Oberseminar Logik - WiSe 24/25

Organizers

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

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

• October 7th: Kyle Gannon (Peking University) : Model Theoretic Events
  
  Abstract: This talk is motivated by the following two soft questions: How do we sample an infinite sequence from a first order structure? What model theoretic properties might hold on almost all sampled sequences? We advance a plausible framework in an attempt to answer these kinds of questions. The central object of this talk is a probability space. The underlying set of our space is a standard model theoretic object, namely the space of types in countably many variables over a monster model. Our probability measure is an iterated Morley product of a fixed Borel-definable Keisler measure. Choosing a point randomly in this space with respect to our distribution yields a random generic type in infinitely many variables. We are interested in which model theoretic events hold for almost all random generic types. Two different kinds of events will be discussed: (1) The event that the induced structure on a random generic type is isomorphic to a fixed structure; (2) the event that a random generic type witnesses a dividing line.
  
• October 14th: No seminar
  
• October 21st: No seminar
  
• Saturday October 26th: Bonn-Münster-Düsseldorf GeSAMT (Gemeinsames Seminar Algebra und Modelltheorie) in SRZ216/217, Orléans-Ring 12, Münster
  
• October 28th: No seminar
  
• November 4th: No seminar
  
• November 11th: No seminar
  
• November 18th: Allen Gehret (Artificial Intelligence Center, Czech Technical University) : On definable subsets of T_log
  
  Abstract: In this talk I will present two theorems concerning definable subsets of the structure T_log: the differential field of logarithmic transseries. One result is that the zero sets of arbitrary nonzero differential polynomials are co-analyzable relative to the constant field R. The other (joint work with Elliot Kaplan and Nigel Pynn-Coates) is a characterization of the "small" definable subsets of the asymptotic couple. In particular, the characterization implies the asymptotic couple is d-minimal, i.e., every definable subset in 1-variable either has interior or is a finite union of discrete sets.
• November 25th: Vincent Bagayoko (Université Paris Cité) : Preliminaries for taming a vector field
  
  Abstract: In the study of dynamical systems, a main obstruction to the classification of local objects (vector fields, diffeomorphisms, substitutions, derivations...) is the phenomenon of Poincaré resonance. Working in the formal (local) setting in dimension 1, we will show how to interpret resonance as a property of the underlying asymptotic differential algebra of the setting. We will then explain how this relates to the search for tame groups of germs.
• December 2nd: Adele Padgett (Universität Wien) : O-minimal definability and non-definability of the Gamma function
  
  Abstract: O-minimality is a model-theoretic property with applications in number theory and functional transcendence. Many important functions are known to be definable in o-minimal structures when restricted to appropriate domains, including the exponential function, the Klein j function, and Weierstrass p functions. I will discuss joint work with P. Speissegger in which we prove that the Gamma function, which was known to be o-minimal when restricted to the positive real numbers, is in fact o-minimal on certain unbounded complex domains.
  
• December 9th: No seminar
  
• December 16th: Stefan Ludwig (Freiburg): Model theory of difference fields with an additive character on the fixed field
  
  Abstract: Motivated by work of Hrushovski on pseudofinite fields with an additive character we investigate the theory ACFA+ which is the model companion of the theory of difference fields with an additive character on the fixed field working in (a mild version of) continuous logic. Building on results by Hrushovski we can recover it as the characteristic 0-asymptotic theory of the algebraic closure of finite fields with the Frobenius-automorphism and the standard character on the fixed field. We characterise 3-amalgamation in ACFA+ and obtain that ACFA+ is simple as well as a description of the connected component of the Kim-Pillay group. If time permits we present some results on higher amalgamation and mention ongoing work on pseudofinite fields equipped with both additive and multiplicative character.
  
• January 6th: Charlotte Bartnick (Freiburg) : On non-elimination of imaginaries
  
  Abstract: Given a theory, every definable set has a canonical parameter after introducing imaginary sorts. Many classical theories eliminate imaginaries so that it suffices to work in the real sort. In this talk, we will present a general criterion that yields the failure of elimination of imaginaries due to equivalence classes arising as cosets of subgroups. We will illustrate the main ideas by considering the specific example of the theory of proper pairs of algebraically closed fields. Pillay and Vassiliev already proved that this theory does not eliminate imaginaries and our criterion gives an alternative proof. If time permits, we will show that the general setting of the criterion with additional assumptions allows to prove that types over real algebraically closed sets are stationary.
  
