Periodic Reporting for period 2 - HSD (Homeomorphisms in symplectic topology and dynamics)
Période du rapport: 2021-07-01 au 2022-12-31
The HSD project is centered around fundamental open questions in the field of continuous symplectic topology.
This is an area of symplectic topology which defines and studies continuous analogues of smooth symplectic objects,
such as symplectic/Hamiltonian homeomorphisms, and asks questions about persistence of various symplectic phenomena under
uniform limits and perturbations.
The aim of the project is to explore, and further develop, continuous symplectic topology and to make progress towards
the resolution of longstanding open problems which have served as motivation for the field. On the one hand, the project
proposes to develop new tools for the advancement of the field via the medium of barcodes which will serve
as a replacement of Floer homology for homeomorphisms. On the other hand, it proposes new approaches
towards several important questions in the field including the symplectic four-sphere problem which asks if
non-symplectic manifolds, such as the four-sphere, could admit the structure of a topological symplectic manifold,
and the simplicity conjecture which asks if the group of compactly supported area-preserving homeomorphisms
of the disc is a simple group.
This is an area of symplectic topology which defines and studies continuous analogues of smooth symplectic objects,
such as symplectic/Hamiltonian homeomorphisms, and asks questions about persistence of various symplectic phenomena under
uniform limits and perturbations.
The aim of the project is to explore, and further develop, continuous symplectic topology and to make progress towards
the resolution of longstanding open problems which have served as motivation for the field. On the one hand, the project
proposes to develop new tools for the advancement of the field via the medium of barcodes which will serve
as a replacement of Floer homology for homeomorphisms. On the other hand, it proposes new approaches
towards several important questions in the field including the symplectic four-sphere problem which asks if
non-symplectic manifolds, such as the four-sphere, could admit the structure of a topological symplectic manifold,
and the simplicity conjecture which asks if the group of compactly supported area-preserving homeomorphisms
of the disc is a simple group.
The project has advanced successfully on several fronts. First and foremost, the strategy outlined in the project for approaching the simplicity conjecture
has led to a successful resolution of the conjecture in a joint article with D. Cristofaro-Gardiner and V. Humilière. The solution to the simplicity conjecture
motivated a number of other works which have resulted in the resolution of several other outstanding problems in symplectic topology and dynamical systems:
1. The resolution of the Kapovich-Polterovich question on the large scale geometry of Hofer's metric on the two-dimensional sphere. This was carried out
jointly with D. Cristofaro-Gardiner and V. Humilière. Another solution to this problem was published simultaneously and independently by Polterovich and Shelukhin.
2. In a joint work with D. Cristofaro-Gardiner, C.-Y. Mak, V. Humilière and I. Smith, we resolved several open questions from topological surface dynamics
and continuous symplectic topology: we show that the group of Hamiltonian homeomorphisms of any compact surface with (possibly empty) boundary is not simple;
we extend the Calabi homomorphism to the group of Hameomorphisms constructed by Oh-Müller; and, we construct an infinite dimensional family of quasimorphisms
on the group of area and orientation preserving homeomorphisms of the two sphere.
Beyond the above results, which were all motivated by the simplicity conjecture, significant progress has also been obtained towards the development of the theory
of barcodes (on negative monotone symplectic manifolds) by Y. Kawamoto who was a doctoral student of the PI.
has led to a successful resolution of the conjecture in a joint article with D. Cristofaro-Gardiner and V. Humilière. The solution to the simplicity conjecture
motivated a number of other works which have resulted in the resolution of several other outstanding problems in symplectic topology and dynamical systems:
1. The resolution of the Kapovich-Polterovich question on the large scale geometry of Hofer's metric on the two-dimensional sphere. This was carried out
jointly with D. Cristofaro-Gardiner and V. Humilière. Another solution to this problem was published simultaneously and independently by Polterovich and Shelukhin.
2. In a joint work with D. Cristofaro-Gardiner, C.-Y. Mak, V. Humilière and I. Smith, we resolved several open questions from topological surface dynamics
and continuous symplectic topology: we show that the group of Hamiltonian homeomorphisms of any compact surface with (possibly empty) boundary is not simple;
we extend the Calabi homomorphism to the group of Hameomorphisms constructed by Oh-Müller; and, we construct an infinite dimensional family of quasimorphisms
on the group of area and orientation preserving homeomorphisms of the two sphere.
Beyond the above results, which were all motivated by the simplicity conjecture, significant progress has also been obtained towards the development of the theory
of barcodes (on negative monotone symplectic manifolds) by Y. Kawamoto who was a doctoral student of the PI.
In the second half of the project, the PI plans to continue the study of Floer theoretic invariants which led to the solution of the simplicity conjecture.
A recent article, written jointly with D. Cristofaro-Gardiner, C.-Y. Mak, V. Humilière and I. Smith, outlines a relation between these Floer theoretic
invariants and a classical invariant of area-preserving maps called the Ruelle invariant. Further, exploration of this direction of research seems very promising.
Furthermore, the PI intends continue studying the open problems outlined in the ERC proposal such as the four-sphere problem, further development of the theory
barcodes and the study of continuous contact geometry.
A recent article, written jointly with D. Cristofaro-Gardiner, C.-Y. Mak, V. Humilière and I. Smith, outlines a relation between these Floer theoretic
invariants and a classical invariant of area-preserving maps called the Ruelle invariant. Further, exploration of this direction of research seems very promising.
Furthermore, the PI intends continue studying the open problems outlined in the ERC proposal such as the four-sphere problem, further development of the theory
barcodes and the study of continuous contact geometry.