Systolic and Diastolic Geometry
IMPA, Rio de Janeiro, Março 17 – 21
Inequalities that establish a connection between the total volume of a manifold and the “size” of objects inside it are usually the manifestation of deep, fundamental geometric phenomena. For instance, the classical isoperi- metric inequality asserts that among planar compact regions of area π, the length of the boundary cannot be larger than 2π.
In this context, a particularly interesting class of objects have a systolic or diastolic character, by which we mean that they are realised as an ab- solute minimum or a min-max critical point of some geometric functional, respectively. Examples of objects that fit into this description are shortest homotopically non-trivial loops (i.e. the classical “systoles”, in the terminol- ogy of Marcel Berger), shortest periodic billiard orbits, symplectic capacities, min-max minimal surfaces and Laplace eigenvalues.
Although this line of investigation has a long history, in the last two decades there have been enormous advances in establishing sharp isosystolic and isodiastolic inequalties related to such objects, and in characterising the optimal shapes. However, despite obvious analogies, some of these develop- ments have been parallel, and one could hope for a convergence, so that ideas that have been useful in one area inspire fruitful approaches in other areas.
We would like to propose an interdisciplinary workshop about this topic, with the aim of joining together leading specialists on such different areas and propitiate a stimulating occasion for the participants to learn the state of the art of several isosystolic and isodiastolic problems (e.g. Calabi-Croke Conjecture), to be inspired by different perspectives on this class of prob- lems, and to advance new conjectures and directions for the future of this fundamental line of investigation.
Registration – TBA
Program – TBA
Speakers – TBA
Organizing Committee
Lucas Coelho Ambrozio (IMPA)
Vinicius Ramos Gripp (IMPA)
Umberto Leone Hryniewicz (Aachen University)
