Oberseminar Geometrie, Topologie und Analysis

Oberseminar Geometrie, Topologie und Analysis

H. Geiges, G. Marinescu, S. Sabatini, D. V. Vu

Sommersemester 2024

Freitag, 10:30-11:30, Seminarraum 2 (Raum 204)

19.4.24 BACH Seminar in Heidelberg

26.4.24 Christian Lehn (Bochum)
Singular varieties with trivial canonical class

Abstract: We will present recent advances in the field of singular varieties with trivial canonical class, obtained in joint work with Bakker and Guenancia, building on work by many others. This includes the decomposition theorem, which says that such a variety is up to a finite cover isomorphic to a product of a torus, irreducible Calabi-Yau and irreducible symplectic varieties. The proof uses a reduction argument to the projective case, which in turn is possible due to advances in deformation theory and a certain result about limits of Kähler Einstein metrics in locally trivial families.

3.5.24 Marcos Salvai (Córdoba)
Tangent ray foliations and their associated outer billiards

Abstract: Let v be a unit vector field on a complete, umbilic (but not totally geodesic) hypersurface N in a space form; for example, on the unit sphere S^2k-1 in R^2k or on a horosphere in hyperbolic space. We give necessary and sufficient conditions on v for the rays with initial velocities v (and -v) to foliate the exterior U of N, or, equivalently, for v to induce an outer billiard map whose billiard table is U. We find relationships among these vector fields and geodesic vector fields. We describe the unit vector fields on N whose associated outer billiard map is volume preserving. We comment on an outer billiard map in the space of oriented lines of hyperbolic 3-space.
Joint work with Yamile Godoy and Michael Harrison.

17.5.24 Simon Vialaret (Paris-Saclay)
Sharp systolic inequalities for invariant contact forms on S¹-principal bundles

Abstract: In Riemannian geometry, a systolic inequality aims to give a uniform bound on the length of the shortest closed geodesic for metrics with fixed volume on a given manifold. This notion generalizes to contact geometry, replacing the geodesic flow by the Reeb flow, and the length by the period. As opposed to the Riemannian case, it is known that there is no systolic inequality for general contact forms on a given contact manifold. In this talk I will state a systolic inequality for invariant contact forms on S¹-principal bundles over the 2-sphere, and will discuss applications to a class of Finsler geodesic flows and to a conjecture of Viterbo.

21.6.24 Floer Lectures in Bochum

5.7.24 BACH Seminar in Bochum

H. Geiges, 15.4.24


19.4.24	BACH Seminar in Heidelberg
26.4.24	Christian Lehn (Bochum) Singular varieties with trivial canonical class Abstract: We will present recent advances in the field of singular varieties with trivial canonical class, obtained in joint work with Bakker and Guenancia, building on work by many others. This includes the decomposition theorem, which says that such a variety is up to a finite cover isomorphic to a product of a torus, irreducible Calabi-Yau and irreducible symplectic varieties. The proof uses a reduction argument to the projective case, which in turn is possible due to advances in deformation theory and a certain result about limits of Kähler Einstein metrics in locally trivial families.
3.5.24	Marcos Salvai (Córdoba) Tangent ray foliations and their associated outer billiards Abstract: Let v be a unit vector field on a complete, umbilic (but not totally geodesic) hypersurface N in a space form; for example, on the unit sphere S^2k-1 in R^2k or on a horosphere in hyperbolic space. We give necessary and sufficient conditions on v for the rays with initial velocities v (and -v) to foliate the exterior U of N, or, equivalently, for v to induce an outer billiard map whose billiard table is U. We find relationships among these vector fields and geodesic vector fields. We describe the unit vector fields on N whose associated outer billiard map is volume preserving. We comment on an outer billiard map in the space of oriented lines of hyperbolic 3-space. Joint work with Yamile Godoy and Michael Harrison.
17.5.24	Simon Vialaret (Paris-Saclay) Sharp systolic inequalities for invariant contact forms on S¹-principal bundles Abstract: In Riemannian geometry, a systolic inequality aims to give a uniform bound on the length of the shortest closed geodesic for metrics with fixed volume on a given manifold. This notion generalizes to contact geometry, replacing the geodesic flow by the Reeb flow, and the length by the period. As opposed to the Riemannian case, it is known that there is no systolic inequality for general contact forms on a given contact manifold. In this talk I will state a systolic inequality for invariant contact forms on S¹-principal bundles over the 2-sphere, and will discuss applications to a class of Finsler geodesic flows and to a conjecture of Viterbo.
21.6.24	Floer Lectures in Bochum
5.7.24	BACH Seminar in Bochum