• January 13th: Giles Gardam (Bonn) : Solving semidecidable problems in group theory
  
  Abstract: Group theory is littered with undecidable problems. A classic example is the word problem: there are finitely presented groups for which there exists no algorithm that can decide if a product of generators represents the trivial element or not. Many problems (the word problem included) are at least semidecidable, meaning that there is a correct algorithm guaranteed to terminate if the answer is "yes", but with no guarantee on how long one has to wait. I will discuss strategies to try and tackle various semidecidable problems computationally using modern solvers for Boolean satisfiability, with the key example being the discovery of a counterexample to the Kaplansky unit conjecture.
  
• January 20th: Elliot Kaplan (MPIM) : Constant power maps on Hardy fields and transseries
  
  Abstract: We study H-fields (certain ordered differential fields generalizing Hardy fields and transseries) equipped with "constant power maps". We show that this class has a model companion, the models of which include the field of LE-transseries and any maximal Hardy field. We study the induced structure on the constant field, prove a relative decidability result, and give some applications to certain systems of differential equations.
  
• January 27th: No seminar
  
• February 17th (SemR 1.008): Christian D’Elbée (Leeds): Two cases of Wilson's conjecture for omega-categorical Lie algebras
  
  Abstract: Recall that a structure (group, Lie algebra, associative algebra, etc) M is omega-categorical if there is a unique countable model of its first-order theory, up to isomorphism. This model theoretic notion has a dynamical definition: M is omega-categorical if and only if there are only finitely many orbits in the component-wise action of Aut(M) on the cartesian power M^n, for all natural number n.
  
  In 1981, Wilson conjectured that any omega-categorical locally nilpotent group is nilpotent. If true, a quite satisfactory decomposition of omega-categorical groups would follow. This conjecture is very much open more than 40 years later. The analogue statement for Lie algebras (every locally nilpotent omega-categorical Lie algebra is nilpotent) is also open and, as it turns out, it reduces to proving that for each n and prime p, every omega-categorical n-Engel Lie algebra over F_p is nilpotent. As for associative algebras, the analogous question was already answered by Cherlin in 1980: every locally nilpotent omega-categorical ring is nilpotent. We see the Wilson conjecture for Lie algebra as a bridge between the result of Cherlin and the original question of Wilson for omega-categorical groups.
  
  The question of Wilson, for groups, for Lie algebras or for associative algebras are connected to classical nilpotency problems such as the Burnside problem, the Kurosh problem or the problem of local nilpotency of n-Engel groups.
  
  Using a classical result of Zelmanov, the Wilson conjecture for omega-categorical Lie algebras is true asymptotically in the following sense: for each n, every n-Engel Lie algebra over F_p is nilpotent for all but finitely many p's. The situation for small values of the pair (n,p) is as follows:
  . Every 2-Engel Lie algebra is nilpotent (Higgins 1954),
  . Every 3-Engel Lie algebra over F_p with p\neq 2,5 is nilpotent (Higgins 1954),
  . Every 4-Engel Lie algebra over F_p is nilpotent for p\neq 2,3,5. (Higgins 1954, Kostrikin 1959),
  . Every 5-Engel Lie algebra over F_p is nilpotent for p\neq 2,3,5,7 (Vaughan-Lee, 2024).
  
  In other words, for (n,p) = (3,2), (3,5), (4,2), (4,3), (4,5),... It is known that n-Engel Lie algebras of char p are not globally nilpotent. Our goal, on the long run, is to prove that for those values of (n,p), omega-categorical n-Engel Lie algebra of characteristic p are nilpotent. We have recently dealt with the cases (n,p) = (3,5) and (n,p) = (4,3), and the proofs are different both in taste and method. The goal of the talk is to present a proof that every omega-categorical 4-Engel Lie algebras of characteristic 3 is nilpotent. Our solution of the case at hand consists in adapting in the definable context some classical tools for studying Engel Lie algebras, appearing earlier in the work of Higgins, Kostrikin, Zelmanov, Vaughan-Lee, Traustason and others. Our solution involves the use of computer algebra.