## Oberseminar Logik - SoSe 2023

### Organizers

- Prof. Dr. Philipp Hieronymi
- Dr. Christian d'Elbée

### 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 3rd: Amador Martin-Pizzaro (Freiburg),
*Corners and stability*

Abstract: Given an abelian group G, a corner is a a subset of pairs of the form (x,y),(x+g,y),(x,y+g) with g non trivial. Ajtai and Szemerédi proved that, asymptotically for finite abelian groups, every dense subset S of G×G contains an corner. Shkredov gave a quantitative lower bound on the density of the subset S. In this talk, we will explain how model-theoretic conditions on the subset S, such as local stability, will imply the existence of corners and of other configurations for (pseudo-)finite abelian groups. We will not assume prior knowledge of geometric stablity theory. This is joint work with D. Palacin (Madrid) and J. Wolf (Cambridge).

- April 10th: No seminar

- Friday April 14th 5pm on Zoom: Chris Schulz (Illinois),
*Definability and decidability for expansions of arithmetic by sets definable from positional numeration systems*

- April 17th: Jinhe Ye (Oxford),
*Curve-excluding fields*

Abstract: Given C a curve over Q with genus at least 2 and C(Q) is empty, the class of fields K of characteristic 0 such that C(K) is empty has a model companion, which we call CXF. Models of CXF have interesting combinations of properties. For example, they provided an example of a model-complete field with unbounded Galois group, answering a question of Macintyre negatively. One can also construct a model of it with a decidable first-order theory that is not ``large'' in the sense of Pop. Algebraically, it provides a field that is algebraically bounded but not ``very slim'' in the sense of Junker and Koenigsmann. Model theoretically, we find a pure field that is strictly NSOP_4.

- April 24th: David Bradley-Williams (Düsseldorf),
*Limits of Betwenness relations*

Abstract: In order to illustrate the wide and wonderful world of Fraissé limits and their variants, in this talk I will discuss joint work with John Truss (Leeds) in which we construct a family of structures called "limits of betweenness relations" as a kind of tree-limit of trees (for appropriate combinatorial meanings of the word "tree"). A crucial part of the construction is an particular instance of categorical Fraissé Theory. 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.

- Oberseminar May 2nd 16:15 Zeichensaal: Matthias Aschenbrenner (Wien),
*A uniform framework for analytic Nullstellensätze*

Abstract: I shall present a technique for establishing Nullstellensätze in rings of power series using model theory. Our approach, in contrast to others, relies on quantifier elimination theorems for valued fields. This leads to new Nullstellensätze and Hilbert’s 17th Problem-type results for p-adic power series, both formal, convergent, and restricted. (Joint work with A. Srhir.) - Hausdorff Colloquium May 3rd 16:45 Lipschitz-Saal: Matthias Aschenbrenner (Wien),
*A transfer principle in asymptotic analysis*

Abstract: Hardy fields form a natural domain for a “tame” part of asymptotic analysis. They may be viewed as one-dimensional relatives of o-minimal structures and have applications to dynamical systems and ergodic theory. In this talk I will explain a recent theorem which permits the transfer of statements concerning algebraic differential equations between Hardy fields and related structures, akin to the “Tarski Principle” at the basis of semi-algebraic geometry, and sketch some applications, including to some classical linear differential equations. (Joint work with L. van den Dries and J. van der Hoeven.)

- May 8th: Tingxiang Zou (Münster)
*Number of rational points of difference varieties in finite difference fields*

Abstract: A difference field is a field with a distinguished automorphism. Automorphisms of a finite field are powers of the Frobenius map. In this talk, I will discuss how to estimate the number of rational points of a difference variety, namely a system of difference equations, in a finite field with a distinguished power of Frobenius. Like algebraic geometry, one can assign a dimension, called transformal dimension, to a difference variety. I will present a result which is a difference version of the Lang-Weil estimate with respect to the transformal dimension. This is joint work with Martin Hils, Ehud Hrushovski and Jinhe Ye.

