Mathematics

The Seidel maps are two maps associated to a Hamiltonian circle action on a convex symplectic manifold, one on Floer cohomology and one on quantum cohomology. We extend their definitions to S 1 -equivariant Floer cohomology and S 1 -equivariant quantum cohomology based on a construction of Maulik and Okounkov. The S 1 -action used to construct S 1 -equivariant Floer cohomology changes after applying the equivariant Seidel map (a similar phenomenon occurs for S 1 -equivariant quantum cohomology). We show the equivariant Seidel map on S 1 -equivariant quantum cohomology does not commute with the S 1 -equivariant quantum product, unlike the standard Seidel map. We prove an intertwining relation which completely describes the failure of this commutativity as a weighted version of the equivariant Seidel map. We will explore how this intertwining relationship may be interpreted using connections in an upcoming paper. We compute the equivariant Seidel map for rotation actions on the complex plane and on complex projective space, and for the action which rotates the fibres of the tautological line bundle over projective space. Through these examples, we demonstrate how equivariant Seidel maps may be used to compute the S 1 -equivariant quantum product and S 1 -equivariant symplectic cohomology.

In this article, we provide an introduction to an algorithm for constructing Weinstein handlebodies for complements of certain smoothed toric divisors using explicit coordinates and a simple example. This article also serves to welcome newcomers to Weinstein handlebody diagrams and Weinstein Kirby calculus. Finally, we include one complicated example at the end of the article to showcase the algorithm and the types of Weinstein Kirby diagrams it produces.

We give a different and simpler proof of a slightly modified (and weaker) variant of a recent theorem of Shelukhin extending Franks' "two-or-infinitely-many" theorem to Hamiltonian diffeomorphisms in higher dimensions and establishing a sufficiently general case of the Hofer-Zehnder conjecture. A few ingredients of our proof are common with Shelukhin's original argument, the key of which is Seidel's equivariant pair-of-pants product, but the new proof highlights a different aspect of the periodic orbit dynamics of Hamiltonian diffeomorphisms.

This is an expanded version of the talk given be the first author at the conference "Topology, Geometry, and Dynamics: Rokhlin - 100". The purpose of this talk was to explain our current results on classification of rational symplectic 4-manifolds equipped with an anti-symplectic involution. Detailed exposition will appear elsewhere.

Recently in symplectic geometry there arose an interest in bounding various functionals on spaces of matrices. It appears that Grothendieck's theorems about factorization are a useful tool for proving such bounds. In this note we present two such applications.

The goal of this paper is to set up the general framework of higher-dimensional Heegaard Floer homology, define the contact class, and use it to give an obstruction to the Liouville fillability of a contact manifold and a sufficient condition for the Weinstein conjecture to hold. We discuss several classes of examples including those coming from analyzing a close cousin of symplectic Khovanov homology and the analog of the Plamenevskaya invariant of transverse links.

Arboreal singularities are an important class of Lagrangian singularities. They are conical, meaning that they can be understood by studying their links, which are singular Legendrian spaces in S 2n−1 std . Loose Legendrians are a class of Legendrian spaces which satisfy an h --principle, meaning that their geometric classification is in bijective correspondence with their topological types. For the particular case of the linear arboreal singularities, we show that constructable sheaves suffice to detect whether any closed set of an arboreal link is loose.

We establish a stability result for canonical models of arboreal singularities. As a main application, we give a geometric characterization of the canonical models as the closure of the class of smooth germs of Lagrangian submanifolds under the operation of taking iterated transverse Liouville cones. The parametric version of the stability result implies that the space of germs of symplectomorphisms that preserve a canonical model is weakly homotopy equivalent to the space of automorphisms of the corresponding signed rooted tree. Hence the local symplectic topology around a canonical model reduces to combinatorics, even parametrically.

We prove an "h-principle without pre-conditions" for the elimination of tangencies of a Lagrangian submanifold with respect to a Lagrangian distribution. The main result states that the tangencies can always be completely removed at the cost of allowing the Lagrangian to develop certain non-smooth points, called Lagrangian ridges, modeled on the corner {p=|q|}⊂ R 2 together with its products and stabilizations.

In this paper, we study various asymptotic behavior of the infinite family of monotone Lagrangian tori T a,b,c in CP 2 associated to Markov triples (a,b,c) described in \cite{Vi14}. We first prove that the Gromov capacity of the complement CP 2 ∖ T a,b,c is greater than or equal to 1 3 of the area of the complex line for all Markov triple (a,b,c) . We then prove that there is a representative of the family { T a,b,c } whose loci completely miss a metric ball of nonzero size and in particular the loci of the union of the family is not dense in CP 2 .