Forschungsseminar 2023
You will find here abstracts of my guests which are invited to the Forschungsseminar of my research group, see here for a complete schedule.
Schedule
08.05.2023, 14:00-15:45, Alexander Schmitt (Freie Universität Berlin),
"The moduli space of singular principal bundles over the moduli space of stable curves"
Abstract: In the study of moduli spaces of vector or principal bundles over smooth projective curves and their properties, one may use degenerations to singular curves. An example for this approach is Gieseker's proof of a conjecture by Newstead and Ramanan in the rank two case. Motivated by this, Bhosle and the speaker constructed moduli spaces of singular principal bundles over irreducible curves with only nodes as singularities. The analog for reducible curves has been considered in the thesis of Angel Munoz Castaneda. For a given semisimple structure group G and genus g at least 2, there is a universal moduli space M(g,G) of semistable principal G-bundles over the moduli space M(g) of smooth curves of genus g. Using the aforementioned results, Munoz Castaneda and the speaker constructed a moduli space of singular principal G-bundles on stable curves which compactifies M(g,G) relative to the moduli space M(g) of stable curves of genus g, generalizing Pandharipande's construction for the structure group GL(n). Compactifications of M(g,G) which are flat over M(g), but do not have a modular interpretation were obtained by Manon and Belkale/Gibney for the structure group G = SL(n), and by Wilson for simple and simply connected complex Lie groups of type A or C, using vector bundles of conformal blocks. Anderson, Esole, Fredrickson, and Schaposnik have raised similar questions for Higgs bundles in view of possible applications to string theory. In this talk, I will briefly make some layman's remarks about the background from theoretical physics. Then, I will review the classification of holomorphic principal bundles over compact Riemann surfaces. I will present the joint work with Munoz Castaneda and mention Wilson's work on the relation between our moduli space and conformal blocks.
16.05.2023, 14:00-15:30, Michael Wiemeler (Universität Münster),
"On 10-dimensional positively curved manifolds with isometric actions of 3-dimensional tori"
Abstract: I will report on joint work in progress with Anusha Krishnan. In this work we compute the cohomology rings of positively curved manifolds of dimension ten with isometric effective actions of three-dimensional tori.
23.05.2023, 14:00-15:30, Stefan Witzel (Justus-Liebig-Universität Gießen),
T.B.A.
Abstract: T.B.A.
06.06.2023, 14:00-15:30, Michael Jung (Vrije Universiteit Amsterdam),
A geometric computation of cohomotopy sets in codegree one.
Abstract: It is a classical fact that for closed manifolds X the homotopy classes of maps X^n\rightarrow S^n are classified by their degree. The Pontryagin-Thom construction provides a similar construction when X and the sphere have different dimensions, and thus generalizes the notion of degree. In particular, the homotopy classes of maps X^{n+1}\rightarrow S^n are in one-to-one correspondence with framed circles up to framed cobordism in X, and the corresponding set comes equipped with a group structure. In this talk, we introduce the Pontryagin-Thom construction and the concept of framed cobordism classes, and we compute the group of homotopy classes X^{n+1}\rightarrow S^n in terms of topological information of X.
04.07.2023, 14:00-15:30, Jonas Lampart (LICB, Dijon),
"The trajectories of the Schrödinger equation"
Abstract: I will discuss some properties of the set of all trajectories that can be obtained from a fixed initial state by varying the potential in the Schrödinger equation. This is related to the control problem, i.e., driving the system from a fixed initial state to a given target, which turns out to be impossible for "typical" target states using bounded, time-dependent potentials.