Veranstaltungsübersicht -- Sommersemester 2022

Montag,
2022-04-25
16:15 Uhr
Raum 125, Ernst-Zermelo-Str. 1
Christian Ketterer: Rigidity of mean convex subsets in non-negatively curved $RCD$ spaces and stability of mean curvature bounds

Kasue showed the following theorem. Let $M$ be a Riemannian manifold with non-negative Ricci curvature and mean convex boundary $\partial M=N$ that is disconnected. Then it follows that $M$ is isometric to $[0,D]\times N$. I present a generalization of Kasue's rigidity result for a non-smooth context. For this purpose a synthetic and stable notion of mean curvature bounded from below of subsets in $RCD$ metric measure spaces is introduced. A consequence is a Frankel-type theorem for mean convex subsets in $RCD$ spaces.

Montag,
2022-05-23
16:15 Uhr
Online / SR 125
Luca Vitagliano: Homogeneous G-structures

G-structures unify several interesting geometries including: almost complex, Riemannian, almost symplectic geometry, etc., the integrable versions of which being complex, flat Riemannian, symplectic geometry, etc. Contact manifolds are odd dimensional analogues of symplectic manifolds but, despite this, there is no natural way to understand them as manifolds with an ordinary integrable G-structure. In this talk, we present a possible solution to this discrepancy. Our proposal is based on a new notion of homogeneous G-structures. Interestingly, besides contact, the latter include other nice (old and new) geometries including: cosymplectic, almost contact, and a curious “homogeneous version” of Riemannian geometry. This is joint work with A. G. Tortorella and O. Yudilevich.

Montag,
2022-05-30
16:15 Uhr
Raum 125, Ernst-Zermelo-Str. 1
Jonas Schnitzer: The strong Homotopy Structure of Phase Space Reduction in Deformation Quantization

A Hamiltonian action on a Poisson manifold induces a Poisson structure on a reduced manifold, given by the Poisson version of the Marsden-Weinstein reduction or equivalently the BRST-method. For the latter there is a version in deformation quantization for equivariant star products, i.e. invariant under the action and admitting a quantum momentum map which produces a star product on the reduced manifold. Fixing a Lie group action on a manifold, one can define a curved Lie algebra whose Maurer-Cartan elements are invariant star products together with quantum momentum maps. Star products on the reduced manifold are Maurer-Cartan elements of the usual DGLA of polydifferential operators. Thus, reduction is just a map between these two sets of Maurer-Cartan elements. In my talk I want to show that one can construct an $L_\infty$-morphism, which on the level of Maurer Cartan elements provides a reduction map. This a joint work with Chiara Esposito and Andreas Kraft (arXiv: 2202.08750).

Montag,
2022-07-04
16:15 Uhr
Raum 125, Ernst-Zermelo-Str. 1
Michal Wrochna: Lorentzian complex powers and spectral zeta function densities

The spectral theory of the Laplace–Beltrami operator on Riemannian manifolds is known to be intimately related to geometric invariants. This kind of relationships has inspired many developments in relativistic physics, but a priori it only applies to the case of Euclidean signature. In contrast, the physical setting of Lorentzian manifolds has remained problematic for very fundamental reasons.

In this talk I will present results that demonstrate that there is a well-posed Lorentzian spectral theory nevertheless, and moreover, it is related to Lorentzian geometry in a way that parallels the Euclidean case to a large extent. In particular, in a recent work with Nguyen Viet Dang (Sorbonne Université), we show that the scalar curvature can be obtained as the pole of a spectral zeta function density. The proof indicates that a key role is played by the dynamics of the null geodesic flow and its asymptotic properties.

The primary consequence is that gravity can be obtained from a spectral action; I will also outline furthermore motivation coming from Quantum Field Theory on curved spacetimes.

Montag,
2022-07-11
16:15 Uhr
Raum 125, Ernst-Zermelo-Str. 1
Fabian Kertels: A Lie algebra constructed from BPS states of a torus conformal field theory

Conformal field theories with extended supersymmetry contain a distinguished subspace of BPS (Bogomol'nyi–Prasad–Sommerfield) states. In 1995, physicists Jeffrey Harvey and Gregory Moore proposed a multiplication map to promote such spaces to algebras. I will present my attempt at formalizing this mathematically, specifically for the example theory emerging in the study of a heterotic string on a torus. The result is a Lie bracket on a space obtained from BPS states by a kind of subquotient construction. I intend to highlight, in particular, the very different roles played by the bosonic and fermionic side of the theory in this definition.

Montag,
2022-07-18
16:15 Uhr
Raum 127, Ernst-Zermelo-Str. 1
Marius Amann: Well-posedness of the Laplacian with pure Neumann boundary conditions on domains with corners and cusps

The Laplacian is invertible/fredholm on a smooth domain with Dirichlet/Neumann boundary conditions on the usual sobolev scale. This is still true for the Dirichlet case on a domain with corners, but no longer holds for the Neumann case. I will show that the statement can be recovered by introducing weighted sobolev spaces and furthermore I want to show that a similar statement holds for cusps.

Montag,
2022-09-26
15:00 Uhr
Raum 404, Ernst-Zermelo-Str. 1
Anthony Giaquinto: Deformations, Twists and Frobenius Lie Algebras

Universal deformations formulas (UDFs), or twists, play an important role in the quantization of associative algebras. The history and basic examples of UDFs will be presented along with their connections to the classical Yang-Baxter equation and (quasi)-Frobenius algebras. Some properties and conjectures related to Frobenius Lie algebras will be given.


Abonnieren

Die Einträge dieser Veranstaltung können im iCalender Format abonniert werden. die URL dazu lautet:
http://wochenprogramm.mathematik.uni-freiburg.de/ical/SS2022/Oberseminar%3A%20Differentialgeometrie.ics