Periodic Reporting for period 6 - CSG (C° symplectic geometry)
Periodo di rendicontazione: 2024-03-01 al 2024-03-31
The project addresses a number of challenges in Symplectic Geometry and Topology, a rapidly developing mathematical discipline providing basic tools for modeling a variety of fundamental physical processes. The proposed research focuses on the study continuous objects, as well as properties of smooth objects under small perturbations and uniform limits, in the field of Symplectic Geometry and Topology.
One circle of questions concerns symplectic and Hamiltonian homeomorphisms, and their rigidity and flexibility properties. One proposed direction is related to the action symplectic homeomorphisms on smooth submanifolds. Another direction suggests to study Hamiltonian homeomorphisms in relation to the celebrated Arnold conjecture from Hamiltonian Dynamics. This direction is closely related to the theory of spectral invariants and their behaviour under small perturbations. An additional question deals with the conjectural behaviour of Hamiltonian diffeomorphisms under uniform limits (the so-called C^0 flux conjecture).
A second circle of questions is about the Poisson bracket operator, and its functional-theoretic properties. One direction deals with the conjectured lower bound for the Poisson bracket invariant of a cover, and is connected to Quantum Mechanics. Another direction aims to study rigidity versus flexibility of the Poisson bracket operator in terms of norms coming from Lebesgue spaces. The project successfully developed new tools and mathematical theorems to address the challenges that these questions pose and advance the state-of-the-art in Symplectic Geometry and Topology.
One circle of questions concerns symplectic and Hamiltonian homeomorphisms, and their rigidity and flexibility properties. One proposed direction is related to the action symplectic homeomorphisms on smooth submanifolds. Another direction suggests to study Hamiltonian homeomorphisms in relation to the celebrated Arnold conjecture from Hamiltonian Dynamics. This direction is closely related to the theory of spectral invariants and their behaviour under small perturbations. An additional question deals with the conjectural behaviour of Hamiltonian diffeomorphisms under uniform limits (the so-called C^0 flux conjecture).
A second circle of questions is about the Poisson bracket operator, and its functional-theoretic properties. One direction deals with the conjectured lower bound for the Poisson bracket invariant of a cover, and is connected to Quantum Mechanics. Another direction aims to study rigidity versus flexibility of the Poisson bracket operator in terms of norms coming from Lebesgue spaces. The project successfully developed new tools and mathematical theorems to address the challenges that these questions pose and advance the state-of-the-art in Symplectic Geometry and Topology.
The PI with Alexander Logunov and Shira Tanny have studied the Poisson bracket invariant of the cover. We introduced a new approach to this invariant, which enabled us to prove the lower bound conjectured by L. Polterovich, in dimension 2. We have also achieved partial results in higher dimensions.
The PI with Vincent Humiliere and Sobhan Seyfaddini proved that the spectral norm on the group of Hamiltonian diffeomorphisms of a symplectic manifold M, introduced in the works of Viterbo, Schwarz and Oh, is continuous with respect to the C^0 topology, when M is symplectically aspherical. This statement was previously proven only in the case of closed surfaces. As a corollary, using a recent result of Kislev and Shelukhin, we obtain C^0 continuity of barcodes on aspherical symplectic manifolds, and furthermore define barcodes for Hamiltonian homeomorphisms. We also present several applications to Hofer geometry and dynamics of Hamiltonian homeomorphisms.
Our second main result is related to the Arnold conjecture about fixed points of Hamiltonian diffeomorphisms. The recent example of a Hamiltonian homeomorphism on any closed symplectic manifold of dimension greater than 2 having only one fixed point, shows that the conjecture does not admit a direct generalization to the C^0 setting. However, we demonstrate that a reformulation of the conjecture in terms of fixed points as well as spectral invariants still holds for Hamiltonian homeomorphisms on symplectically aspherical manifolds.
Maksim Stokic (a former PhD student of the PI) constructed a C^0 counterexample to the Lagrangian Arnold conjecture in the cotangent bundle of a closed manifold. Additionally, he proved a quantitative h-principle for subcritical isotropic embeddings in contact manifolds, and provide an explicit construction of a contact homeomorphism which takes a subcritical isotropic curve to a transverse one. On the rigid side, he gave another proof of the Dimitroglou Rizell and Sullivan theorem which states that Legendrian knots are preserved by contact homeomorphisms, provided that their image is smooth. Moreover, his method gives related examples of rigidity in higher dimensions as well.
The PI with Shira Tanny studied a local-to-global inequality for spectral invariants of Hamiltonians whose supports have a “large enough” tubular neighborhood on semipositive symplectic manifolds. In particular, we present the first examples of such an inequality when the Hamiltonians are not necessarily supported in domains with contact type boundaries, or when the ambient manifold is irrational. This extends a series of previous works studying locality phenomena of spectral invariants. A main new tool is a lower bound, in the spirit of Sikorav, for the energy of Floer trajectories that cross the tubular neighborhood against the direction of the negative-gradient vector field.
The PI with Emmanuel Opshtein proved a quantitative h-principle statement for subcritical isotropic embeddings. As an application, we constructed a symplectic homeomorphism that takes a symplectic disc into an isotropic one in dimension at least 6.
Maksim Stokic (a former PhD student of the PI) constructed a compactly supported contact homeomorphism of R^5, with the standard contact structure, which maps a Legendrian disc to a smooth nowhere Legendrian disc.
The PI gave a mini-course and invited lectures on the topics of the grant at many venues including seminars and conference presentations.
