Systolic and diastolic estimates in geometry

ended

The project is devoted to two classes of metric invariants of Riemannian and contact manifolds: systoles and diastoles. Riemannian systoles are shortest non-contractible curves (and higher-dimensional cycles) on manifolds and other spaces. Diastoles measure longest curves in an optimal slicing of a manifold into a family of curves (or cycles). Symplectic/contact systoles measure the least action on closed characteristics (integral curves of the Reeb flow). The goals of the project are to extend and complement the results of M. Gromov, M. Freedman, C. Viterbo, and others, by relating all these invariants to the volumes of the corresponding spaces and other metric quantities. A key tool for investigating the systolic freedom (measured as the behavior of several systoles compared to the volume) is its connections with quantum error correction codes, which are a promising rich source of spaces of great systolic freedom. The diastolic geometry, which is the study of waists of slicings/foliations/sweepouts via the methods of geometric analysis, has applications to the open Buser pants decomposition problem, as well as connections with persistence and dimensionality reduction in manifold learning. The symplectic isosystolic/isodiastolic conjecture of Viterbo, which will be studied via extensions of billiard approach, has exciting implications in convexity, namely the longstanding Mahler conjecture on the volume product. My experience in waist and width estimates, combined with the expertise of Prof. Hugo Parlier in systolic geometry and Buser's problem, will help me to carry out this project at the University of Luxembourg. The secondment at Freie Universität Berlin in the quantum group of Prof. Jens Eisert will help me to bring closer the systolic and quantum topics of research as well as the communities of researchers.