Oberseminar Logik - SoSe 26

Organizers

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

Time and location

Unless stated otherwise: Mondays 17.00-18.00 in SemR 1.008, 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 13th: No seminar
  
• Saturday April 18th: GeSAMT — Gemeinsames Seminar Algebra und Modelltheorie in Münster
  
• April 20th: Ulla Karhumäki (Lyon): Primitive pseudo-finite permutation groups of finite SU-rank
  
  Abstract: A (definably) primitive permutation group (G,X) is a group G together with a faithful action on a set X such that there are no proper nontrivial (definable) G-invariant equivalence relations on X. A rough classification of primitive permutation groups of finite Morley rank was proven by Macpherson and Pillay and, using that, Borovik and Cherlin showed that if (G,X) is a definably primitive permutation group of finite Morley rank, then RM(G) can be bounded in terms of RM(X). We show an analogue of this result in pseudo-finite finite SU-rank context. Namely, we show that if (G,X) is a pseudo-finite definably primitive permutation group of finite SU-rank then SU(G) can be bounded as a function of SU(X). This is joint work with Nick Ramsey.
  
• April 27th: No seminar
  
• May 4th: No seminar
  
• May 11th: No seminar
  
• May 18th: No seminar, but informal talk by Leon Chini (Bonn) at 4pm (c.t.)
  
• May 21st 10am (c.t.) Zeichensaal: Informal talk by Leon Chini (Bonn)
  
• May 25th: No seminar
  
• June 1st: Margarete Ketelsen (Bonn): Definable henselian valuations in positive residue characteristic
  
  Abstract: Valuations arise as non-archimedean generalizations of absolute values, with examples coming from all over number theory and algebraic geometry. In many cases, henselian valuations turn out to be definable, meaning that their valuation ring is definable by a first-order formula in the language of rings. In the case where the residue characteristic of the so-called canonical henselian valuation is zero, Jahnke and Koenigsmann gave a full characterization of when a field admits a definable non-trivial henselian valuation. In joint work with Simone Ramello and Piotr Szewczyk, we extended their to fields with positive residue characteristic. More recently, in joint work with Gessica Alecci, Ihsane Hadeg, Franziska Jahnke and Isabella Negrini, we applied these results to perfectoid fields and showed that definability of the perfectoid valuation tilts and untilts.
  
• June 8th: No seminar
  
• June 15th: informal talk by Leon Chini (Bonn) at 5pm (s.t.)
  
• June 22nd: Moreno Invitti (Lyon): Skew braces of finite Morley rank
  
• June 29th: No seminar
  
• July 6th: Blaise Boissonneau (Düsseldorf): Is Fp((Q)) NTP2?
  
  Abstract: years ago in Oxford, Sylvy Anscombe and I asked this question, which is part of the general effort to try and understand the model theory of henselian valued fields through dividing lines. In 2024, Sylvy Anscombe and Franziska Jahnke completely classified NIP henselian valued fields. Their methods can be extended, with the help of works of Chernikov, Kaplan and Simon and of Kuhlmann and Rzepka, to NTP2 henselian valued fields, obtaining the following: – if a henselian valued field is NTP2, then it is semitame and its residue field is NTP2; – if a henselian valued field is separably algebraically maximal Kaplansky and its residue field is NTP2, then it is NTP2. This covers a large class of fields, but there is still a gap. Notably, Fp((Q)) is in the middle: it is semitame but not Kaplansky. To answer this question, we studied so called tame henselian fields with finite residue field, and derived quantifier elimination results, namely, we prove that any formula in the language of valued fields reduces to a formula of the form (∃y f(x,y)=0) ∧ φ(v(x)) ∧ ψ(res(x)), where φ and ψ are formulas in the language of ordered groups and of rings, respectively. In Fp((Q)) specifically, the valuation ring itself is definable with a diophantine formula (ie of the form ∃y f(x,y)=0), reducing further our quantifier elimination result. Finally, a large chunk of these formulas are known to be NTP2: when f(x,y) is additive in y, the formula ∃y f(x,y)=z is NTP2 (with respect to x and z). Unfortunately, that does not cover all formulas, so the answer to the titular question is still unknown.
  
• July 13th: Mira Tartarotti (Oxford): Incidence bounds and relative distality
  
  Abstract: The Szemerédi-Trotter theorem gives bounds for the number of incidences between finite sets of points and lines in the real plane. A generalization due to Chernikov, Galvin, and Starchenko establishes similar bounds for bipartite graphs definable in distal structures. In particular, this recovers incidence bounds for algebraic graphs over any field of characteristic zero. In positive characteristic, these bounds fail in general. Bays and Martin proved that Szemerédi–Trotter-type bounds hold in fields admitting a valuation with finite residue field, such as (\mathbb{F}_q(t)), though their bounds depend on the parameter $q$. We show that this dependence can be removed: Szemerédi–Trotter-type bounds hold uniformly across all fields admitting a valuation with finite residue field. To prove this, we formalize the intuition that every algebraically closed valued field is distal “relative to its residue field” in a way that allows the methods of Chernikov, Galvin, and Starchenko to apply when attention is restricted to a subfield with finite residue field. I will review the relevant model-theoretic and combinatorial background, introduce the notion of relative distality, and present proofs as time permits.
  
• July 20th: No talk