Seminar on Metric and Differential Geometry

The seminar takes place on Monday,  16:15 - 17:30, room MA B1 524.


November 11 : Guillaume Buro (EPFL)

 Géométrie Finslérienne de basse régularité.

Résumé :  Un résultat classique, démontré en 1941 par H. Busemann et W.  Mayer, et fréquemment cité en géométrie Finslerienne, affirme qu'une structure Finslerienne sur une variété est déterminée par la fonction distance associée.  Malheureusement l'article original de Busemann-Mayer est d'une lecture difficile et la preuve ne semble jamais avoir l'objet d'une réfaction plus moderne et/ou plus pédagogique.

Le but de cet exposé sera de revisiter le théorème de Busemann-Mayer et de faire le lien avec des recherches actuelles en géométrie métrique et en géométrie Finslerienne de basse régularité.

Nous  montrerons en particulier que  la convexification d’une métrique pré-Finslérienne semi-continue supérieurement induit la même distance que la métrique pré-Finslerienne elle même. Nous montrerons aussi des résultats sur la dérivée métrique et la régularité des courbes minimisantes pour une métrique Finslérienne de basse régularité.

October 28: Judith Abardia (Frankfurt)

 Integral geometry of flag area measures. 

Abstract:  Kinematic formulas are one of the main object of study in integral geometry. They express the average of a geometric functional over a group acting on the space of convex bodies, in terms of some other geometric functionals. In the classical kinematic formulas, the intrinsic volumes are considered and the integral can be expressed in terms of all intrinsic volumes only.

In this talk, I shall present a joint work with Andreas Bernig, where we obtain additive kinematic formulas for smooth flag area measures. A flag area measure on a Euclidean vector space is a continuous and translation-invariant valuation (additive functional from the space of convex bodies) with values in the space of signed measures on a fixed flag manifold.

After stating the existence of such additive kinematic formulas, I will consider the particular case of the flag manifold consisting of a unit vector and a linear subspace of fixed dimension which contains the unit vector. We will first give a basis of these flag area measures and interpret geometrically its elements. The kinematic formulas will be obtained after moving to the dual space of flag area measure and studying its structure of algebra.

October 21: Anna Kausamo (Jyväskylä) 

Optimal mass transportation: from Kantorovich to Monge, from two to many marginals. 

Abstract: Once upon a time there was a French mathematician called Gaspard Monge who set out to explore the problem of transporting mass from one place to another place in an optimal way.  More than 100 years later, a Russian mathematician called Leonid Kantorovich studied the duality between minimizing the cost and maximizing the benefits of the transport. Today we study 'the Monge problem', 'the Kantorovich Duality', and 'The Monge-Kantorovich problem', named in honor of the two founding fathers of the field. In the most classical formulation of the problem, we move mass from one place (formally: from one 'marginal' measure) to another one, and the transporting gets more expensive when the transportation distance increases. But what happens if we have more than two marginals? What changes if the cost function is repulsive, i.e. increases when the distance of the points to be coupled decreases? Why, in particular, does the Monge problem become so difficult when we move from two to many marginals? And what is this Monge problem in the first place ?

 October 14 : Marcos Cossarini (Paris-Est, Marne la Vallée)

Discrete surfaces with length and area and minimal fillings of the circle.

Abstract: We propose to imagine that every Riemannian metric on a surface is discrete at the small scale, made of curves called walls. The length of a curve is its number of crossings with the walls, and the area of the surface is the number of crossings between the walls themselves. We show how to approximate a Riemannian or self-reverse Finsler metric by a wallsystem.

This work is motivated by Gromov's filling area conjecture (FAC) that the hemisphere has minimum area among orientable Riemannian surfaces that fill isometrically a closed curve of given length. (A surface fills its boundary curve isometrically if the distance between each pair of boundary points measured along the surface is not less than the distance measured along the boundary.) We introduce a discrete FAC: every square-celled surface that fills isometrically a 2n-cycle graph has at least n(n-1)/2 squares. This conjecture is equivalent to the FAC extended to surfaces with self-reverse Finsler metric.

If the surface is a disk, the discrete FAC follows from Steinitz's algorithm for transforming curves into pseudolines. This gives a new, combinatorial proof that the FAC holds for disks with Riemannian or self-reverse Finsler metric.

If time allows, we also discuss how to discretize a directed metric on a surface using a triangulation with directed edges. The length of each edge is 1 in one way and 0 in the other way, and the area of the surface is the number of triangles. These discrete surfaces are dual to Postnikov's plabic graphs.