Parallel spinors, Calabi-Yau manifolds, and special holonomy
Montag, 2.11.20, 16:15-17:15, vSR318 (Kasparov)
Nu-Invariants of Extra Twisted Connected Sums
Montag, 9.11.20, 16:15-17:15, Virtueller SR 318 (Kasparov)
We analyse the possible ways of gluing twisted products of circles\n with asymptotically cylindrical Calabi-Yau manifolds to produce\n manifolds with holonomy \(G_2\),\n thus generalising\n the twisted connected sum construction of Kovalev and Corti,\n Haskins, Nordström, Pacini.\n We then express the extended \(\bnu\)-invariant\n of Crowley, Goette, and Nordström in terms of fixpoint and gluing\n contributions, which include different types\n of (generalised) Dedekind sums.\n Surprisingly, the calculations\n involve some non-trivial number-theoretical arguments connected with\n special values of the Dedekind eta-function and the theory of complex \n multiplication.\n One consequence of our computations is\n that there exist compact \(G_2\)-manifolds that are not \(G_2\)-nullbordant.
Noncommutative differential forms
Freitag, 20.11.20, 14:15-15:15, vSR TF4 (Krush)
Starting with a ring (possibly noncommutative), how can one develop calculus in such a way that, if we start with the ring of functions on an algebraic variety, we get the usual calculus of differential forms? We will do this from the very beginning and without requiring any prior knowledge. Namely, we will start with the basic construction of noncommutative differential forms and explain what has to be added to get a nontrivial theory. We will recover Hochschild and cyclic homology of rings, both in their original version and in the version of Ginzburg and Schedler. We will also show the connection with crystalline cohomology and its generalisation to noncommutative rings. \n
A geometric model for weight variations and wall-crossing on moduli spaces of parabolic Higgs bundles over the Riemann sphere
Montag, 23.11.20, 16:15-17:15, vSR318 (Kasparov)
In this talk I will describe an ongoing project that aims to reconstruct the hyperkähler geometry of Hitchin metrics on moduli spaces of parabolic Higgs bundles over the Riemann sphere in terms of explicit geometric models. By\ndefinition, these moduli spaces depend on a polytope of real parameters called parabolic weights. This dependence induces wall-crossing phenomena, whose incarnation in the models is structurally analogous to a problem of variation of non-reductive GIT-quotients as introduced by Berczi-Jackson-Kirwan. In the smallest possible dimension, these ideas are suited to study the hyperkähler geometry of gravitational instantons of ALG type in terms of the work of Fredrickson–Mazzeo–Swoboda–Weiss.
On SU(2)-bundles on 1-connected spin 7-manifolds
Montag, 30.11.20, 16:15-17:15, vSR318 (Kasparov)
Equivariant Cerf theory and perturbative SU(n) Casson invariants
Montag, 7.12.20, 16:00-17:00, vSR 404 (Lasker)
In 1985, Casson introduced an invariant for integer homology 3-spheres by counting SU(2) representations of the fundamental groups. Boden-Herald generalized the Casson invariant to SU(3) by considering the critical orbits of perturbed Chern-Simons functionals. In this talk, we will present a construction of perturbative SU(n) Casson invariant for all n. The construction is based on an equivariant transversality argument of Wendl. This is joint work with Shaoyun Bai.
Homogenität und Anisotropie in der ART
Montag, 14.12.20, 16:15-17:15, Kasparov
In diesem Vortrag werden wir sehen wie stark uns die\nForderung nach Homogenität und Isotropie bei der Suche nach Lösungen\nder Einsteinschen Feldgleichungen begrenzt. Besonders die Forderung\nnach Isotropie schränkt uns dabei ein und wir werden sehen was für\nModelle des Universums wir erhalten wenn wir diese fallen lassen.\n\n
Monopoles and Landau-Ginzburg Models
Montag, 11.1.21, 16:00-17:00, vSR318 (Kasparov)
Multiplication of BPS states in VOAs from string theory
Freitag, 15.1.21, 14:15-15:15, vSR217 (Steinitz)
In the first part of the talk, I will state some generalities about vertex operator algebras (VOAs). This includes a brief outline of how studying these mathematical objects is justified by their importance to conformal field theory.\n\nThe second part will contain segments of my PhD thesis project. This will make use of a generalized version of VOAs, which is needed, for instance, to formalize those field theories occurring in string theory. The project aims at a mathematically rigorous definition of an algebra structure on states of minimal energy -- so-called Bogomol'nyi-Prasad-Sommerfield (BPS) states --, which was first introduced by Harvey and Moore. While a significant amount of generalization is still work in progress or beyond the scope of the talk, I will try and demonstrate the main concept in the case of torus compactifications.
Simple Singularities and Their Symmetries
Montag, 18.1.21, 16:15-17:15, bbb Konferenzraum 1 (PW Konferenz3210)
We will study simple singularities from various points of view.\nIn the first part, I will give an introduction to the theory of unfoldings. We will see how to use unfoldings to analyse and resolve singularities. An important tool therein will be the Jacobian algebra. \nThen, we will review blowups which provide a different method to resolve singularities. Here, the type of a singularity is determined by the appearance of its exceptional divisor. \nIn both cases, the associated objects allow for actions of symmetry transformations. In the last part of the talk, we will study how to translate between the different perspectives.
On certain lattice polarized K3 surfaces
Montag, 25.1.21, 16:15-17:15, vSR318 (Kasparov)
Let M be an even non-degenerate lattice of signature (1,t). A complex K3 surface X is M-polarized, if there exists a primitive lattice embedding of M into its Picard group Pic(X). \n\nVia such a polarization of the Picard group one is able to encode certain properties of the members of the family of M-polarized K3 surfaces. In this talk we will focus on Kummer surfaces which correspond to the product of two elliptic curves. We will discuss which kind of polarization, i.e. which lattice M, leads to those special Kummer surfaces. \n\nThe bigger picture that these polarized K3 surfaces fit into was described by Dolgachev’s influential paper „Mirror symmetry for lattice polarized K3 surfaces“. We will sketch some of Dolgachev’s insights and give an idea how they can be applied to the Kummer surfaces mentioned above.
tba
Montag, 1.2.21, 16:15-17:15, vSR318 (Kasparov)
Spectral theory of infinite-volume hyperbolic manifolds
Montag, 1.2.21, 16:15-17:15, vSR318 (Kasparov)
In this talk, we define a twisted Laplacian on an orbibundle over a hyperbolic surface (that might be of infinite volume). We prove the meromorphic continuation of the resolvent to the entire complex plane and prove an upper bound on the number of resonances. Additionally, we introduce the corresponding scattering matrix and prove an explicit formula for its determinant in terms of the Weierstrass product over the resonances.\n\nThis is a joint work with M. Doll and A. Pohl.\n\nP.S. The announcement is duplicated, because I have forgotten the password for the previous announcement.
The Laplace on unbounded domains with mixed boundary conditions
Montag, 8.2.21, 16:15-17:15, vSR318 (Kasparov)
In the first part we are going to talk about basic preliminaries to show existence of solutions of the Possion problem.\nIn the second part we will see the invertibility of the Laplace with mixed boundary conditions on manifolds with finite width and bounded geometry. I will also adress the problems in generalising the former proof to the problem with pure Neumann conditions.
Gelfand-Tripel
Montag, 15.2.21, 16:15-17:15, vKasparov
Masse von Sternen und die TOV-Gleichung
Donnerstag, 4.3.21, 12:00-13:00, vKasparov