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The Research Seminar meets Wednesdays at 2.30pm (unless otherwise stated) in the HIM lecture hall (Poppelsdorfer Allee 45, Bonn). The seminar is organized by Philipp Hieronymi, Jennifer Park, and Florian Pop.
Recordings of past seminars are at the bottom of this page.
Speaker: Massoud Pourmahdian (IPM)
Date: November 19th
Title: TBD
Abstract: TBD
Speaker: Anna de Mase (Konstanz)
Date: Monday November 24th 11am
Title: TBD
Abstract: TBD
Speaker: Gunther Cornelissen (Utrecht)
Date: November 26th
Title: TBD
Abstract: TBD
Speaker: TBD
Date: December 3rd
Title: TBD
Abstract: TBD
Date: December 10th
No seminar due to the conference
Speaker: Pierre Touchard (Dresden)
Date: September 3rd
Title: On Groups Recognizing Coordinates
Abstract: The classical result of Feferman-Vaught describes a language in which a product of ℒ-structures ∏ᵢMᵢ eliminates quantifiers. It is, however not always clear whether this language is interpretable in the original language ℒ. To investigate this question, one can first ask whether ∏ᵢMᵢ, or any so-called reduced product, recognizes coordinates (namely, whether the product structure “knows” that it is a product). After providing all the necessary definitions, we will discuss this question in the context of various first order structures and for particular classes of groups. This is joint work with Ilijas Farah and Kyle Gannon.
Speaker: Douglas Cenzer (Florida)
Date: September 10rd
HomogeneousTalkRevised.pdfTitle: Homogeneity and Categoricity of Linear Orderings
Abstract: We introduce the notion of sp-homogeneous and weakly sp-homogeneous linear orderings --- linear orderings which become homogeneous or weakly homogeneous when expanded by partial functions for successor and predecessor.
We demonstrate that these orderings are always relatively $\Delta_4$ categorical and determine exactly which ones are (uniformly) relatively $\Delta_3$ categorical.
We also provide a classification for sp-homogeneity and weak sp-homogeneity. This classification is optimal in that the set of sp-homogeneous linear orderings is $\Pi_5^0$-complete and that the set of weakly sp-homogeneous linear orderings is $\Sigma_6^0$-complete. This result is obtained in the setting of computability theory and also in the setting of descriptive set theory.
This work is joint with Wesley Calvert, David Gonzalez, Valentina Harizanov, Keng Meng Ng.
Date: September 17th
No seminar, due to the introductory school
Speaker: Adele Padgett (Wien)
Date: September 24th
Title: A partial Ax-Lindemann-Weierstrass theorem for the Gamma function
Abstract: It is natural to study the algebraic properties of interesting transcendental holomorphic functions. For example, the complex exponential function is a group homomorphism. Moreover, one can characterize precisely which algebraic varieties are mapped coordinatewise by exp to other algebraic varieties. Analogous characterizations are known for other functions that, like exp, satisfy differential equations. These results are proved using techniques that exploit the differential equations and do not adapt cleanly to differentially transcendental functions. The Gamma function, which extends the factorial function to complex numbers, is an important differentially transcendental function. I will characterize the algebraic varieties mapped by Gamma to other algebraic varieties of the same dimension. This is joint work with Sebastian Eterović and Roy Zhao.
Speaker: Toghrul Karimov (MPI-SWS)
Date: October 1st 11am
Title: Arithmetic Predicates and Decidability of Logical Theories
Abstract: I will discuss some recent results as well as ongoing work, including the following results.
--The first-order theory of <Z; <, n -> tau(n)> is undecidable, where tau is the Ramanujan tau function.
--The existential fragment of the first-order theory of <N; +, {a_1^n: n >=0}, {a_2^n: n >= 0}> is decidable, where a_1,a_2 are positive integers.
--The monadic second-order theory of <N; <, {a_1^n}, ..., {a_k^n}> is decidable, where a_1,...,a_k are positive integers.
Speaker: Sylvy Anscombe (U Paris-Cité)
Date: October 8th
Title: Existentially definable henselian valuations
Abstract: Thematically following Blaise's seminar from last week, I will describe what is known about existential definability of henselian valuation rings within their fields of fractions, working in the language of rings, sometimes allowing parameters. There is a story---somewhat parallel to the one Blaise recounted---in which one may classify such valuation rings (at least in equal characteristic) by a first-order property of their residue field. I will try to explain the key ingredients and examples, together with some open questions. This is based on non-recent work, some joint with Arno Fehm, and some also with Philip Dittmann.
Speaker: Guang Hu (Dresden)
Date: October 15th
Title: Anisotropic quadratic forms over global fields are diophantine
Abstract: We prove that the set of anisotropic quadratic forms over a global field of characteristic different from two is Diophantine. Our method follows the framework of Koenigsmann, Poonen, Jennifer Park, and Eisenträger–Morrison, by classifying the completions of the global field using class field theory and providing a uniform Diophantine description on each class.
Date: October 22nd
No seminar, due to the workshop
Speaker: Pantelis Eleftheriou (Leeds)
Date: October 29th
Title: On the global linear Zarankiewicz problem
Abstract: The global Zarankiewicz's problem for hypergraphs asks for anupper bound on the number of edges of a hypergraph, whose edge relation is induced by a fixed hypergraph E that has no sub-hypergraphs of a given size. Basit-Chernikov-Starchenko-Tao-Tran (2021) obtained linear Zarankiewicz bounds in the case of a semilinear E, namely E definable in a linear o-minimal structure. We extend this theorem to a broader range of “linear-like” structures, in o-minimal, Presburger Arithmetic and stability theoretic settings. Some of the methods involved include (a) a reduction of the problem to the case of arbitrary subgroups E of powers of groups, and (b) an abstract version of Zarankiewicz problem in the saturated setting.
Joint work with Aris Papadopoulos
Speaker: John Alexander Cruz Morales (Universidad Nacional de Colombia)
Date: November 5th
Title: Mirror maps through logically perfect structures
Abstract: Mirror symmetry is an interesting phenomenon in both mathematics and physics. Emerging in physics more than 30 years ago, as a kind of duality between symplectic and complex geometry, has called the attention of mathematicians who have proposed several programs to understand it. One of the central objects in mirror symmetry is the so-called mirror map which is still a bit mysterious. On the other hand, from the model theory side, Boris Zilber has proposed a program for both mathematics and physics based on the idea of logically perfect structures, a notion close to the idea of categoricity and a kind of extension of his program in abstract algebraic geometry based on the idea of Zariski geometries.
In this talk, I will propose a set of ideas looking for understanding mirror maps from the point of view of logically perfect structures. The main idea behind is to explore how model theoretic tools could help to understand the way different geometries are related.
Speaker: Pablo Andújar Guerrero (Valencia)
Date: Tuesday November 11th 11am
Title: Tame topology in NTP2 structures
Abstract: NTP2 is one of the central dividing lines for theories introduced by Shelah within the program of neostability. It generalizes both NIP and simple theories, encompassing for example o-minimal theories and all theories of ordered abelian groups. We investigate the connection between the combinatorial notion of NTP2 and topological tameness. Our main result shows that any NTP2 expansion by closed sets of the ordered group of reals or of the p-adic fields defines only finite Boolean combinations of closed sets. This yields a good notion of dimension and generic continuity of functions definable in these structures.
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