Bogomolov-Sommese Vanishing on log canonical pairs
Freitag, 2.11.12, 10:00-11:00, Raum 404, Eckerstr. 1
Gromov Witten Invariants for the Hilbert scheme of points of a K3 surface
Freitag, 9.11.12, 10:00-11:00, Raum 404, Eckerstr. 1
The Yau-Zaslow formula gives an expression of the number of nodal rational curves on a K3 surface in terms of a modular form. In this talk we explain how to extend their result to the Hilbert scheme of 2 points of a K3 surface. In particular, we will present the generating series for the reduced genus 0 GW Invariants which will be given by a weak Jacobi Form.\n
Drinfeld modules and their application to factoring polynomials
Freitag, 16.11.12, 10:00-11:00, Raum 404, Eckerstr. 1
Major works done in Function Field Arithmetic show a strong analogy between the ring of integers Z and the ring of polynomials over a finite field Fq[T]. While an algorithm has been discovered to factor integers using elliptic curves, the discovery of Drinfeld modules, which are analogous to elliptic curves, made it possible to exhibit an algorithm for factorising polynomials in the ring Fq[T]. \nIn this talk, we introduce the notion of Drinfeld modules, then we demonstrate the analogy between Drinfeld modules and Elliptic curves. Finally, we present an algorithm for factoring polynomials over a finite field using Drinfeld modules.\n
Motives of Deligne-Mumford Stacks
Freitag, 23.11.12, 10:00-11:00, Raum 404, Eckerstr. 1
For every smooth and separated Deligne-Mumford stack F,\nwe will associate a motive M(F) in Voevodsky's category of mixed motives with rational coecients DM^eff(k; Q). For F proper over a field of characteristic 0, we will compare M(F) with the Chow motive associated to F by Toen. Without the properness condition we will show that M(F) is a direct summand of the motive of a smooth quasi-projective\nvariety. Then we will generalize a motivic decomposition theorem due to Karpenko to relative geometrically cellular Deligne-Mumford stacks.\nThis will depend on a vanishing result of Voevodsky. Even in the classical case, our method yields a simpler and more conceptual proof of Karpenko's result.
Rational volume of varieties over complete local fields and Galois extensions
Freitag, 30.11.12, 10:00-11:00, Raum 404, Eckerstr. 1
We are interested in the following question: When does a given variety over a complete local field K have a rational point? The rational volume is a motivic invariant of a K-variety X vanishing if X has no K-rational point. For a tame Galois extension L over K, we will compare the rational volume of a K-variety X and of its base change XL to L. To do so, we construct out of a given weak Néron model of XL with an action of the Galois group of L over K a weak Néron model of X with some universal property. As an application, we will show that some varieties over K with potential good reduction have K-rational points.
Homology stability for special linear group and scissors congruence groups.
Freitag, 7.12.12, 10:00-11:00, Raum 404, Eckerstr. 1
The Scissors Congruence group - or pre-Bloch group - P(F),\nof a field F is a group presented by generators and relations which derive from the five-term functional equation of the dilogarithm.\n\nThe Bloch group is a subgroup of P(F) which, by a result of Suslin, is naturally a quotient of the indecomposable K3 of F, and this in turn is a quotient of the group H3(SL(2,F),Z). Up to some possible 2-torsion, the kernel of the map H3(SL(2,F),Z) to K3^ind(F) coincides with the kernel of the stabilization map from H3(SL(2,F),Z) to H3(SL(3,F),Z). We will describe how, for fields with valuations, lower bounds - and even exact computations - of this latter kernel can be expressed as direct sums of pre-Bloch groups of residue fields.
L2-dbar-cohomology of singular complex spaces
Freitag, 14.12.12, 10:00-11:00, Raum 404, Eckerstr. 1
We will explain how one can use a resolution of singularities to understand the L2-Dolbeault-cohomology of a singular Hermitian complex space \(X\)\n(in the sense of finding a smooth model for the cohomology). The central tool is an L2-resolution for the Grauert-Riemenschneider canonical sheaf of \(X\).\nWe obtain particularly nice results if X is a Gorenstein space with canonical singularities, including an L2-representation of the cohomology of the structure sheaf.\nTo attack the more general case, we introduce a new kind of canonical sheaf, namely the canonical sheaf of square-integrable holomorphic\ntop-degree-forms with some (Dirichlet) boundary condition at the singular set of X. An L2-resolution for this sheaf allows to describe the L2-cohomology of arbitrary isolated singularities.
Pulling back differential forms
Freitag, 21.12.12, 10:00-11:00, Ort noch nicht bekannt
The Hele Shaw Flow and the Moduli of Holomorphic Discs
Freitag, 11.1.13, 10:00-11:00, Raum 404, Eckerstr. 1
The Hele-Shaw Flow is a model for describing the propagation of\nfluid in a cell consisting of two parallel places separated by a small\ngap. This model has been intensely studied for over a century, and is a\nparadigm for understanding more complicated systems such as the flow of\nwater in porous media, melting of ice and models of tumor growth.\n\nIn this talk I will discuss how this flow fits into the more general\nframework of "inverse potential theory" through the idea of complex\nmoments. I will then discuss joint work with David Witt Nystrom that\nconnects to the moduli space of holomorphic discs with boundary in a\ntotally real manifold. Using this we prove a number of short time\nexistence/uniqueness results for the flow, including the case of the Hele\nShaw flow with varying permeability starting from a smooth Jordan domain,\nand for the Hele Shaw flow starting from a single point.
Modified surgery theory - Application to Bott manifolds
Freitag, 18.1.13, 10:00-11:00, Raum 404, Eckerstr. 1
One of the fundamental questions in geometric topology is the question whether two manifolds are diffeomorphic. Surgery theory translated that question as follows: Assuming they are simply connected it now reads: Are our manifolds h-cobordant? But for certain settings yet another transformation of our question is useful leading to modified surgery theory. The key idea here is to compare controlled bordism classes of manifolds.\n\nIn my talk I will explain how one can apply this theory to problems related to Bott manifolds. Bott manifolds are a very nice and explicite class of manifolds given as iterated CP^1-fiber bundles. They are of special interest since, conjecturally, they are diffeomorphic if and only if their cohomology rings are isomorphic.
Canonical degree of curves on varieties of general type
Freitag, 25.1.13, 10:00-11:00, Raum 404, Eckerstr. 1
Bogomolov-Sommese vanishing on log canonical pairs
Freitag, 1.2.13, 10:00-11:00, Raum 404, Eckerstr. 1
Freitag, 8.2.13, 10:00-11:00, tba
Tilting and mutations
Freitag, 8.2.13, 10:00-11:00, Raum 404, Eckerstr. 1
Freitag, 15.2.13, 10:00-11:00, Raum 404, Eckerstr. 1
Log Terminal Singularities (joint work with A. Chiecchio)
Freitag, 8.3.13, 10:00-11:00, Raum 404, Eckerstr. 1
Inspired by the work of de Fernex and Hacon on singularities of normal varieties (2009), we define a new notion for Log Terminal singularities. In this context we prove that the relative canonical ring is finitely generated and that log terminal varieties are klt in the usual sense if and only if the anti-canonical ring is finitely generated. We deduce a relation to the Minimal Model Program and some interesting features on defect ideals.
Transitivity of automorphism groups of Gizatullin surfaces
Dienstag, 12.3.13, 10:00-11:00, Raum 404, Eckerstr. 1