The PI with Vincent Humiliere and Sobhan Seyfaddini proved that the spectral norm on the group of Hamiltonian diffeomorphisms of a symplectic manifold M, introduced in the works of Viterbo, Schwarz and Oh, is continuous with respect to the C^0 topology, when M is symplectically aspherical. This statement was previously proven only in the case of closed surfaces. As a corollary, using a recent result of Kislev and Shelukhin, we obtain C^0 continuity of barcodes on aspherical symplectic manifolds, and furthermore define barcodes for Hamiltonian homeomorphisms. We also present several applications to Hofer geometry and dynamics of Hamiltonian homeomorphisms.
Our second main result is related to the Arnold conjecture about fixed points of Hamiltonian diffeomorphisms. The recent example of a Hamiltonian homeomorphism on any closed symplectic manifold of dimension greater than 2 having only one fixed point, shows that the conjecture does not admit a direct generalization to the C^0 setting. However, we demonstrate that a reformulation of the conjecture in terms of fixed points as well as spectral invariants still holds for Hamiltonian homeomorphisms on symplectically aspherical manifolds.
Maksim Stokic (a former PhD student of the PI) constructed a C^0 counterexample to the Lagrangian Arnold conjecture in the cotangent bundle of a closed manifold. Additionally, he proved a quantitative h-principle for subcritical isotropic embeddings in contact manifolds, and provide an explicit construction of a contact homeomorphism which takes a subcritical isotropic curve to a transverse one. On the rigid side, he gave another proof of the Dimitroglou Rizell and Sullivan theorem which states that Legendrian knots are preserved by contact homeomorphisms, provided that their image is smooth. Moreover, his method gives related examples of rigidity in higher dimensions as well.
The PI with Shira Tanny studied a local-to-global inequality for spectral invariants of Hamiltonians whose supports have a “large enough” tubular neighborhood on semipositive symplectic manifolds. In particular, we present the first examples of such an inequality when the Hamiltonians are not necessarily supported in domains with contact type boundaries, or when the ambient manifold is irrational. This extends a series of previous works studying locality phenomena of spectral invariants. A main new tool is a lower bound, in the spirit of Sikorav, for the energy of Floer trajectories that cross the tubular neighborhood against the direction of the negative-gradient vector field.
The PI with Emmanuel Opshtein proved a quantitative h-principle statement for subcritical isotropic embeddings. As an application, we constructed a symplectic homeomorphism that takes a symplectic disc into an isotropic one in dimension at least 6.
Maksim Stokic (a former PhD student of the PI) constructed a compactly supported contact homeomorphism of R^5, with the standard contact structure, which maps a Legendrian disc to a smooth nowhere Legendrian disc.
The PI gave a mini-course and invited lectures on the topics of the grant at many venues including seminars and conference presentations.
Progress was made beyond the state-of-the-art and expected results of the project.
The PI with Alexander Logunov and Misha Sodin constructed a Riemannian metric on the 2-dimensional torus, such that for infinitely many eigenvalues of the Laplace-Beltrami operator, a corresponding eigenfunction has infinitely many isolated critical points.
Ood Shabtai (a former PhD student of the PI and Leonid Polterovich) showed that the operator norm of an arbitrary bivariate polynomial, evaluated on certain spectral projections of spin operators, converges to the maximal value in the semiclassical limit. We contrast this limiting behavior with that of the polynomial when evaluated on random pairs of projections. The discrepancy is a consequence of a type of Slepian spectral concentration phenomenon, which we prove in some cases.
Joint work with Jordan Payette (a former postdoc of the PI), Iosif Polterovich, Leonid Polterovich, Egor Shelukhin and Vukašin Stojisavljević:
Over the years, there have been various attempts to find an appropriate generalization of the Courant theorem statement in different directions. We propose a new take on this problem using ideas from topological data analysis. We show that if one counts the nodal domains in a coarse way, basically ignoring small oscillations, Courant's theorem extends to linear combinations of eigenfunctions, to their products, to other operators, and to higher topological invariants of nodal sets. We also obtain a coarse version of the Bézout estimate for common zeros of linear combinations of eigenfunctions. We show that our results are essentially sharp and that the coarse count is necessary, since these extensions fail in general for the standard count. Our approach combines multiscale polynomial approximation in Sobolev spaces with new results in the theory of persistence modules and barcodes.
The PI with Alexander Logunov and Misha Sodin constructed a Riemannian metric on the 2-dimensional torus, such that for infinitely many eigenvalues of the Laplace-Beltrami operator, a corresponding eigenfunction has infinitely many isolated critical points.
Ood Shabtai (a former PhD student of the PI and Leonid Polterovich) showed that the operator norm of an arbitrary bivariate polynomial, evaluated on certain spectral projections of spin operators, converges to the maximal value in the semiclassical limit. We contrast this limiting behavior with that of the polynomial when evaluated on random pairs of projections. The discrepancy is a consequence of a type of Slepian spectral concentration phenomenon, which we prove in some cases.
Joint work with Jordan Payette (a former postdoc of the PI), Iosif Polterovich, Leonid Polterovich, Egor Shelukhin and Vukašin Stojisavljević:
Over the years, there have been various attempts to find an appropriate generalization of the Courant theorem statement in different directions. We propose a new take on this problem using ideas from topological data analysis. We show that if one counts the nodal domains in a coarse way, basically ignoring small oscillations, Courant's theorem extends to linear combinations of eigenfunctions, to their products, to other operators, and to higher topological invariants of nodal sets. We also obtain a coarse version of the Bézout estimate for common zeros of linear combinations of eigenfunctions. We show that our results are essentially sharp and that the coarse count is necessary, since these extensions fail in general for the standard count. Our approach combines multiscale polynomial approximation in Sobolev spaces with new results in the theory of persistence modules and barcodes.