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.
