Scalar curvature of definable Alexandrov spaces.
Sharp isoperimetric inequalities and model spaces for the Curvature-Dimension-Diameter condition
We obtain new sharp isoperimetric inequalities on a Riemannian manifold equipped with a probability measure, whose generalized Ricci curvature is bounded from below (possibly negatively), and generalized dimension and diameter of the convex support are bounded from above (possibly infinitely). Our inequalities are sharp for sets of any given measure and with respect to all parameters (curvature, dimension and diameter). Moreover, for each choice of parameters, we identify the model spaces which...
Simplicial nonpositive curvature
We introduce a family of conditions on a simplicial complex that we call local k-largeness (k≥6 is an integer). They are simply stated, combinatorial and easily checkable. One of our themes is that local 6-largeness is a good analogue of the non-positive curvature: locally 6-large spaces have many properties similar to non-positively curved ones. However, local 6-largeness neither implies nor is implied by non-positive curvature of the standard metric. One can think of these results as a higher...
Some results on curvature and topology of Finsler manifolds
We investigate the curvature and topology of Finsler manifolds, mainly the growth of the fundamental group. By choosing a new counting function for the fundamental group that does not rely on the generators, we are able to discuss the topic in a more general case, namely, we do not demand that the manifold is compact or the fundamental group is finitely generated. Among other things, we prove that the fundamental group of a forward complete and noncompact Finsler n-manifold (M,F) with nonnegative...
Some results on the geometry of convex hulls in manifolds of pinched negative curvature.
Stability problems in a theorem of F. Schur.
Stable norms of non-orientable surfaces
We study the stable norm on the first homology of a closed non-orientable surface equipped with a Riemannian metric. We prove that in every conformal class there exists a metric whose stable norm is polyhedral. Furthermore the stable norm is never strictly convex if the first Betti number of the surface is greater than two.
Structure quasi-conforme et dimension conforme d'après P. Pansu, M. Gromov et M. Bourdon
Suites de flots de Ricci en dimension 3 et applications
Dans cet article, on passe en revue certains résultats dus à Miles Simon sur le flot de Ricci de certains espaces métriques de dimension 3 exposés dans [28] et [26].On commence par voir le lien entre théorèmes de rigidité et convergence des variétés sur un exemple dû à Berger et Durumeric. On remarque ensuite que pour obtenir de tels théorèmes de rigidité en utilisant le flot de Ricci, il faut être capable de construire le flot pour des espaces peu lisses.Les deux dernières partie sont consacrées...
Sur des problèmes de la géométrie systolique
Sur le problème de Bennequin : un contre-exemple (d'après Alexander)
Symplectic Topology and Extemal Rays.
Systolic invariants of groups and -complexes via Grushko decomposition
We prove a finiteness result for the systolic area of groups. Namely, we show that there are only finitely many possible unfree factors of fundamental groups of -complexes whose systolic area is uniformly bounded. We also show that the number of freely indecomposable such groups grows at least exponentially with the bound on the systolic area. Furthermore, we prove a uniform systolic inequality for all -complexes with unfree fundamental group that improves the previously known bounds in this dimension....