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The Junior Research Seminar meets Fridays at 2:30pm (unless otherwise stated) in the HIM lecture hall (Poppelsdorfer Allee 45, Bonn) and will start after the introductory school. The seminar is organized by Guang Hu and The Gia Hoang.
Speaker: Margarete Ketelsen (Münster)
Date: December 5th
Title: Definable Henselian Valuations in Positive Residue Characteristic
Abstract: In many cases, henselian valuations are definable in the language of rings, a phenomenon famously observed by Julia Robinson in the context of the p-adics. Building on this, Jahnke and Koenigsmann gave a full characterization of when a field admits a non-trivial definable henselian valuation for the case where the canonical henselian valuation has residue characteristic zero. In my talk, I will show how their result can be extended to fields with positive residue characteristic.
The talk includes joint work with Simone Ramello and Piotr Szewczyk.
Speaker: Leo Gitin (University of Oxford)
Date: September 26th
Title: Canonical constructions in valuation theory
Abstract: There are many reasons why one would like to find and study canonical constructions within a given mathematical framework: they can be a source of toy models, invariants, or structure theory—the more natural they are, the more useful they tend to be. In valuation theory, among the most useful are the Standard Decomposition, the canonical henselian valuation, and the canonical S-compatible valuation. In this talk, the speaker will present these constructions and discuss their incredible usefulness for the structure theory of valued fields.
Speaker: Sebastian Eterovic (Wien)
Date: Monday September 29th, 2.30pm
Title: Algebraic dependencies of Gamma on linear polynomials
Abstract: I will talk about the possible algebraic dependencies that arise among a set of non-constant functions of the form $\Gamma(az+b)$, both in the case where the parameters $a$ and $b$ are all rational numbers, and in the case where they are arbitrary complex numbers. This is joint work with Adele Padgett and Roy Zhao.
Speaker: Blaise Boissonneau (Düsseldorf)
Date: Wednesday October 1st 2.30pm
Title: Enlarge your spine
Abstract: This team of four mathematicians has discovered a SECRET about ordered abelian groups! Attend this talk to learn about their one weird trick and its surprising consequences: They classify augmentable linear orders! They transfer augmentability from spines to groups! They define valuations! Attend this talk to learn about their method for FREE; then, you'll finally be able to find, for any ordered abelian group GG, another ordered abelian group HH such that G⪯H⊕GG⪯H⊕G, or (subject to terms and conditions) such that G⪯G⊕HG⪯G⊕H.
Speaker: Anna Dmitrieva (University of Manchester)
Date: Friday October 10th
Title: Generic functions and quasiminimality
Abstract: In 2002, Zilber introduced the theory of a generic function on a field, axiomatized in first-order logic by versions of the Schanuel property and existential closedness. In this talk we discuss the motivation behind this theory and its model-theoretical properties, including quasiminimality.
Speaker: Yilong Zhang (Bonn)
Date: October 17th
Title: Hrushovski construction in ordered fields
Abstract: The Hrushovski construction is a variant of amalgamation methods. It was invented to construct new examples of strongly minimal theories. The method was later adapted to expansions of fields, including colored fields and powered fields. In this talk, I will present my attempt to apply the Hrushovski construction to ordered fields. I will construct an expansion of RCF by a dense multiplicative subgroup (green points). The construction induces a back-and-forth system, enabling us to study the dp-rank and the open core of this structure. I will also introduce my recent progress on powered fields, an expansion of RCF by "power functions" on the unit circle, and my plan to axiomatize expansions of the real field using the Hrushovski construction.
Speaker: Nick Rome (TU Graz)
Date: October 31st
Title: Magic Squares of Squares
Abstract: I will discuss a joint work with Shuntaro Yamagishi (now at the University of Bonn), in which we show that for at least n>=4, there exists an n x n magic square whose entries are all distinct squares of integers.
Speaker: Daebeom Choi (U Penn)
Date: November 7th
Title: Introduction to Anabelian Geometry
Abstract: Anabelian geometry is a branch of arithmetic geometry that studies varieties over arithmetically meaningful fields through their étale fundamental groups. In this talk, I will give an introduction to important theorems and conjectures in anabelian geometry, viewed through their analogues in geometry and topology—namely, surface bundles and mapping class groups.
Speaker: Henry Klatt (GWU)
Date: November 14th
Title: Introduction to Cohesive Products
Abstract: Cohesive products are an analog to ultraproducts within computable structure theory. Unlike an ultraproduct, cohesive powers are countable and admit presentations that “aren’t too complicated,” but they only have a fragment of Łoś’ theorem. In this talk, we will go over the definition of a cohesive product of computable structures, prove some preliminary results about cohesive powers of ℕ, ℚ, and number fields K, and state possible directions for future research that are potentially of interest to attendees of the Trimester Program.
Speaker: Leon Chini (Bonn)
Date: November 21st
Title: Model Theory of Generic Vector Space Endomorphisms
Abstract: Let T be a model-complete L-theory that defines a vector space over a fixed field K. I will present a characterization of the existentially closed models of a certain expansion of the L \cup {theta}-theory
T \cup {``theta is an endomorphism of the vector space''},
and provide a criterion for when a model companion exists. I then give examples of theories that satisfy this criterion, followed by some results concerning reducts and neostability.
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