Oberseminar Logik - SoSe 2024

Organizers

• Prof. Dr. Philipp Hieronymi
• Dr. Tingxiang Zou

Time and location

Unless stated otherwise: Mondays 17.00-18.00 in SemR N0.003, Endenicher Allee 60.

The participants of the seminar are welcome for coffee and tea in room 4.005 (office Hieronymi) at 16.30 before the talks.

Subscribe to the mailing lists for the Oberseminar and other logic activities in Bonn: https://listen.uni-bonn.de/wws/subscribe/logic.

Talks

• April 8th: No seminar
  
• April 15th: No seminar
  
• April 22nd: Martin Hils (Münster)
  

  Title: Model theory of valued fields with non-standard Frobenius automorphism
  
  Abstract: In the talk, we will give an overview of the model theory of the limit - where p tends to infinity - of the Frobenius automorphism acting on an algebraically closed non-trivially valued field of characteristic p,
  and similarly in the Witt-Frobenius case. We will in particular discuss
  i) the classification of imaginaries by the geometric sorts (from a recent joint work with Rideau-Kikuchi);
  ii) model-theoretic tameness results (obtained in joint work with Chernikov), and
  iii) an application to pseudofinite difference fields concerning estimates on the number of rational points (obtained in joint work with Hrushovski, Zou and Ye).

• April 29th: Simone Ramello (Münster)
  

  Title: Model theory of endomorphisms of valued fields
  
  Abstract: We tackle the problem of understanding theories of valued fields endowed with a possibly non-surjective endomorphism,
  the most prominent example being the limit of the action of Frobenius on non-trivially valued separably closed valued fields of positive characteristic.
  We establish an embedding theorem for such valued difference fields, building on previous work of Pal (in the inversive case) and Dor-Halevi (in the contracting case),
  and deduce Ax-Kochen/Ershov-style principles.

• May 6th: Konstantinos Kartas (IMJ-PRG/Sorbonne)
  

  Title: On Ci fields
  
  Abstract: Given a natural number i, a field k is called Ci if every homogeneous polynomial over k of degree d in more than d^i variables has a non-trivial zero.
  Emil Artin had famously conjectured that Qp is C2. While this was refuted by Terjanian, an appropriate asymptotic version for p where p tends to infinity was proved by Ax-Kochen.
  In a somewhat orthogonal direction, we fix p but instead let the ramification go to infinity. We show that any maximal totally ramified extension of Qp is C1.
  The two main ingredients are Esnault’s result on degenerations of rationally connected varieties over finite fields and F.-V. Kuhlmann’s theory of tame fields.

• May 13th: Nigel Pynn-Coates (Vienna)
  

  Title: Tame pairs of transseries fields
  
  Abstract: The elementary theory of the differential field of (logarithmic-exponential) transseries has been completely axiomatized by Aschenbrenner--van den Dries--van der Hoeven, and this theory is model complete. This talk concerns pairs of models of this theory such that one is a tame substructure of the other. Tameness here is meant in the sense of a tame pair of real closed fields, which goes back to van den Dries--Lewenberg and, ultimately, Macintyre and Cherlin--Dickmann. I will describe work in progress on the model theory of such transserial tame pairs, including a model completeness result for them. An example comes from the differential field of hyperseries, constructed by Bagayoko--Kaplan--van der Hoeven and shown to be an elementary extension of the differential field of transseries by Bagayoko, equipped with an exponentially bounded subfield.

• May 20th: No seminar
  
• May 27th: Matt Foreman (Irvine/MPIM)
  

  Title: Filter games and precipitous ideals
  
  Abstract: Large cardinals posit elementary embeddings j of the universe V into transitive classes M. The first ordinal moved by j must be a measurable cardinal, hence much above cardinals such as |R|. Precipitous ideals are a method of getting such elementary embeddings with critical points as small as omega_1, by loosening the requirement that the elementary embedding be definable over V to the requirement that the elementary embedding be defined in a generic extension of V. Building on work of Holy and Schlicht, Welch and Neilsen proposed a method for proving the existence of a precipitous ideal using games where plays are increasing sequences of filters. This talk is about joint work with Magidor and Zeman that shows that player II winning the games of various lengths is equivalent to the existence of precipitous ideals whose quotients have different structural properties. Very recent work with Eshkol and Magidor use ineffability to extend these techniques to get precipitous ideals that concentrate on given sets, such as the set of cardinals of high Mitchell order.

