Systoles-diastoles · Systolic and diastolic estimates in geometry

Systolic and diastolic estimates in geometry

The project was dedicated to two families of geometric invariants, ubiquitous in different areas of theoretical mathematics, as well as computer science, and the physics of quantum computations. These are called systoles and diastoles. Very informally: systoles measure sizes of geometrical objects that cannot be contracted (in a given ambient space) while preserving their topology; diastoles measure sizes of the effective slicings/foliations of a given space. The names are given metaphorically to reflect a (loose) analogy with the heart contraction cycles, but there is no direct connection there. The objectives of the project are split into three groups, based on the underlying mathematics as well as distinct pathways to impact. The first objective was to study intersystolic inequalities to close up the gap of our understanding left after works of M. Gromov and M. Freedman. The relevant impact here is based on the connection of these inequalities with quantum error-correcting codes. Without going into details, a brief way to say what these are is as follows: these codes allow for storing information in a quantum computer, while avoiding decoherence, that is, the loss of entanglement, which is essential for quantum information, for physical reasons. The second objective was to study a family of diastoles, or waists, and even more specifically the waist invariant known as the Urysohn width, and relate them with other metric invariants. On top of various purely geometrical applications (such as Weyl’s law for non-linear spectrum, minimal surfaces, pants decompositions), one relevant point of impact here is based on the connection of the waist with the dimensionality reduction in theoretical computer science. Loosely speaking, in a variety of applications, the questions arise whether a given data point cloud in a high-dimensional space can be approximated by a lower-dimensional geometric shape. One way of formalizing this question leads to waist invariants. The third objective brings together the study of systoles and diastoles in a particular setting of symplectic geometry of the standard even-dimensional phase space. This setting takes its roots in Hamilton’s description of movement, goes a long way throughout the 20th century, through a number of breakthroughs such as Gromov’s non-squeezing phenomenon, and leads to the concept of symplectic capacities, one of which has a systolic nature. The “isoperimetric” question for this capacity has been a widely open conjecture of C. Viterbo until last year, when it was disproved. Yet the remaining follow-up questions allow for applications in convex geometry, namely, for the longstanding Mahler conjecture on the minimality of the volume product. The goal of this part of the project called for the study of the convex bodies whose boundaries satisfy the “systole=diastole” condition; these are expected to be symplectic balls in a certain sense.

Текст от CORDIS, на английски · Данни: CORDIS, © Европейски съюз