Zeta regularized determinant and Functorial QFT
Montag, 8.5.17, 16:15-17:15, Raum 404, Eckerstr. 1
An attempt to axiomatize locality of path integrals leads to the notion of functorial quantum field theory (usually known as Atiyah-Segal field theory). In this talk, we will review this notion and briefly indicate how it predicts the gluing relation for the zeta regularized determinant of Laplacian. We will also discuss how to construct a functorial quantum field theory for the scalar field theory. Time permitting, we will outline a construction of functorial quantum field theory arising from two dimensional perturbative quantum scalar field theories.
Spin geometry II
Montag, 15.5.17, 16:15-17:15, Raum 404, Eckerstr. 1
In this talk, we explain how to use Kirby calculus and characteristic sublinks to examine the spinnability of 4-dimensional cobordisms. We will illustrate this approach with some concrete examples.
Spin structures on 3- and 4-manifolds
Montag, 15.5.17, 16:15-17:15, Raum 404, Eckerstr. 1
In this talk, we explain how to use Kirby calculus and characteristic sublinks to describe spin structures on 3-manifolds and the obstruction to extending a given spin structure on the boundary of a 4-dimensional cobordism. We will illustrate this approach with some concrete examples.
Klt varieties with trivial canonical class - holonomy, differential forms, and fundamental groups
Montag, 22.5.17, 16:15-17:15, Raum 404, Eckerstr. 1
We investigate the holonomy group of singular Kähler-Einstein metrics on klt varieties with numerically trivial canonical divisor. Finiteness of the\nnumber of connected components, a Bochner principle for holomorphic tensors,\nand a connection between irreducibility of holonomy representations and stability\nof the tangent sheaf are established. As a consequence, we show that up to finite\nquasi-étale covers, varieties with strongly stable tangent sheaf are either Calabi-Yau (CY) or irreducible holomorphic symplectic (IHS). Finally, finiteness properties of fundamental groups of CY and IHS varieties are established.
Homotopy Theory for Connective Spaces
Montag, 29.5.17, 16:15-17:15, Raum 404, Eckerstr. 1
Connective spaces are a generalization of both graphs and topological spaces.\nThey carry a structure that is somewhat weaker than a topology, yet strong\nenough to support a sort of "algebraic topology". We shall look at their\nproperties, define suitable morphisms and introduce a homotopy theory.\nIn the end we will see a theorem that allows us to directly compare the\nhomotopy groups of manifolds and graphs.\n\n
Mirror symmetry of Calabi-Yau manifolds looked from the moduli spaces
Montag, 12.6.17, 16:15-17:15, Raum 404, Eckerstr. 1
Mirror symmetry of Calabi-Yau manifolds was discovered from physics in 90's. Since then, one way to describe the symmetry is to look at suitable moduli spaces of Calabi-Yau manifolds. In this talk, I will start with a brief summary\nof mirror symmetry, and then I will show two interesting examples of Calabi-Yau manifolds given as complete intersections. In these examples, I will observe that\nbirational geometry of Calabi-Yau manifolds are nicely encoded in the moduli spaces of mirror Calabi-Yau manifolds in terms of monodromy properties. In particular, I will identify Picard-Lefschetz type monodromy which corresponds to flops. This is based on collaborations with Hiromichi Takagi.
Moment maps: from symplectic geometry to G_2 and Spin(7)
Montag, 19.6.17, 16:15-17:15, Raum 404, Eckerstr. 1
After reviewing the classical notions of moment maps in symplectic and hyperkähler geometry, we discuss several generalizations to multisymplectic geometry, where a closed differential form higher degree takes the place of the symplectic form. We describe how these generalizations are related and give further examples for moment maps on manifolds with G_2 or Spin(7)-structures.
On the Dirac equation in Condensed Matter Physics
Montag, 26.6.17, 16:15-17:15, Raum 404, Eckerstr. 1
The Dirac equation has been widely used to build up \nrelativistic models of particles. Recently it made its (somehow \nunexpected) appearance in Condensed Matter Physics. New two-dimensional \nmaterials possessing Dirac fermions low-energy excitations have been \ndiscovered, the most famous being the graphene (2010 Nobel Prize in \nPhysics awarded to Geim-Novoselov). In this talk I will give an overview \nabout the role of the Dirac operator in some condensed matter systems, \nwith particular emphasis on some models and related analytical \nproblems.
Self-Adjoint Fredholm Operators in K-Theory
Montag, 3.7.17, 16:15-17:15, Raum 404, Eckerstr. 1
In this talk we introduce certain ways to think about the group K^1(X):\nFor a compact space X, the group K^1(X) is the Grothendieck group of the monoid of fnite rank complex vector bundles over the suspension SX. \nThere are several classifying spaces for the gorup K^1(X). We study here two of them: the infinite dimensional unitary group and the space of selfadjoint fredholm operator. Both of them are connected via the Cayley transform. \nFinally we will consider the kernel dimension of self-adjoint fredholm operators. Using the Fredholm operators as classifying space of K^1 we get an obstruction for high kernel dimension.
Compactification of a moduli space of lattice polarized K3 surfaces
Montag, 10.7.17, 16:15-17:15, Raum 404, Eckerstr. 1
After a short introduction to hermitian symmetric spaces, I will explain the classical statement that the moduli space of complex elliptic curves is isomorphic to the Siegel modular variety of genus 1. In analogy Clingher and Doran proved that the moduli space of certain lattice polarized K3 surfaces is isomorphic to the Siegel modular variety of genus 2. Finally I will introduce a compactification for this moduli space and show that it is isomorphic to the Baily-Borel compactification of the Siegel modular \nvariety of genus 2.
One can hear the corners of a drum
Montag, 17.7.17, 16:15-17:15, Raum 404, Eckerstr. 1
Analytically computing the spectrum of the Laplacian is impossible\nfor all but a handful of classical examples. Consequently, it can be tricky\nbusiness to determine which geometric features are spectrally determined; such\nfeatures are known as geometric spectral invariants. Weyl demonstrated in 1912\nthat the area of a planar domain is a geometric spectral invariant. In the\n1950s, Pleijel proved that the perimeter is also a spectral invariant. Kac,\nand McKean & Singer independently proved in the 1960s that the Euler\ncharacteristic is a geometric spectral invariant for smoothly bounded domains. \nAt the same time, Kac popularized the isospectral problem for planar domains in\nhis article, "Can one hear the shape of a drum?'' Colloquially, one says that\none can "hear'' spectral invariants. Hence the title of this talk in which we\nwill show that the presence, or lack, of corners is spectrally determined. \nThis talk is based on joint work with Zhiqin Lu. \n
The large scale geometry of the Higgs bundle moduli space
Montag, 24.7.17, 16:15-17:15, Raum 404, Eckerstr. 1
In this talk I will explain recent joint work with Rafe Mazzeo, Hartmut Weiß and Frederik Witt on the asymptotics of the natural L2-metric on the moduli space of rank-2 Higgs bundles over a Riemann surface as given by the set of solutions to the so-called self-duality equations \nfor a unitary connection and a Higgs field. \n\nExtended abstract (including formulas) can be found using the link above.