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)