The Loewner equation for multiple slits
Wednesday, 8.4.15, 14:00-15:00, Raum 404, Eckerstr. 1
For which arbitrary set Γ ⊆ C can one measure the growth of Γ (via a differential equation)? One partial answer to this geometrical question can be provided by Loewner Theory.\n\n\nIf Γ ⊆ H admits a homeomorphism γ : [0,1] → Γ such that γ([0,1)) = Γ holds, then we call Γ a slit (in H). We will derive a Loewner equation for finitely many slits. By using [Roth und Schleißinger(2014)] one can now encode the grow of finitely many slits into a differential equation. Hereby we will extend results of [del Monaco und Gumenyuk(2013)]. Furthermore will discuss possible generalizations. \n\n\nLiteratur\n\n[del Monaco und Gumenyuk(2013)] del Monaco, A. und P. Gumenyuk (2013): Chordal Loewner Equation. eprint arxiv:1302.0898v2.\n\n[Roth und Schleißinger(2014)] Roth, O. und S. Schleißinger (2014): The Schramm-Loewner Equation for multiple slits. eprint arxiv:1311.0672v2.
Metric measure spaces with lower Ricci curvature bounds - An introduction
Monday, 27.4.15, 16:15-17:15, Raum 404, Eckerstr. 1
Minimizers of the Allen-Cahn equation on hyperbolic groups
Monday, 18.5.15, 16:15-17:15, Raum 404, Eckerstr. 1
An Index Theorem for Parameterized Euler Characteristic
Monday, 1.6.15, 16:15-17:15, Raum 404, Eckerstr. 1
In this talk, I will first recall some basis of algebraic K-theory, and review an index theorem for unparameterized Euler characteristic. Then by introducing a generalised Poincare duality and Euler class, I will sketch the index theorem for parameterized Euler characteristic.
Universal Dirac bundles, conformal extensions and Dirac-Maxwell-Einstein Theory
Monday, 8.6.15, 16:15-17:15, Raum 404, Eckerstr. 1
Dirac-Maxwell-Einstein Theory is the canonically coupled combination\nof classical electromagnetism and gravity. In order to apply notions like\nmaximal Cauchy developments to it and to allow an interpretation of the theory\nas a variational theory, one has to overcome the problem that spinors for\ndifferent metrics are formally defined as sections of different bundles. The\napproach to this problem presented in this talk is via the definition of a\nfunctor that assigns to every manifold M a vector bundle (though not over M)\nthat carries the information of the spinor bundle. It will turn out to be a\nuniversal objects in various aspects, e.g. as universal Dirac bundle. On the\nother hand, the talk will explore the recent use of conformal extensions to\nobtain results about global existence of Maxwell-Dirac Theory.
The extended Crowley-Nordström nu invariant
Monday, 15.6.15, 16:15-17:15, Raum 404, Eckerstr. 1
Topological G2 structures on a closed 7-manifold M are classified by the Crowley-Nordström nu invariant in Z/48 if the topology of M is sufficiently simple. For metrics with holonomy G2, there is a lift of this invariant to Z. This lift can be used to detect G2 holonomy metrics in different connected components of the moduli space of G2 holonomy metrics on some 7-manifolds M.
Quaternionic lattices and the neighbourhood operator
Monday, 15.6.15, 16:15-17:15, Raum 404, Eckerstr. 1
Kneser's neighbourhood method, a global construction changing the completions of a lattice only at one prime, has frequently been used for classifying lattices up to isometry. I will present a version of this method for quaternionic lattices, i.e. lattices with an additional structure as a module over a maximal order in a definite quaternion algebra over the rationals. Moreover, I will present results obtained by applying this method to even unimodular lattices. These lattices are of special interest since their quaternionic (resp. Siegel) theta series are modular forms for the full quaternionic (resp. Siegel) modular group.\nIn connection with the neighbour method, the neighbourhood operator, an endomorphism of the space of formal linear combinations of lattices of a given genus, is defined. This operator can also be regarded as an operator on the space spanned by the Siegel theta series of degree n of the considered lattices. It follows from formulas describing the effect of certain Hecke operators on theta series that this operator is a Hecke operator. I will present corresponding results in the quaternionic case, but only in the case of the full modular group.\n
Hitchin systems
Monday, 22.6.15, 16:15-17:15, Raum 404, Eckerstr. 1
In 1987 Hitchin introduced a remarkable integrable system on the moduli space of so-called Higgs bundles. Since then it has been extensively studied and generalized in different directions by the work of Faltings, Simpson, Corlette, Donagi, Markman and others. In this talk we confine ourselves to outline the construction of the Hitchin system for a semisimple complex Lie group G and discuss the abelianization process. The latter is at the heart of the integrability of the Hitchin system.
A mirror conjecture and counting of rational curves
Monday, 29.6.15, 16:15-17:15, Raum 404, Eckerstr. 1
Traces of singular moduli
Monday, 6.7.15, 16:15-17:15, Raum 404, Eckerstr. 1
A CM-point is a solution of a quadratic equation (with rational coefficients) in the upper half-plane. Singular moduli are j-values of those CM-points. They have interesting properties, for example they are algebraic integers. This talk will be about traces of singular moduli and Don Zagiers proof of an equation that was first proved by Borcherds in which those traces have a crucial part. All the necessary facts about complex elliptic curves, modular forms of half-integral weight and number theory will be explained in the talk.
Chiral de Rham complex of tori and orbifolds
Monday, 13.7.15, 16:15-17:15, Raum 404, Eckerstr. 1
Metric measure spaces with variable lower curvature bounds
Monday, 20.7.15, 16:15-17:15, SR 404