Oberseminar Algebra und Geometrie
Summer term 2024: Mixed topics
Chaired by I. Halupczok, H. Kammeyer and B. Klopsch.
All talks take place on Fridays at 14:30 in 25.22.03.73.
This term, instead of having a fixed topic, we will have mixed topics, mainly with guest speakers. Please get it touch if you have a suggestion on whom to invite. Note also the changed time of the seminar.
If you want get announcements about the seminar, please get in touch with I. Halupczok so that your address is added to the mailing list.
Infos für Studierende
Im SoSe 2024 wird das Seminar (hauptsächlich?) aus in sich abgeschlossenen Einzel-Vorträgen bestehen, größtenteils von externen Gästen, und größtenteils über aktulelle Forschungsthemen. Das Oberseminar richtet sich an alle, die einen Einblick in aktuelle Forschung erhalten möchten, ist aber tendenziell eher für fortgeschrittene Studierende geeignet (ab Master). Besonders empfohlen wird die Teilnahme an Oberseminaren, wenn Sie sich vorstellen können zu promovieren.Wenn Sie intessiert sind, können Sie sich einfach (ohne Anmeldung) ins Seminar reinsetzen - gerne auch nur zu einzelnen Vorträgen, die Sie interessieren.
Die Vorträge in diesem Oberseminar sind auf englisch. Üblicherweise nehmen an Oberseminaren auch viele Doktorand:innen, Postdocs und Professor:innen teil. Wahrscheinlich werden Sie nicht alles verstehen; das passiert aber auch den fortgeschritteneren Teilnehmer:innen, und auch, wenn man nicht alles verstanden hat, hat man doch am Ende oft einen interessanten Einblick in ein neues Thema erhalten.
Schedule
- 12.4.: Johannes Ebert (Uni Münster): Tautological Classes and Higher Signatures
Abstract: For a bundle of smooth oriented $d$-manifolds $\pi: E \to X$, a map $f:E \to BG$ to the classifying space of a discrete group G and $u \in H^p(B;\mathbb{Q})$, one defines the tautological classes associated to the higher signatures by the formula $\kappa_{\mathcal{L}_m,u} (E,f):= \pi_! (\mathcal{L}_m(T_v E) \cup f^* u) \in H^{4m+p-d}(X;\mathbb{Q})$, where $\mathcal{L}_m$ is the $m$th component of the Hirzebruch $L$-class. When $4m+p-d=0$, this gives the Novikov higher signatures. It is well-known that when $G=1$ and $d$ is odd, these classes are always $0$. We shall show that the question whether these classes vanish or not depends sensitively on the group $G$.
- 19.4.: Sira Busch (Uni Münster): Projectivity groups of spherical buildings
Abstract: Buildings are combinatorial and geometric structures that were introduced by Jacques Tits for the study of semi-simple algebraic groups. They turned out to be very helpful and interesting objects of study on their own. When we see the structure of a building and want to work with it, it can sometimes be useful to know, whether or not non-trivial automorphisms of the building exist. In my talk I would like to explain what a building is and under which circumstances we know for sure that our building has a rich automorphism group. Given a building with a rich automorphism group, interesting questions are, what kind of automorphisms we can find, if automorphisms with certain properties exist and if there is a relation to other known concepts. At the moment Hendrik Van Maldeghem, Jeroen Schillewaert and me are working on determining the special and general groups of projectivities of all (thick, irreducible) spherical buildings. In my talk I would also like to give some insight into that; specifically for the example of buildings of type A_n (which correspond to projective spaces).
- (26.4.: no seminar, due to the Felix Klein Colloqium on the 25th)
- 3.5.: Immi: Riso-stratifications
Abstract: Fix your favourite notion of "geometric sets", e.g. subsets $X$ of $\mathbb{R}^n$ (or of $\mathbb{C}^n$) given as zero sets of polynomials. Such sets may have singularities, and some singularities are worse than others. A stratification is a partition of $X$ which yields some control of how many how bad the singularities there are. I will present a new notion of stratification which is more intuitive than classical ones and which in some sense is also more precise. There will be a lot of pictures. This is joint work with David Bradley-Williams.
