Periodic Reporting for period 1 - Systoles-diastoles (Systolic and diastolic estimates in geometry)
Reporting period: 2023-09-01 to 2025-08-31
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.
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.
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.
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.
1) The novel intersystolic inequalities as well as the “dictionary” between systolic and quantum notions extend the reach of systolic rigidity phenomena and provide a framework for transferring well-developed combinatorial and geometric tools into the quantum realm, a connection that had not been systematically explored before.
2) By linking classical notions of width with invariants from persistent homology, the project establishes a new bridge between geometric analysis and topological data analysis. This connection is two-fold enriching: first, it provides a new set of tools for controlling geometric complexity in data-driven contexts; second, it provides new bounds on widths coming from a previously unexpected source—applied topological data analysis. The first steps are done towards a unifying conceptual framework of the homological perspective on diastoles, which can guide future research well beyond the current state of the art.
3) The forthcoming results on Viterbo’s conjecture represent a major advance: the near-resolution of the 4‑dimensional case of lagrangian products, and the reduction of higher-dimensional cases to convex packing problems. The unexpected analogy with the Voronoi conjecture on space-tiling polytopes introduces a novel interdisciplinary perspective, connecting symplectic geometry with convex geometry and the study of quadratic forms. This conceptual leap has the potential to reshape approaches to long-standing open problems in both fields.
Potential impacts and needs for further uptake:
- Scientific impact: new avenues for cross-fertilisation between systolic geometry, quantum physics, and data analysis.
- Further research: formalisation of the multifaceted homological framework for width, deeper exploration of Donaldson divisors in higher-degree systolic inequalities, and the extension of convex packing strategies to broader classes of symplectic manifolds.
- Internationalisation and collaboration: the project has already fostered collaborations across institutions and disciplines; further international partnerships will be essential to fully exploit the interdisciplinary potential.
2) By linking classical notions of width with invariants from persistent homology, the project establishes a new bridge between geometric analysis and topological data analysis. This connection is two-fold enriching: first, it provides a new set of tools for controlling geometric complexity in data-driven contexts; second, it provides new bounds on widths coming from a previously unexpected source—applied topological data analysis. The first steps are done towards a unifying conceptual framework of the homological perspective on diastoles, which can guide future research well beyond the current state of the art.
3) The forthcoming results on Viterbo’s conjecture represent a major advance: the near-resolution of the 4‑dimensional case of lagrangian products, and the reduction of higher-dimensional cases to convex packing problems. The unexpected analogy with the Voronoi conjecture on space-tiling polytopes introduces a novel interdisciplinary perspective, connecting symplectic geometry with convex geometry and the study of quadratic forms. This conceptual leap has the potential to reshape approaches to long-standing open problems in both fields.
Potential impacts and needs for further uptake:
- Scientific impact: new avenues for cross-fertilisation between systolic geometry, quantum physics, and data analysis.
- Further research: formalisation of the multifaceted homological framework for width, deeper exploration of Donaldson divisors in higher-degree systolic inequalities, and the extension of convex packing strategies to broader classes of symplectic manifolds.
- Internationalisation and collaboration: the project has already fostered collaborations across institutions and disciplines; further international partnerships will be essential to fully exploit the interdisciplinary potential.