their role in Diophantine geometry.
Abstract: The goal of this talk is to state and sketchily prove the difference version of Newton's lemma. I will start with some preliminaries in difference valued fields. Roughly speaking, a difference valued field is a valued field together with an automorphism. Then, I will go through a lemma and a theorem. Suppose $f$ is a difference polynomial; if $b$ is a point that is not a root of $f$, the lemma guarantees the existence of a better estimation of a possible root around $b$. In the theorem, by using this lemma, we build a PC-sequence whose pseudolimit is a root of $f$. Throughout this lemma and theorem, I will end up with the difference Newton lemma.
Abstract: I will give an introduction to the topic of my PhD thesis. Raf Cluckers and Immanuel Halupczok showed that in the case of $p$-adic integration an analogue of the Kontsevich and Zagier period conjecture is true: Equalities between semi-algebraic sets come from a set of very easy transformation rules. While this does not provide an approach to the real case, it naturally leads to the question whether something similar holds in other non-archimedean integration theories. I will give a general (and vague) introduction to motivic integration in the style of Cluckers-Loeser over $K((t))$ for a field $K$ of characeristic 0 using the $p$-adics as a guiding principle to discuss similarities and differences between the $p$-adic case and $K((t))$ and provide a model-theoretic view on motivic integration.
Abstract of the 2nd talk: In this (second) talk we will revisit Hironaka-style resolution of singularities. After a short overview on strategy and key steps, we shall focus on questions arising in the process of passing from a constructive proof through an algorithmic proof to an implementable algorithm.