- 10.5.: (Brückentag)
- 17.5.: Philip Möller (Uni Münster): Profinite rigidity of crystallographic groups and affine Coxeter groups
Abstract: A crystallographic group is a discrete, cocompact subgroup of the isometry group of Euclidean space. In this talk, I want to talk about the question of profinite rigidity, that is, whether a group is determined by its finite quotients. Of particular interest for this question are affine Coxeter groups, a well-behaved subclass of the class of crystallographic groups. This is joint work with Sam Corson, Sam Hughes and Olga Varghese.
- 24.5.: Ralf Köhl (CAU Kiel): Covolumes of lattices in simple Lie groups
Abstract: Siegel proved directly via geometric arguments that the hyperbolic (2,3,7) triangle provides a lattice of minimal covolume in SL(2,R). Later Kazhdan-Margulis established that in any simple Lie group the covolumes of lattices are bounded away from 0 (other than in (R^n,+), for example!). In groups for which Margulis arithmeticity holds (e.g., higher rank or quaternionic) one can now look for this minimal lattice = arithmetic group via number-theoretic methods. A key role is played by Rohlf's maximality criterion and the Prasad covolume formula; moreover, ingenious computations by Borel-Prasad can be put to use (for instance by artificially introducing the regulator and then generally bounding it).
- 31.5.: (Brückentag)
- 7.6.: Stavroula Makri (Nicolaus Copernicus University of Torun): Strong Nielsen equivalence on the punctured disc
Abstract: A classical problem related to Nielsen theory deals with the question of determining the minimum number of fixed points among all maps homotopic to a given continuous map f from a compact space to itself. This problem motivated the definition of an equivalence relation on the set of fixed points of f separating the fixed points into Nielsen equivalence classes. In this talk we will focus on the strong Nielsen equivalence, which is a "stronger" equivalence relation concerning periodic points of a surface homeomorphism. We will focus on orientation-preserving homeomorphisms of the 2-disc D2 with certain properties. We will study the strong Nielsen equivalence of periodic points of such homeomorphisms f and we will present its connection with braid groups. An introduction to the braid group theory will be given as well.
- 14.6.: Robin Sroka (Uni Münster): Homological stability and K-theory
Abstract: Homological stability is a well-established and powerful tool for studying and, eventually, computing group homology. In the first part of this talk, I will introduce this technique and outline its connection to K-theory. In the second part, I will showcase the flexibility of this tool and survey recent works applying it in different contexts, such as the classical setting of algebraic K-theory, the study of bounded cohomology, the context of scissor congruence, and the study of augmented algebras.
- 21.6.: tba
- 28.6.: Ilaria Castellano (Uni Bielefeld): Finiteness properties for locally compact groups and the Euler-Poincaré characteristic
Abstract: Finiteness properties of groups provide various generalisations of the properties of being "finitely generated" and "finitely presented". I will give a brief overview of the different types of finiteness properties for (abstract) groups, and then focus on locally compact groups. The aim is to introduce an Euler-Poincaré characteristic for unimodular totally disconnected locally compact (= t.d.l.c.) groups of type FP over the rationals. In this context the characteristic is no longer just a number, but a rational multiple of a Haar measure. In many cases it happens that we can define a meromorphic function of the complex plane whose value in -1 detects, miraculously, the Euler-Poincaré characteristic of the unimodular t.d.l.c. group. Moreover, I will sketch how this invariant can be used to deal with accessibility problems for t.d.l.c. groups.
