Reshetikhin-Turaev representations as Kähler local systems
Friday, 6.5.22, 10:30-11:30, Hörsaal II, Albertstr. 23b
From a joint work, partially in progress, with Louis Funar. In Orbifold Kähler Groups related to Mapping Class groups, arXiv:2112.06726, we constructed certain orbifold compactifications of the moduli stack of stable pointed curves labelled by an integer p such that the corresponding Reshetikhin Turaev representation of the mapping class group descends to a representation of the orbifold fundamental group. I will explain the construction of that orbifold and why it is uniformizable. I will then report on a work in progress on the uniformization of these orbifolds. I will sketch a proof of the steiness of its universal covering p odd large enough. An interesting new quantum topological consequence is that the image of the fundamental group of the smooth base of a non isotrivial complex algebraic family of smooth complete curves of genus greater than 2 by the Reshetikhin-Turaev representation is infinite (generalizing the Funar-Masbaum and the Koberda-Santharoubane-Funar-Lochak infiniteness theorems).
Lifting globally F-split surfaces over the Witt vectors
Friday, 20.5.22, 10:30-11:30, Hörsaal II, Albertstr. 23b
Given a projective variety X over an algebraically closed field k of characteristic p, it is natural to understand the possible geometric and arithmetic obstructions to the existence of a lifting to characteristic zero. Motivated by the case of abelian manifolds and K3 surfaces, a folklore conjecture claims that ordinary Calabi-Yau manifolds should admit a lifting over the ring of Witt vectors W(k). I will report a joint work with I. Brivio, T. Kawakami and J. Witaszek where we show that globally F-split surfaces (which can be thought of as log Calabi-Yau surfaces that behave arithmetically well) are liftable over W(k) and we deduce several geometric consequences (as the Bogomolov bound on the number of singular points of klt del Pezzo F-split surfaces).
Global properties of period maps at infinity
Friday, 3.6.22, 10:30-11:30, Hörsaal II, Albertstr. 23b
Orbit Method: From Matrices to Unitary Representations
Tuesday, 14.6.22, 10:30-11:30, Raum 218, Ernst-Zermelo-Str. 1
The talk is intended as a leisurely introduction to one of the fundamental tasks of representation theory: the construction of irreducible unitary representations. I will first discuss two major sources of unitary representations of Lie groups, one from Symplectic Geometry (Kirillov theory) and another from Number Theory (Arthur’s conjecture). I will then introduce a constructive method called theta lifting which has been fruitful for representations of classical groups and discuss some recent applications of this method to unitary representation theory.\n
The derived category of permutation modules
Friday, 24.6.22, 10:30-11:30, Hörsaal II, Albertstr. 23b
To a field k and a finite group G one associates the derived\ncategory of kG-modules, an important invariant that is difficult to\nunderstand in general. At least, its tensor-triangulated structure\nadmits a familiar description in terms of the support variety.\n\nWe propose to study a refinement, the derived category of G-permutation\nmodules over k. It has interesting interpretations in algebraic\ngeometry, representation theory and equivariant homotopy theory. We\nwill say a few things we know about its tensor-triangulated structure. \nThis is based on joint work, mostly in progress, with Paul Balmer.
Geometrical aspects of singular surfaces and their smoothings
Friday, 1.7.22, 10:30-11:30, Hörsaal II, Albertstr. 23b
Degenerations of nonsingular algebraic surfaces into surfaces with only cyclic quotient singularities (c.q.s.) are relevant for the study of the Kollár--Shepherd-Barron--Alexeev (KSBA) compactification of the moduli of surfaces of general type, for the existence of surfaces with given invariants, etc, and lately to find semi-orthogonal decompositions (s.o.d.) of the derived categories of the surfaces involved. Thanks to the work of Kollár--Shepherd-Barron (1988), the local picture of these degenerations is well-understood via P-resolutions of c.q.s. (partial resolutions with only T-singularities and positive relative canonical class), which can be replaced in a one-to-one correspondence by M-resolutions (partial resolutions with only Wahl singularities and nonnegative relative canonical class). Hence arbitrary degenerations with only c.q.s. can be replaced by Q-Gorenstein smoothings of Wahl surfaces, i.e. surfaces with only Wahl singularities. The purpose of this talk is to show how geometry works in this setting, for example how to find minimal models via flips and divisorial contractions (in the joint work "Flipping surfaces" with P. Hacking and J. Tevelev). Particularly I hope to be able to state what the N-resolution of an M-resolution is, how to find it via antiflips, and some consequences on particular s.o.d. of the derived category of the singular and the nonsingular fibers. This is about the joint work with Jenia Tevelev https://arxiv.org/abs/2204.13225.
The wild ramification locus
Friday, 8.7.22, 10:30-11:30, Hörsaal II, Albertstr. 23b
We study the notions of wild and tame ramification in arithmetic geometry. Wildly ramified morphisms tend to behave very differently from what we know about ramification phenomena in characteristic zero. We discuss several approaches to define tame covering spaces and explain how valuative spaces such as adic spaces or Berkovich spaces naturally enter the picture. Points of these spaces are certain valuations, such as discrete valuations coming from a divisor. But in general these valuations tend to be complicated. By analytic methods we show, however, that we can check tameness on divisors.
Towards automatic diagram chasing
Friday, 22.7.22, 10:30-11:30, Hörsaal II, Albertstr. 23b
The goal of this presentation is to explain what I learned\nand what I did during my internship in Freiburg.\n\nDiagram chasing in abelian categories is commonly used as a routine technique. However, as the better way to explain such a proof, is "do the only thing you could do", thins kind of proof is not going to be accepted by a proof checker.\n\nI will then explain how to go in the direction of automatic diagram chasing, with a particular attention to what worked and didn't worked in my actual implementation. The internship was focused on finding proofs, having the result checked by a proof assistant is future work.
Excision in algebraic K-theory and applications
Friday, 29.7.22, 10:30-11:30, Hörsaal II, Albertstr. 23b
Algebraic K-groups of rings or schemes are interesting invariants, which appear in different areas of mathematics, e.g. in number theory, algebraic geometry, or topology. Unfortunately, computations of algebraic K-groups are usually quite hard. One reason for this is that useful tools, familiar from singular homology, like homotopy invariance or some long exact Mayer-Vietoris sequences, are missing in algebraic K-theory. In the talk I will give an introduction to algebraic K-theory and in particular discuss the following question about excision': When does a given cartesian square of rings give rise to a long exact sequence of algebraic K-groups? Surprisingly, the answer turns out bealmost always’. I will explain this result and some of its consequences. (Based on joint work with Markus Land)