26th October 2022.

8:30 Meeting in front of Seminar Raum 4 (Mathematikon)

9:00-9:45 Tania Bossio (University of Padova)

Title: On the volume of (half)-tubular neighborhoods of surfaces in sub-Riemannian geometry

In 1840 Steiner proved that the volume of the tubular neighborhood of a convex set in Rn is polynomial of degree n in the ”size” of the tube. The coefficients of such a polynomial carry information about the curvature of the set. In this talk, we investigate the validity of Steiner-like formulas in the case of half-tubes built over surfaces embedded in a three-dimensional sub- Riemannian contact manifold with respect to the sub-Riemannian distance. We extend the known results in the Heisenberg group, deriving an asymptotic expansion for the volume of the half-tube and provide a geometric interpretation of the coefficients in terms of sub-Riemannian curvature objects. This is a joint work with Davide Barilari.

Caffe break (Besprechungsraum 03.414)

10:30-11:15 Ana Chavez Caliz (Heidelberg University)

Title: Projective self-dual polygons

In his book "Arnold's Problems," Vladimir Arnold shares a collection of questions without answers formulated during seminars in Moscow and Paris for over 40 years. One of these problems, stated in 1994, goes as follows: find all projective curves projectively equivalent to their duals. The answer seems to be unknown even in projective 2-dimensional space. Motivated by this question, in their paper "Self-dual polygons and self-dual curves" from 2009, D. Fuchs and S. Tabachnikov explore a discrete version of Arnold's question in 2-dimensions. If P is an n-gon with vertices A1, A3, A5, ..., then its dual polygon P* has vertices B*2, B*4, ..., where B*i is the line connecting two consecutive vertices of P. Given an integer m, a polygon P is m-self-dual if there is a projective transformation f such that f(Ai) = B{i+m}. In this talk, I will discuss how we can generalize Fuchs and Tabachnikov's work to polygons in higher dimensions. I will include some conjectures about the pentagram map, supported by computational results.

11:45-12:30 Cèdric Oms (ENS Lyon)

Title: From convexity of contact structures to singular ones

Convexity played since its introduction by Giroux in the 90's a fundamental tool for the study of contact structures. In this talk we will review the basic notions and will see how to use convexity to study the homotopical classification of b-contact structures in a neighbourhood of a convex surface. We will then use to establish an h-principle for a generalization of contact structures, so called b-contact structures. This talk is based on joint work with Robert Cardona.

Common Lunch (Besprechungsraum 03.414)

14:15-15:00 Steffen Schmidt (Heidelberg University)

Title: A (super)geometric investigation

In representation theory we study vector spaces on which a symmetry group of a geometric object can act on. While the classification can often be reduced to pure algebraic considerations one hopes to give an explicit geometric realization. A famous result for complex semisimple Lie groups is the Borel-Weil-Bott theory which realizes the finite-dimensional irreducible representations as suitable sheaf cohomology groups. In this talk we use the BWB theory together with its supergeometric extension to study an interesting class of representations, the unitary representations of the complex Lorentz group and the superunitary representations of its smallest supersymmetric extension. From a pure representation theoretical point of view it is a remarkable observation that the complex Lorentz group has no finite-dimensional unitary representations while its smallest supersymmetric extension has finite-dimensional (super)unitary representations. In this talk I will translate the representation theoretical observations to the existence and non-existence of suitable geometric structures and explain how in the view of (super) representation theory supergeometry is not more than geometry.

Caffe break (Besprechungsraum 03.414)

15:45-16:30 Adam Chalumeau (Strasbourg University)

Title: Kobayashi hyperbolicity in Lorentzian Geometry.

Kobayashi has introduced a notion of "hyperbolicity" for complex manifolds. Hyperbolic complex manifolds are ones that can be endowed with a distance which turns biholomorphic mappings into isometries. This can be used to go from a "complex-geometric problem" to a "metric space problem". This notion of hyperbolicity is also well defined in projective geometry and in flat conformal riemannian geometry. For flat conformal riemannian geometry, Apanasov, Kulkarni and Pinkall have proved a stunning regularity result : they produce a C^{1,1} conformally invariant metric. In my talk, I will present a similar notion of hyperbolicity for flat conformal lorentzian manifolds and I will give examples of such manifolds. To do so, I will present the geometry of Einstein's static universe and I will describe the dynamics of it's conformal group PO(2,n).