1) The main achievement for the Objective 1 (intersystolic inequalities): there was a new intersystolic inequality obtained, upper-bounding the product of the systole in degree 1 and the cosystole in degree 1 under _macroscopic_ assumptions. First of all, together with Hannah Alpert and Larry Guth, we proposed a reasonable definition of macroscopically bounded local geometry. Then, we proved an inequality that is parallel to (but independent from) our earlier result in the continuous realm. The inequality is applicable for excluding systolic freedom (and establishing systolic rigidity) in a variety of coarse geometric contexts, broader than it was known before.
As for degrees greater than 1, the main result as of today is the deepened understanding of the phenomena that could force (or provide counter-examples to) intersystolic inequalities in degree 2. While there is no direct analogue for the main tool in degree 1—the Schoen-Yau descent—the closest to that in degree 2 might be the machinery of Donaldson divisors in symplectic geometry.
Over the secondment in Freie Universität Berlin, in collaboration mostly with Armanda Quintavalle, we figured out the “dictionary” translating between systolic and quantum notions. This allows for using in the quantum world the already existing methods developed in the geometric and combinatorial setting—the commonly used tools for (co)isoperimetric inequalities, waist inequalities, and graph expansion.
2) The main achievement for the Objective 2 (bounds on width): a publication in the Transactions of the American Mathematical Society, joint with Baris Coskunuser and Facundo Memoli, dedicated to various methods of geometric control over persistence. Briefly speaking, the machinery of persistent homology was invented to capture the topology of imperfect data point clouds assuming they originate from embedded manifolds. Among the results that we established there are bounds of various widths (Urysohn’s, Alexandrov’s, Kolmogorov’s, as well as other related notions of size), in terms of the lifespans of topological features as seen in the persistence diagram. With the risk of an oversimplification, one can say that we provide bounds for the Urysohn width in terms of quantities in the spirit of Gromov’s filling radius.
More generally, I developed a multifaceted view on the underlying reasons for the width to be bounded: these include a variety of (co)homological reasons (including ones forced by fundamental classes, Steenrod squares, essentiality, and equivariance); this is yet to be formalized properly and written down in a future treatise.
I briefly mention approaches that failed to produce results on width bounds as of now: curve-shortening, hyperbolic geometry considerations, basic geometric measure theory methods. Several other approaches remain to be investigated further.
3) The main achievement for the Objective 3 (symplectic systoles): an almost finished series of works in progress, which is expected to produce two preprints by the end of this year, joint with I. Mitrofanov and A. Polyanskii. One preprint will introduce novel methods for proving special cases of Viterbo’s conjecture, essentially closing for the most part the 4-dimensional case of lagrangian products, which was an initial goal of the project. One of our methods is heavily inspired by billiard dynamics.
The second preprint will consider a higher-dimensional scenario and draw a novel and surprising analogy with the widely open Voronoi conjecture on space-tiling polytopes. I developed a new strategy reducing Viterbo’s conjecture to a convex packing problem, and I described how this strategy applies for the case of the lagrangian products of a simplex with any convex set. I identified which sets give rise to the equality, and the answer is a rich and sophisticated family of space-tiling zonotopes corresponding to the primitive type of quadratic forms in Voronoi’s arithmetic classification.
I mention among the other activities performed that the attempts to generalize the existing work on the type A Toda lattice hasn’t produced results yet; this is might be due to my lack of qualification in the domain of integrable systems, which will be addressed in my future research.
