Tag - Symplectic geometry

Symplectic implosion was developed to solve the problem that the symplectic cross-section of a Hamiltonian K-space is usually not symplectic, when K is a compact Lie group. The symplectic implosion is a stratified symplectic space, introduced in a 2002 paper of the speaker with Guillemin and Sjamaar. I survey some examples showing how symplectic implosion has been used. I describe the universal imploded cross-section, which is the imploded cross-section of the cotangent bundle of a compact Lie group. Imploded cross-sections are normally not smooth manifolds. We describe some invariants (for example intersection homology) which replace homology for singular stratified spaces.

Triangulated categories play an important role in symplectic topology. The aim of this talk is to explain how to combine triangulated structures with persistence module theory in a geometrically meaningful way. The guiding principle comes from the theory of Lagrangian cobordism.

Let X be a compact symplectic manifold, and D a normal crossings symplectic divisor in X. We give a criterion under which the quantum cohomology of X is the cohomology of a natural deformation of the symplectic cochain complex of X \ D. The criterion can be thought of in terms of the Kodaira dimension of X (which should be non-positive), and the log Kodaira dimension of X \ D (which should be non-negative). The crucial tool is Varolgunes' relative symplectic cohomology.

Viterbo conjectured that a normalized symplectic capacity, on convex domains of a given volume, is maximized for the ball. A stronger version of this conjecture asserts that all normalized symplectic capacities agree on convex domains. Since convexity is not symplectomorphism invariant, one can also ask to what extent these statements still hold for non-convex domains. We survey some special cases and examples around these questions, including recent joint works with Julian Chaidez, and with Jean Gutt and Vinicius Ramos.

I will start by explaining Takahashi's homological mirror symmetry (HMS) conjecture regarding invertible polynomials, which is an open string reinterpretation of Berglund-Hubsch-Henningson mirror symmetry. In joint work with A. Polishchuk, we resolved this HMS conjecture in the chain type case up to rigorous proofs of general statements about Fukaya-Seidel categories. Our proof goes by showing that the categories in both sides are obtained from the category Vect(k) by applying a recursion. I will explain this recursion categorically and sketch the argument for why it is satisfied on the A-side assuming the aforementioned foundational results. If time permits, I will also mention what goes into the proof in the B-side.

The Arnold conjecture about fixed points of Hamiltonian diffeomorphisms was partly motivated by the celebrated Poincare-Birkhoff fixed point theorem for an area-preserving homeomorphism of an annulus in the plane. Despite the fact that the Arnold conjecture was formulated in he smooth setting, several attempts to return to the continuous setting of homeomorphisms and to study the conjecture in this setting has been made afterwards. In this talk I will describe some old and more recent results on the subject.

This talk reports on joint work with Maria Bertozzi, Tara Holm, Emily Maw, Grace Mwakyoma, Ana Rita Pires, and Morgan Weiler on a WiSCon project to investigate the embedding capacity function of the one-point blow up of ℂP². We found three new families of staircases, that are related by symmetries and have other interesting structural features. This talk will explain our findings and our conjectures.

This talk will discuss a new algebraic structure called triangulated persistence category (TPC). It combines the triangulated category structure with the persistence module structure. This algebraic structure can be used to associate a metric topology on the object-set of a triangulated category, which leads to various dynamical questions on a pure algebraic set-up. Many examples are naturally endowed with the TPC structure, for instance, derived Fukaya category, Tamarkin category, etc. In this talk, we will illustrate one algebraic example in depth via extending the Bondal-Kapranov's classical pre-triangulated dg-category to a filtered version. This talk is based on an in-progress project joint with Paul Biran and Octav Cornea.