• June 3rd: No seminar
  
• June 10th: Adrian De Lon (Bonn)
  

  Title: Formalizing set-theoretic mathematics in the Naproche-ZF theorem prover
  
  Abstract: Naproche-ZF is a new experimental natural-language-oriented theorem prover based on set theory and classical logic; formalizations in Naproche-ZF are written in a controlled natural language embedded into LaTeX and proof gaps are filled with automated theorem provers. Naproche-ZF aims to approximate informal mathematics in a practical manner. I will talk about the motivations behind Naproche-ZF, its syntax and semantics, its design and implementation, and first impressions from formalization experiments.

• June 17th: Pablo Suárez-Serrato (UNAM/MPIM)
  

  Title: Turing Complete Flows: Recent Perspectives on Dynamical Systems
  
  Abstract: We will explore the concept of Turing completeness in the context of continuous-time flows, revealing a deep connection between the dynamics of nonlinear systems and the fundamental limits of computation. Moore first explained how to construct a universal flow that simulates the behavior of any Turing machine. Flows can exhibit Turing-complete behavior, capable of solving any computable problem. Recently, Tao proposed using Turing complete flows to approach the potential existence of finite time singularities in the Navier-Stokes equations. In related work, Cardona--Miranda--Peralta-Salas--Presas showed the existence of Turing complete flows that satisfy the Euler equations of fluid motion on any compact 3-manifold. Building on these previous ideas, I will explain my construction of volume-preserving, Turing complete flows on smooth 4-manifolds. Contemplating the implications of these findings for our comprehension of complex systems is not only fascinating but also inspiring. It underscores the potential for flows to execute arbitrary computations and tackle intricate problems, opening up new avenues for research and innovation.

• June 24th: Margarete Ketelsen (Münster)
  

  Title: Model-theoretic tilting
  
  Abstract: The tilting construction (as introduced by Fontaine) provides a way to transfer theorems between the worlds of characteristic zero and positive characteristic. Classically, this was done for perfectoid fields: for each perfectoid field of characteristic zero, we can obtain its tilt - a perfectoid field of positive characteristic. Perfectoid fields are complete non-discretely valued fields of rank 1 that satisfy some perfectness condition. In my talk, I will tell you how we can extend the tilting construction to certain valued fields of higher rank using model theory. The model-theoretic tilt we obtain is only defined up to elementary equivalence, so we tilt the theories rather than the fields themselves. To understand a valuation of higher rank, we can decompose it into simpler parts. I will introduce the standard decomposition and tell you about ongoing joint work with Sylvy Anscombe and Franziska Jahnke, where we prove for certain valued fields that the theories of the fields in the standard decomposition only depend on the theory of the valued field we started with. This work is crucial for the well-definedness of the model-theoretic tilt.

• July 1st: Yilong Zhang (Bonn)
  

  Title: Hrushovski construction in fields
  
  Abstract: Hrushovski construction is a variant of amalgamation methods. It was invented to construct new examples of strongly minimal theories. In 2000s, Zilber proposed the study of expansions of fields using Hrushovski construction. In this talk, I will present our study on the expansion of the real field by dense logarithmic spirals. I will introduce Kirby's result on quasiminimality of the complex field expanded by exponential or power functions, and discuss the situation in the real field.

• July 8th: No seminar
  
• Hausdorff Colloquium - Wednesday July 17th 15:00, Lipschitz-Saal: Jeremy Avigad (CMU)
  

  Title: A formal perspective on mathematical structures
  
  Abstract: Reasoning about axiomatically characterized abstract structures has been central to mathematics since the early twentieth century. Mathematicians today are using the Lean interactive proof assistant to build a formal library called Mathlib, and the ability of the system to manage a complex network of such structures has been essential to its success. In this talk, I will discuss some of the challenges that structural reasoning brings and how they are addressed in Lean and Mathlib.