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.
3. In another article with D. Cristofaro-Gardiner, C.-Y. Mak, V. Humilière and I. Smith, we studied the subleading asymptotics of link spectral invariants and discovered a supervising connection to
the classical Ruelle invariant. Much of this was generalized to higher genus surfaces by Mak and Trifa, a current PhD student.
4. There has been significant progress on understanding of C^0 contact geometry by Baptiste Serraille, a current PhD student.
5. In a different direction, related to the thematics of the ERC proposal, some progress has been obtained towards the smooth closing lemma in higher dimensions, in two articles:
one joint with Erman Cineli, a current postdoc, and another joint with Erman Cineli and Shira Tanny.
6. 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. Vukasin Stojisavljevic, a postdoc of the PI, was involved in
a series of outstanding results on the theory of barcodes and its connections to geometry.