- 5.7.: Sonia Petschick (Uni Wuppertal): Towards the Equivariance Property of the inductive Galois-McKay Condition
Abstract: Character theory has proven to be a valuable tool in the representation theory of finite groups, with significant recent advancements in the local-global conjectures. Navarro's refinement of the McKay conjecture suggests that the bijection from the original conjecture can be refined to be equivariant under specific Galois automorphisms. Similar to the McKay conjecture, Navarro-Späth-Vallejo reduced this to quasi-simple groups in 2019. In this talk, we will start with some basics on character theory, discuss the reduction theorem, and explore techniques for computing character values of Sylow normalizers in type A working towards the equivariance property of the inductive conditions.
- 12.7.: Quy Thuong Lê (Hanoi): Motivic integration, motivic Milnor fiber, and singularity theory
Abstract: Since 1995, motivic integration has been a powerful tool in algebraic geometry and other branches of mathematics. In particular, it has many important applications to singularity theory. For instance, Denef-Loeser around 2000 gave a breakthrough point of view in the study of singularities, by introducing the so-called motivic Milnor fiber, with the philosophy that this is a motivic incarnation of the classical Milnor fiber. One shows that many singularity invariants can be easily recovered from motivic zeta function and motivic Milnor fiber employing an appropriate Hodge realization. Furthermore, there are important problems concerning singularity theory such as monodromy conjecture, the integral identity conjecture, and the Thom-Sebastiani theorem that are waiting for new methods in motivic integration to have a solution.
In this talk, we will describe some surprising interactions between motivic integration, model theory and singularity theory that lead to our proofs for the integral identity conjecture, and the motivic Thom-Sebastiani theorem, as well as other applications to singularities. The talk will avoid technical aspects and emphasize key ideas in motivic integration and singularity theory, which may be friendly to a general audience. - 19.7.: Jamshid Derakhshan (Oxford): Model Theory of the Adeles
Abstract: The ring of adeles of a global field, introduced by Andre Weil and independently Artin and Whaples in the 1940’s has become a part of the basic language and formalism of modern number theory. It is a locally compact ring constructed from all the completions of the global field and has enabled deep local to global transfers in many contexts.
It is a natural question as to what extent the model-theoretic properties of the local fields transfer or have analogs for the adeles. In this talk I shall present results of this nature.
I will first present joint results with Angus Macintyre on quantifier elimination and definability in adele rings and some applications including solution to a problem of James Ax formulated in 1968 on decidability of the class of all the rings $\mathbb{Z}/m\mathbb{Z}$, for all $m>1$, and results on elementary equivalence in adele rings.
I will then present joint results with Ehud Hrushovski where we describe the imaginary sorts of various infinite enriched or reduced products in terms of imaginary sorts of the factors. As a special case, using the uniform $p$-adic elimination of imaginaries (due to Hrushovski-Martin-Rideau and later Hils-Rideau-Kikuchie) we find the imaginary sorts of the ring of adeles and prove weak elimination of imaginaries for the adeles. Our methods include the use of the Harrington-Kechris-Louveau Glimm-Efros dichotomy in descriptive set theory and we introduce for the first time methods from geometric stability theory and binding groups in the model theory of products of structures and adele rings.
Archive
WS 2023/24: Central Simple Algebras
SS 2023: Knot theory and quandles
WS 2022/23: Combinatorics and Commutative Algebra
SS 2022: Discrete Groups, Expanding Graphs and Invariant Measures
WS 2021/22: Superrigidity
SS 2021: Group cohomology
SS 2020 and WS 20/21: cancelled due to pandemic
WS 2019/20: Intersection theory
SS 2019: Knots and primes
WS 2018/19: The Grothendieck group of varieties and stacks
SS 2018: Arithmetic Groups - Basics and Selected Applications
WS 2017/18: Algebraic K-theory
SS 2017: Berkovich spaces
WS 16/17: Resolution of singularities and alterations
SS 2016: Modular Representation Theory
WS 15/16: The Milnor Conjectures
SS 2015: Rationality
WS 14/15: Essential Dimension
SS 2014: Varieties of Representations
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