## Final Report Summary - SSGHD (Studies in Symplectic Geometry and Hamiltonian Dynamics)

The project SSGHD is composed of several studies in symplectic geometry and Hamiltonian dynamics.

Below is a detailed description of the main results achieved during the project.

One of the main objectives of the project was to answer the following long-standing question regarding the geometry of the group of Hamiltonian diffeomorphisms, which was raised by Eliashberg and Polterovich in 1993: Is Hofer’s metric the only bi-invariant Finsler metric on the group of Hamiltonian diffeomorphisms. In a joint work with Lev Buhovsky (affiliated with the University of Chicago at the time of the work), we investigated this question, and were able to fully answer it and show that Hofer's metric is indeed unique. More precisely, we proved that for a closed symplectic manifold M, any bi-invariant Finsler pseudo metric on the group of Hamiltonian diffeomorphism, obtained by a pseudo norm on the Lie algebra that is continuous is either identically zero or equivalent to Hofer's metric. On top of settling a longstanding question in symplectic geometry, this work sheds new light on the geometric structure of the group of Hamiltonian diffeomorphisms.

In a joint project with Shiri Artstein-Avidan (Tel-Aviv university), we studied Hamiltonian dynamics on singular energy surfaces. This interdisciplinary work aimed at using symplectic tools in order to approach questions in billiard dynamics. In particular, we applied the theory of symplectic capacities to study Finsler billiard dynamics, which from the point of view of geometric optics, describe the propagation of waves in a nonhomogeneous,anisotropic medium containing perfectly reflecting mirrors. A major ingredient in this work was to extend several results from symplectic geometry regarding smooth energy surfaces to the case of singular hypersurfaces. After achieving this goal, we were able to prove that under convexity assumptions,the length of the shortest periodic Finsler billiard trajectory is a symplectic capacity, and use this result to obtain several bounds and inequalities comparing this quantity and other geometric quantities of the billiard table such as volume, diameter, and inradius.

In a joint work with S. Artstein-Avidan and R. Karasev we obtained a highly surprising result in which we related Mahler's conjecture from the field of asymptotic geometric analysis - a 70-years-old open problem regarding the lower bound of an affine invariant associated with a normed space - with a seemingly different open conjecture, the Viterbo symplectic isoperimetric inequality. We showed that a special case of the later conjecture implies the former. This intriguing result points at a revolutionary and promising direction toward solving Mahler's long-standing conjecture by using symplectic methods.

Together with my student Oded Badt we studied periodic billiard trajectories in hyperbolic geometry. This work is motivated by a remarkable connection between certain polyhedral billiards in the hyperbolic space and solutions to the vacuum Einstein equations in the vicinity of a space-like singularity, which was uncovered in a series of works starting with the pioneering papers of Belinskii, Khalatnikov and Lifshitz. In our work we established the existence of a (n+1)-periodic billiard trajectory inside an n-dimensional regular simplex in the hyperbolic space, which hits the interior of every facet exactly once.

Finally, together with S. Artstein-Avidan and D. Florentin we proved certain inequalities for mixed discriminants of positive semi-definite matrices, and mixed volumes of compact convex sets in R^n. Moreover, we discuss how the latter are related to the monotonicity of an information functional on the class of convex bodies, which is a geometric analogue of the classical Fisher information.

The main impact of the above results is that they open several new promising directions to tackle long-standing conjectures in various fields of mathematics, such as the Mahler conjecture, which is a 70-year old open question in convex geometry.

I have started the SSGHD project in my first year as a senior lecturer on a tenure track at Tel-Aviv University's School of Mathematics, after spending several years as a postdoc in the States (MIT and the Institute for Advanced Study in Princeton). The support provided by this grant has greatly aided my integration in Tel-Aviv. It has allowed the attendance of conferences, working with collaborators abroad and inviting them to working sessions in Israel, as well as supporting students.

The results obtained in this project were published in leading peer-reviewed journals, such as Duke and IMRN, and were presented in several international conferences. In particular, the results have been presented in an invited talk at the International Congress of Mathematics (ICM) 2014, in Korea.

