Hermitian and Unitary representations for affine graded Hecke algebras
Friday, 17.10.14, 15:00-16:00, Raum 127, Eckerstr. 1
Affine graded Hecke algebras and their representations play a\nrole in the representation theory of p-adic groups and special functions. To be able to talk about hermitian modules, we need a star operation. There are two related ones, "star" and "bullet". This talk will discuss their\nrole, in particular the unitary dual for bullet and its relation to work on ladder representations for GL(n).
Overconvergent de Rham-Witt connections
Friday, 7.11.14, 10:15-11:15, Raum 404, Eckerstr. 1
For a smooth scheme over a perfect field of characteristic p>0, we generalise a definition of Bloch and introduce overconvergent de Rham-Witt connections. This provides a tool to extend the comparison morphisms of Davis, Langer and Zink between overconvergent de Rham-Witt cohomology and Monsky-Washnitzer respectively rigid cohomology to coefficients.\nIn this talk I will describe the main constructions and explain how the comparison theorems can be adapted.
Variation of Moduli Spaces of Gieseker-Maruyama-semistable sheaves
Friday, 14.11.14, 10:15-11:15, Raum 404, Eckerstr. 1
Moduli spaces of semistable sheaves over polarized projective manifolds of dimensions greater than one have been constructed by Gieseker and Maruyama using Geometric Invariant Theory. In dimension two their variation as the polarization varies has been thoroughly investigated. In dimension three already irrational polarizations appear in an essential way, for which not even the construction of a corresponding moduli space was known. \nIn this talk we present a joint work together with Daniel Greb and Julius Ross in which we introduce and study a new stability notion allowing to solve the construction and variation problems at least in dimension three. The new moduli spaces are obtained as subschemes in moduli spaces of representations of appropriate quivers.
Degenerate flags and Schubert varieties
Friday, 21.11.14, 10:15-11:15, Raum 404, Eckerstr. 1
Introduced in 2010 by E. Feigin, degenerate flag varieties are degenerations of flag manifolds. It has been proven that, in type A and C, they share many properties with Schubert variety. In this talk I will first recall the classical setting (flag and Schubert varieties) and then discuss joint work with Cerulli Irelli, where we prove a surprising fact about degenerate flags.
Ample subschemes and two conjectures of Hartshorne
Friday, 28.11.14, 10:15-11:15, Raum 127, Eckerstr. 1
The talk will survey geometric properties of subvarieties and cycles with\nvarious positivity properties. We also discuss related conjectures of\nHartshorne and Peternell about subvarieties with ample normal bundle.
Conjugacy classes of \(n\)-tuples in semi-simple Jordan algebras
Friday, 5.12.14, 10:15-11:15, Raum 404, Eckerstr. 1
Let \(J\) be a (complex) semi-simple Jordan algebra, and consider the action of the automorphism group acts\non the \(n\)-fold product of \(J\) via the diagonal action. In the talk, geometric properties of this action\nare studied. In particular, a characterization of the closed orbits is given.\n\nIn the case of a complex reductive linear algebraic group and the adjoint action on the \(n\)-fold product\nof its Lie algebra, a result of R.W. Richardson characterizes the closed orbits. A similar condition can\nbe found in the case of Jordan algebras. It turns out that the orbit through an \(n\)-tuple \(x=(x_1,\bldots, x_n)\)\nis closed if and only if the Jordan subalgebra generated by \(x_1,\bldots, x_n\) is semi-simple.\n\nFor the proof, the existence of certain one-parameter subgroups of the automorphism group is important. Those\none-parameter subgroups have special properties with respect to a given subalgebra of the Jordan algebra \(J\).
On automorphic forms for Calabi-Yau threefolds
Friday, 12.12.14, 10:15-11:15, Raum 404, Eckerstr. 1
I will present a novel approach to relate Hodge theory of elliptic curves to quasimodular forms. Then we consider its generalization to the Hodge theory of Calabi-Yau threefolds, leading to the appearance of a new family of Lie algebras.
Walled Brauer algebra and higher Schur-Weyl duality
Friday, 19.12.14, 10:15-11:15, Raum 404, Eckerstr. 1
Good Reduction of K3 Surfaces
Friday, 16.1.15, 10:15-11:15, Raum 404, Eckerstr. 1
By a classical theorem of Serre and Tate, extending previous results of Néron, Ogg, and Shafarevich, an Abelian variety over a p-adic field has good reduction if and only if the Galois action on its first l-adic cohomology is unramified. In this talk, we show that if the Galois action on second l-adic cohomology of a K3 surface over a p-adic field is unramified, then the surface has admits an ``RDP model'' over the that field, and good reduction (that is, a smooth model) after a finite and unramified extension. (Standing assumption: potential semi-stable reduction for K3's.) Moreover, we give examples where such an unramified extension is really needed. On our way, we establish existence existence and termination of certain semistable flops, and study group actions of models of varieties. This is joint work with Yuya Matsumoto.\n
Motives, nearby cycles and Milnor fibers
Friday, 30.1.15, 10:15-11:15, Raum 404, Eckerstr. 1
Let k be a field of characteristic zero, and f a regular function on a smooth quasi-projective algebraic k-variety. By analogy to the work of Igusa, Denef and Loeser have associated with the function f a Zeta function which is a power series with coefficients in a Grothendieck ring of varieties. Using motivic integration, they have shown that this power series is rational and defined, in the Grothendieck ring, an element viewed as a motivic version of the Milnor fiber. An analytic avatar in rigid geometry of the Milnor fiber has also been introduced by Nicaise and Sebag. \n\n In this talk I will explain how the theory of motives and stable homotopy theory may be used to recover these Milnor fibers and relate them. These results are joint work with J. Ayoub and J. Sebag. I will also discuss and illustrate the advantages obtained by working with motives instead of Grothendieck rings via some open questions in birational geometry.
Ramification theory for D-modules in positive characteristic
Friday, 6.2.15, 10:15-11:15, Raum 404, Eckerstr. 1
Abstract: On a smooth variety in positive characteristic, a\nvector bundle carrying an action of the sheaf of\ndifferential operators is called a stratified bundle. I\nwill give a brief introduction into the theory of these\nobjects and I will explain the notion of regular\nsingularity for stratified bundles. This notion is closely\nrelated to tame ramification of étale coverings. For\nstratified bundles on a curve, I will sketch the beginning\nof a higher ramification theory of stratified bundles,\nanalog to higher ramification theory of étale coverings.
Codimension one foliation with a compact leaf
Friday, 13.2.15, 10:15-11:15, Raum 404, Eckerstr. 1
Abstract: In this talk (based on a joint work with J. Pereira, F. Loray and F. Touzet), we will be interested in the study of codimension one foliations on compact Kähler manifold having a compact leaf. This leaf is then an embedded hypersurface whose normal bundle is topologically torsion and its holonomy representation reflects a major part of the information concerning the transversal dynamic of the foliation. We will be concerned with the following issues: existence of foliations having as a leaf a given hypersurface and foliations with abelian holonomy. Most of these results are stated in terms of Ueda theory and we will spend some time reviewing it.