Geometry Day in Créteil

November 5th, 2021

• Gilles Courtois, Sorbonne Université
  
• Thi Dang, Heidelberg University
  
• Thi Hanh Vo, University of Luxembourg
  

On a compact Riemannian manifold, the Cheeger's inequality relates the first non zero eigenvalue of the Laplacian of functions with an isoperimetric constant of the manifold. J. Cheeger asked if an analogous inequality would hold for the first non zero eigenvalue of differential forms. We will discuss the case of 1-differential forms. (Joint work with Adrien Boulanger).

In this talk, I will present a work in progress with Jialun Li. Regular Weyl chamber flows of a higher rank symmetric space of non-compact type generalize in a Lie group action sense, the geodesic flow of a hyperbolic surface. For compact hyperbolic surfaces, equidistribution of closed geodesics is due to Bowen and Margulis (1969).
In higher rank and in the compact case, periodic orbits of regular Weyl chamber flow live on tori. Using orbital counting results of Gorodnik-Nevo (2009) and an adaptation of Roblin's ideas, we obtain an equidistribution result of these tori.

We consider the set of closed geodesics on cusped hyperbolic surfaces. Given any non-negative integer k, we are interested in the set of closed geodesics with at least k self-intersections. Among these, we investigate those of minimal length. In this talk, we will discuss their self-intersection numbers.

• Metro 8, stop Créteil-Université: exit to the right and follow the directions for the university;
• RER D, stop Vert-de-Maisons: take the square Maurice Dufourmantelle, cross rue Jean-Jaures and then follow avenue des Petites Haies;
• Bus 181 or 281, stop Créteil-Université.