Geometrical aspects of singular surfaces and their smoothings
Freitag, 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
Freitag, 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
Freitag, 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
Freitag, 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)