Oberseminar Logik - SoSe 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

• April 7th: Martin Bays (Oxford): Asymmetric Elekes-Rónyai
  Abstract: Elekes and Rónyai showed that a bivariate real polynomial f(x,y) is expanding, |f(A,A)| >= c|A|^{1+η} for all finite A, unless f can be written as the composition of addition or multiplication with univariate polynomials. The proof can be seen as going via the group configuration theorem of model theory. I will talk about recent work with Tingxiang Zou, in which we consider a more general setup where we can apply a homogeneous space version of this group configuration theorem, and yet still subsequently reduce to an abelian group. We deduce asymmetric expansion |f(A,B)| >= c|A|^{1+η} even when B is allowed to be drastically smaller than A. Moreover, we obtain a similar result when y is allowed to be a tuple. Allowing x to also be a tuple introduces new phenomena, and if time permits I may mention some partial results in this case.
  
• April 14th: No seminar
  
• April 21th: No seminar
  
• April 28th: Gunther Cornelissen (MPIM/Utrecht): Automata and complexity of finite order power series
  Abstract: The Nottingham group is the group of formal power series t+… with coefficients from a finite field of order p (prime), under composition. It plays a role in the theory of wild ramification and deformation of Galois covers of curves and in the general (model) theory of p-groups. It turns out to be very hard to construct explicit elements of finite order in that group. Elements of order p were classified by Klopsch, and we know an element of order 4. We look at this construction problem from the point of view of automata: since such elements satisfy an algebraic equation over a function field (that can be found using Witt vectors), their coefficient list is an automatic sequence, by a theorem of Christol. We can turn this into an algorithm and construct more elements and finite p-subgroups of the Nottingham group, and we can also transfer complexity notions from automata theory to such elements, like state complexity, the growth dichotomy of Cobham, synchornisability, and Gowers uniformity. (Joint work with Jakub Byszewski and Djurre Tijsma.)
  
• May 5th: No seminar
  
• May 12th: Neer Bhardwaj (KU Leuven): Effective and approximate Pila-Wilkie type counting with complex-analytic sets
  Abstract: The Pila-Wilkie point counting theorem has had striking applications to arithmetic geometry, functional transcendence, and Hodge theory. This statement cannot be made effective in general, but various effective versions of the theorem exist for zero sets of functions satisfying certain forms of differential equations.
  In joint work with Binyamini, we establish an effective point-counting statement of Pila-Wilkie type for the zero-set of any computable complex-analytic function over a ball. For our proof, we develop an approximate counting result to cover rational points close to being a solution of a family of complex-analytic functions in a ball. This second theorem is unique among results of Pila-Wilkie type in that it applies uniformly to any family of functions.
  
• May 19th: Carlo Pagano (MPIM/Concordia): Hilbert 10 via additive combinatorics
  Abstract: I will overview recent joint work with Peter Koymans, where we introduced a new method to construct elliptic curves over general number fields with positive but constrained rank. This comes in two flavors: either positive rank that is unchanged under quadratic extensions (which we established in 2024) or rank exactly equal to 1 (ongoing work in progress). Our main idea is to combine 2-descent with additive combinatorics. With the first of these two cases, we were able to settle Hilbert's tenth problem in the negative for all finitely generated infinite commutative rings. I will overview the reductions of Hilbert tenth's problem to such an elliptic curve question, and the main steps in our method.
  
• May 26th: Philip Welch (MPIM/Bristol): When cardinals determine the power set: inner models and strong definability logics
  Abstract: In joint work with Jouko Väänänen we discuss the relationship between the predicate representing the class of all infinite Cardinals, and that of the graph of the power set function. This work relates to the notion of inner models defined à la Gödel but using definability in languages stronger than first order which determined, L, the constructible sets.
• June 2nd: No seminar
  
• June 9th: No seminar
  
• June 16th: Michele Serra (TU Dortmund)
  
• June 23rd: No seminar
  
• June 30th: Mariana Vicaria (Münster)
  
• July 7th: Nadja Hempel (Düsseldorf)
  
• July 14th: No seminar