Frontiers of Quantitative Symplectic and Contact Geometry

A fundamental interest driving symplectic geometry research is the dichotomy between rigidity and flexibitility: is the field more like differential geometry or topology? The goal of this workshop is to focus on quantifiable rigid aspects of this dichotomy.

$C^0$-symplectic topology is a prime example. If a sequence of symplectomorphisms $C^0$-converges to a diffeomorphism, then the diffeomorphism is symplectic. This type of rigidity is surprising because the symplectic condition is on the differential, for which the $C^0$-convergence seems to say nothing. Nonetheless, many related results have since been proven. In particular, new invariants for approximating Hamiltonian dynamics have been constructed, and long-standing simplicity conjectures in dimension 2 are now answered. These results broadly fall under the field of quantitative symplectic geometry, which also studies the displacement energy and Gromov width of Lagrangian submanifolds and symplectic domains.

For contact geometry, the odd-dimensional analogue of symplectic geometry, one can ask the same set of questions. Contact geometry has overtwisted contact structures and loose Legendrians on the flexible side, and tight contact structures and fillable Legendrians on the rigid side. $C^0$-convergence questions, displacement energy, Gromov width, contact Hamiltonian/Reeb mechanics, etc are natural ideas in this contact set-up. But there is less progress: while the symplectic form is canonical, the contact form is not; the even number of dimensions in symplectic geometry naturally connect it to the powerful and long-established tools of complex geometry. Still, there is much activity and success answering these questions, such as translated points, positive loops and orderability, some $C^0$-rigidity, Legendrian displacement, and Lagrangian cobordism length/widths.

The tools behind these results rest on pseudo-holomorphic curves. But unlike usual Gromov-Witten-Floer-type theories which package these curves into algebraic invariants, the curves also keep track of the symplectic action of underlying dynamical objects, namely, the Reeb and Hamiltonian orbits. So in a sense, these results rely on a more refined filtered version of the more-studied Floer theories. These filtrations can be captured by symplectic/contact capacities to obstruct symplectic embeddings or to define spectrums. And more recently, these curves produce the barcodes popularized by data science. A related current trend is to combine the triangulated structure encoded in Fukaya type categories with the filtered nature of the relevant Floer type structures and define new algebraic structures and measurements.

While the questions are diverse, the commonality of the tools that can answer them will bring together the different experts on the cutting-edge of developments and progress in this field.