Towards a motivic spectral sequence for hermitian K-theory
Freitag, 24.4.15, 10:15-11:15, Raum 404, Eckerstr. 1
The motivic spectral sequence can be seen as an analogue of the Atiyah-Hirzebruch spectral sequence in (complex) topological K-theory. Postulated by Quillen and Beilinson around 1980, it was finally shown to exist by a number of related results of Voevodsky, Grayson and Suslin in the 90s and early 2000s.\n\nBut the story doesn’t end here: For hermitian K-theory, often said to correspond to real topological K-theory, things are far from clear. In my talk I will present results from my dissertation that generalise Grayson’s ideas on this topic.\n\nI will begin with the basics and revise a certain construction of algebraic and hermitian K-theory. I will then explain conceptionally how spectral sequences can arise from a filtration of the K-theory space.\n\nFinally I will show how Grayson uses tuples of commuting elements of the general linear group to construct his tower and how their role is taken by orthogonal, symplectic, symmetric and antisymmetric matrices in the hermitian realm.
Curves in Hilbert modular varieties
Freitag, 8.5.15, 10:15-11:15, Raum 404, Eckerstr. 1
Abstract: We study curves in Hilbert modular varieties from the point of view of the Green-Griffiths-Lang conjecture claiming that entire curves in complex projective varieties of general type should be contained in a proper subvariety. Using holomorphic foliations theory, we establish the Second Main Theorem in this context as well as a function field analogue of Vojta's conjecture. We also establish the strong Green-Griffiths-Lang conjecture for Hilbert modular varieties up to finitely many possible exceptions. (Joint work with F. Touzet)
Higher Contou-Carrère symbol
Freitag, 15.5.15, 10:15-11:15, Raum 404, Eckerstr. 1
The talk is based on a joint work with D. Osipov.\n\nWe define a higher-dimensional generalization of the Contou-Carrère symbol and discuss its universal property and an explicit formula for it. We also mention a related result on the tangent space to Milnor K-groups.\n\nHigher Contou-Carrère symbol is a far generalization of the Hilbert tame symbol and is intimately related to higher local class field theory.
The period map of certain families of singular hypersurfaces
Freitag, 12.6.15, 10:15-11:15, Raum 404, Eckerstr. 1
This is a joint work with Philippe Eyssidieux. We consider a natural Deligne-Mumford stack parametrizing degree \(d\) hypersurfaces of \(\bmathcal P^n\) with ADE singularities, and prove an infinitesimal Torelli property along the stacky strata. This construction gives rise to examples of smooth projective varieties with interesting fundamental groups and universal covers. If time permits, I will discuss the Toledo and Shafarevich conjecture for these examples.
Locally compact abelian groups and their big friends
Freitag, 19.6.15, 10:15-11:15, Raum 404, Eckerstr. 1
I will discuss how to do homological algebra with\nlocally compact abelian groups (which do not form an\nabelian category, so it's not totally obvious), why one\nwould like to do this without local compactness, or without\ntopology, and ideas of Kato and Kapranov how to actually\nachieve this. Recent advances. This is by the way related\nto Drinfeld's ideas on infinite-dimensional vector bundles\nbecause he proposes to model good fibre spaces by a similar\nmechanism.
On the motive of some hyperKaehler varieties
Freitag, 26.6.15, 10:15-11:15, Raum 404, Eckerstr. 1
I will explain why it is expected that the Chow ring of hyperKaehler varieties has a similar structure as the Chow ring of abelian varieties. Examples of hyperKaehler varieties are given by K3 surfaces, and Hilbert schemes of length-n subschemes on K3 surfaces and their deformations. In fact I will introduce the notion of ``multiplicative Chow-Kuenneth decomposition'' and provide examples of varieties that can be endowed with such a decomposition. In the case of curves, or regular surfaces, this notion is intimately linked to the vanishing of a so-called "modified diagonal cycle". For example, a very general curve of genus >2 does not have vanishing modified diagonal cycle, but a result of Ben Gross and Chad Schoen establishes the vanishing of a modified diagonal cycle for hyperelliptic curves.
Cellular structures using tilting modules
Freitag, 3.7.15, 10:15-11:15, Raum 404, Eckerstr. 1
Classical Schur-Weyl duality says that the actions of the symmetric group Sd and of GL(n) on a tensor power of Cn commute and generate each others commutant. In particular, one can recover the symmetric group algebra as endomorphism algebra of a gl(n) representation. On the other hand, by taking n=2, one recovers another classical algebra: the\nTemperley-Lieb algebra.\nThis is just the tip of an iceberg of a huge class of algebras called centralizer algebras. We discuss\na general method to study their representation theory for the case where gl(n) is replaced by the quantum group acting on the tensor power of a tilting module. That is, we show that the endomorphism algebra of a tilting module is equipped with a cellular basis. The aim of this talk is to explain this approach (from the very beginning) and\nsome consequences of it.\nWe give plenty of examples along the way.
Singular Todd classes of tautological sheaves on Hilbert schemes of points on a smooth surface
Freitag, 10.7.15, 10:15-11:15, Raum 125, Eckerstr. 1
Let \(X\) be a quasi-projective smooth complex algebraic surface, with\n \(X^[n]\) the Hilbert scheme of \(n\) points on \(X\), so that the (rational)\n cohomology of all these Hilbert schemes together can be generated by the\n cohomology of \(X\) in terms of Nakajima creation operators. Given an\n algebraic vector bundle \(V\) on \(X\), there exist universal formulae for\n the characteristic classes of the associated tautological vector bundles\n \(V^[n]\) on \(X^[n]\) in terms of the Nakajima creation operators and the\n corresponding characteristic classes of \(V\). But in general the\n corresponding coefficients are not known. Based on the derived\n equivalence of Bridgeland-King-Reid and work of Haiman and Scala, we\n give an explicit formula in case of the singular Todd classes, but in\n terms of Nakajima creation operators of the delocalized equivariant\n cohomology of all \(X^n\) with its natural \(S_n\)-action.
On the conjecture of Birch and Swinnerton-Dyer
Freitag, 17.7.15, 10:15-11:15, Raum 404, Eckerstr. 1
We report on the classical conjecture of Birch and Swinnerton-Dyer for Abelian varieties over global fields and present (partly conditional) results on this conjecture for Abelian schemes over higher dimensional bases over finite fields.
Dworks Beweis für die Rationalität von Zeta-Funktionn von Varietäten über endlichen Körpern
Freitag, 31.7.15, 10:15-11:15, Raum 404, Eckerstr. 1
Weil ordnete jeder algebraischen Funktion über einem endlichen Körper eine formale Potenzreihe zu. In den Koeffizienten ist die Zahl der Lösungen über F_{q^n} für jedes n kodiert. Er zeigte im Fall von Kurven (und vermutete\nallgemeine), dass es sich um rationale Funktionen handelt.\n\nDies wurde von Dwork in einem spektakulären Beweis in der\nSprache der p-adischen Analysis bewiesen.\n\nZiel des Vortrags ist es, diesen Beweis vorzustellen.