- Tuesday May 9th 10.30am
**(N0.003)**: Matt Foreman (UC Irvine)*Independence results in dynamical systems*

Abstract: In this talk we discuss the classical*time forwards*vs*time backwards*problems in dynamical systems. Using techniques developed to show this question is not Borel, a very general method is presented that gives independence results for smooth dynamical systems. The independent statements are Sigma^1_1--hence covered by the Schoenfeld absoluteness theorem. An example is given of a computable diffeomorphisms of the 2-torus that is conjugate to its inverse if and only if*ZFC*is consistent. Another that is conjugate to its inverse if and only if*Riemann Hypothesis*holds, and finally an example that is conjugate to its inverse if and only if the twin prime conjecture holds. - May 15th: No seminar

- May 22nd: Chieu-Minh Tran (National University of Singapore),
*Measure doubling of small sets in SO(3,R)*

Abstract: The following question was described in Ben Green’s list of open problems, where it was also attributed to discussions with Emmanuel Breuillard: When A \subseteq SO(3, R) has sufficiently small normalized Haar measure, do we have \mu(AA) > 3.99 \mu(A)? This problem is often considered the first serious obstacle in generalizing results of additive combinatorics to nonabelian settings when we also want sharp bounds. In this talk, I will present a recent joint work with Jing Yifan and Zhang Ruixiang where we combine techniques from model theory, probability theory, and harmonic analysis to answer it positively. I will also discuss how ideas from neostable group theory play an important role behind the stage. - May 29th: No seminar

- June 5th: No seminar

- June 12th: Elliot Kaplan (McMaster),
*Hilbert polynomials for finitary matroids*

Abstract: Eventual polynomial growth is a common theme in combinatorics and commutative algebra. The quintessential example of this phenomenon is the Hilbert polynomial, which eventually coincides with the linear dimension of the graded pieces of a finitely generated module over a polynomial ring. A later result of Kolchin shows that the transcendence degree of certain field extensions of a differential field is eventually polynomial. More recently, Khovanskii showed that for finite subsets A and B of a commutative semigroup, the size of the sumset A+tB is eventually polynomial in t. I will present a common generalization of these three results in terms of finitary matroids (also called pregeometries). I’ll discuss other instances of eventual polynomial growth (like the Betti numbers of a simplicial complex) as well as some applications to bounding model-theoretic ranks. This is joint work with Antongiulio Fornasiero. - Tuesday June 13th 10.30am
**N.003**: Alexi Block Gorman (McMaster),*Expansions of the reals by Büchi-automatic sets: choose-your-own-adventure*

Abstract: There are compelling and long-established connections between automata theory and model theory, and this talk will explore some of those connections for expansions of the real additive group. Büchi automata are the natural extension of DFAs and NFAs to a model of computation that accepts infinite-length inputs. We say a subset X of the reals is Büchi-automatic if there some natural number r and some Büchi automaton that accepts (one of) the base-r representations of every element of X, and rejects the base-r representations of each element in its complement. We can analogously define Büchi-automatic subsets of higher arities, and these sets exhibit intriguing behavior from the perspectives of both fractal geometry and model theory. In this talk, we will have the opportunity to discuss how expansions of the reals solely by Büchi-automatic sets fit into the framework of tame geometry, particularly in the context of the Tetrachotomy Theorem of Hieronymi and Walsberg. - Friday June 16th 4pm on Zoom: Eion Blanchard (Illinois),
*Decidability bounds for extensions of Presburger arithmetic*

- June 19th: Adele Padgett (McMaster),
*Existential closedness for the Gamma function*

Abstract: The existential closedness problem for a function f is to show that a system of complex polynomials in 2n variables always has solutions in the graph of f, except when there is some geometric obstruction. Special cases have been proven for exp, Weierstrass p functions, the Klein j function, and other important functions in arithmetic geometry using a variety of techniques. Recently, some special cases have been proven for well-known solutions of difference equations, like the zeta and Gamma functions. I will show how the methods developed for exp can be used to expand on these results. - Tuesday June 20th 10.30am
**N0.003**: Sebastian Eterović (Leeds),*Generic solutions to systems of equations involving functions from arithmetic geometry*

Abstract: Arithmetic geometry has many important transcendental functions exhibiting interesting algebraic properties. Perhaps the most famous is the complex exponential function, which satisfies the definition of a group homomorphism. More generally, for a given function of interest one would like to understand the “existential closedness problem”: when does an algebraic variety intersect the graph of the function in a Zariski dense set? In this talk I will present results about a strengthening of the problem where we look for points in the intersection of the algebraic variety and the graph of the function which are generic in the algebraic variety. - June 26th: No Seminar

- July 3rd: No seminar

- July 10th: No seminar