Periodic Reporting for period 4 - SYMPLECTIC (Symplectic Measurements and Hamiltonian Dynamics)
Berichtszeitraum: 2020-09-01 bis 2022-02-28
The field of symplectic geometry historically arose from classical mechanics, where symplectic manifolds provide a natural geometric framework in which to discuss Newton’s equation of motion. In fact, a vast variety of physical processes – from planetary motions in celestial mechanics, to interactions of complex quantum fields – can be described as Hamiltonian dynamical systems, and consequently treated with symplectic tools. Over the past three decades, due in part to motivating questions from theoretical physics, symplectic geometry has evolved into a major research field, and nowadays, symplectic geometry combines a broad spectrum of interrelated disciplines lying in the mainstream of modern mathematics.
The SYMPLECTIC project is composed of several studies in symplectic geometry and Hamiltonian dynamics on topics related to symplectic embedding questions, the geometry of the group of Hamiltonian diffeomorphisms, Lagrangian rigidity questions, and the theory of symplectic measurements. The project’s objectives are twofold. First, to solve several open research questions, which are consider to be pivotal in the field of symplectic geometry. Some of these questions have already been studied intensively, and progress toward solving them would be of considerable significance. Second, some of the studies in this proposal are interdisciplinary by nature, and use symplectic tools in order to address major open questions in other fields, such as the famous Mahler conjecture in convex geometry. The project aims to deepen the connections between symplectic geometry and these fields, thus creating a powerful framework that will allow the consideration of questions currently unattainable.
The SYMPLECTIC project is composed of several studies in symplectic geometry and Hamiltonian dynamics on topics related to symplectic embedding questions, the geometry of the group of Hamiltonian diffeomorphisms, Lagrangian rigidity questions, and the theory of symplectic measurements. The project’s objectives are twofold. First, to solve several open research questions, which are consider to be pivotal in the field of symplectic geometry. Some of these questions have already been studied intensively, and progress toward solving them would be of considerable significance. Second, some of the studies in this proposal are interdisciplinary by nature, and use symplectic tools in order to address major open questions in other fields, such as the famous Mahler conjecture in convex geometry. The project aims to deepen the connections between symplectic geometry and these fields, thus creating a powerful framework that will allow the consideration of questions currently unattainable.
A significant part of the work in the project was devoted to the study of symplectic capacities of domains in the classical phase space. In a joint work with E. Gluskin we
partially settled an old open question regarding to the equivalence of symplectic capacities which states that symplectic capacities are, in fact, indistinguishable on the class of convex domains.
In another joint work together with E. Gluskin we studied the expected value of certain symplectic capacities of randomly rotated convex bodies. This is analogous with similar
situations in other contexts (e.g. the study of Banach-Mazur compactum in asymptotic geometric analysis), where explicit computations of certain quantities are challenging and often
impossible, and thus one applies a probabilistic approach towards exploring them.
Jointly with Shiri Artstein-Avidan, Daniel Florentin, and Daniel Rosen we studied some interrelations between characteristic foliation on the singular hypersurcases in the classical phase
space and caustics in Minkowski billiard dynamics. The latter model of billiard dynamics is partially motivated by the study of light patterns observed in liquid crystal layers. We
established certain unexpected duality results between caustics curves and their dual counter-part when the rule of the table and the Minkowski geometry are switched.
One of the main objectives of the project was to tackle symplectic embedding questions. In a joint work with Vinicius Ramos (IMPA, Brazil) we explored symplectic embedding questions related to the Lagrangian $p$-sum of two discs. We determine precisely when the Lagrangian p-product of two discs can be symplectically embedded into a ball, and vice versa. In particular, we demonstrated new kinds of rigidity and flexibility phenomena in symplectic geometry. This work was published in the “Journal of Topology and Analysis”.
Finally, together with my Ph.D student P. Haim-Kislev, we developed new tools to estimate symplectic capacities based on certain decomposition schemes of convex bodies. Moreover, we extended the classical product-formula of certain symplectic invariants to the case of $p$-products. These novel results hold promise in tackling the well-known Viterbo’s volume-capacity conjecture from a new perspective.
partially settled an old open question regarding to the equivalence of symplectic capacities which states that symplectic capacities are, in fact, indistinguishable on the class of convex domains.
In another joint work together with E. Gluskin we studied the expected value of certain symplectic capacities of randomly rotated convex bodies. This is analogous with similar
situations in other contexts (e.g. the study of Banach-Mazur compactum in asymptotic geometric analysis), where explicit computations of certain quantities are challenging and often
impossible, and thus one applies a probabilistic approach towards exploring them.
Jointly with Shiri Artstein-Avidan, Daniel Florentin, and Daniel Rosen we studied some interrelations between characteristic foliation on the singular hypersurcases in the classical phase
space and caustics in Minkowski billiard dynamics. The latter model of billiard dynamics is partially motivated by the study of light patterns observed in liquid crystal layers. We
established certain unexpected duality results between caustics curves and their dual counter-part when the rule of the table and the Minkowski geometry are switched.
One of the main objectives of the project was to tackle symplectic embedding questions. In a joint work with Vinicius Ramos (IMPA, Brazil) we explored symplectic embedding questions related to the Lagrangian $p$-sum of two discs. We determine precisely when the Lagrangian p-product of two discs can be symplectically embedded into a ball, and vice versa. In particular, we demonstrated new kinds of rigidity and flexibility phenomena in symplectic geometry. This work was published in the “Journal of Topology and Analysis”.
Finally, together with my Ph.D student P. Haim-Kislev, we developed new tools to estimate symplectic capacities based on certain decomposition schemes of convex bodies. Moreover, we extended the classical product-formula of certain symplectic invariants to the case of $p$-products. These novel results hold promise in tackling the well-known Viterbo’s volume-capacity conjecture from a new perspective.
I plan to continue my research on the theory of symplectic measurements, as well as to complete the research projects suggested in my ERC project
proposal. In particular, I plan to study characteristic foliation on the boundary of convex polytopes in the classical phase space, aiming at having better understanding on how to read
“symplectic information" from the “combinatorial data" of a polytope. In the long run we hope that this will shed light on computational aspects of Hamiltonian dynamics and symplectic
invariants, and lead to breakthroughs in some of the open conjectures concerning symplectic invariants, either by being a source for possible counterexamples, or by providing extra
information in order to tackle them.
proposal. In particular, I plan to study characteristic foliation on the boundary of convex polytopes in the classical phase space, aiming at having better understanding on how to read
“symplectic information" from the “combinatorial data" of a polytope. In the long run we hope that this will shed light on computational aspects of Hamiltonian dynamics and symplectic
invariants, and lead to breakthroughs in some of the open conjectures concerning symplectic invariants, either by being a source for possible counterexamples, or by providing extra
information in order to tackle them.