Cellular structures using tilting modules
Friday, 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
Friday, 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
Friday, 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
Friday, 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.