Below is a detailed description of the main results achieved during the project.

One of the main objectives of the project was to answer the following long-standing question regarding the geometry of the group of Hamiltonian diffeomorphisms, which was raised by Eliashberg and Polterovich in 1993: Is Hofer’s metric the only bi-invariant Finsler metric on the group of Hamiltonian diffeomorphisms. In a joint work with Lev Buhovsky (affiliated with the University of Chicago at the time of the work), we investigated this question, and were able to fully answer it and show that Hofer's metric is indeed unique. More precisely, we proved that for a closed symplectic manifold M, any bi-invariant Finsler pseudo metric on the group of Hamiltonian diffeomorphism, obtained by a pseudo norm on the Lie algebra that is continuous is either identically zero or equivalent to Hofer's metric. On top of settling a longstanding question in symplectic geometry, this work sheds new light on the geometric structure of the group of Hamiltonian diffeomorphisms.

In a joint project with Shiri Artstein-Avidan (Tel-Aviv university), we studied Hamiltonian dynamics on singular energy surfaces. This interdisciplinary work aimed at using symplectic tools in order to approach questions in billiard dynamics. In particular, we applied the theory of symplectic capacities to study Finsler billiard dynamics, which from the point of view of geometric optics, describe the propagation of waves in a nonhomogeneous,anisotropic medium containing perfectly reflecting mirrors. A major ingredient in this work was to extend several results from symplectic geometry regarding smooth energy surfaces to the case of singular hypersurfaces. After achieving this goal, we were able to prove that under convexity assumptions,the length of the shortest periodic Finsler billiard trajectory is a symplectic capacity, and use this result to obtain several bounds and inequalities comparing this quantity and other geometric quantities of the billiard table such as volume, diameter, and inradius.

In a joint work with S. Artstein-Avidan and R. Karasev we obtained a highly surprising result in which we related Mahler's conjecture from the field of asymptotic geometric analysis - a 70-years-old open problem regarding the lower bound of an affine invariant associated with a normed space - with a seemingly different open conjecture, the Viterbo symplectic isoperimetric inequality. We showed that a special case of the later conjecture implies the former. This intriguing result points at a revolutionary and promising direction toward solving Mahler's long-standing conjecture by using symplectic methods.

Together with my student Oded Badt we studied periodic billiard trajectories in hyperbolic geometry. This work is motivated by a remarkable connection between certain polyhedral billiards in the hyperbolic space and solutions to the vacuum Einstein equations in the vicinity of a space-like singularity, which was uncovered in a series of works starting with the pioneering papers of Belinskii, Khalatnikov and Lifshitz. In our work we established the existence of a (n+1)-periodic billiard trajectory inside an n-dimensional regular simplex in the hyperbolic space, which hits the interior of every facet exactly once.

Finally, together with S. Artstein-Avidan and D. Florentin we proved certain inequalities for mixed discriminants of positive semi-definite matrices, and mixed volumes of compact convex sets in R^n. Moreover, we discuss how the latter are related to the monotonicity of an information functional on the class of convex bodies, which is a geometric analogue of the classical Fisher information.

The main impact of the above results is that they open several new promising directions to tackle long-standing conjectures in various fields of mathematics, such as the Mahler conjecture, which is a 70-year old open question in convex geometry.

I have started the SSGHD project in my first year as a senior lecturer on a tenure track at Tel-Aviv University's School of Mathematics, after spending several years as a postdoc in the States (MIT and the Institute for Advanced Study in Princeton). The support provided by this grant has greatly aided my integration in Tel-Aviv. It has allowed the attendance of conferences, working with collaborators abroad and inviting them to working sessions in Israel, as well as supporting students.

The results obtained in this project were published in leading peer-reviewed journals, such as Duke and IMRN, and were presented in several international conferences. In particular, the results have been presented in an invited talk at the International Congress of Mathematics (ICM) 2014, in Korea.