Locally conformal symplectic (LCS) geometry is a variant of symplectic geometry in which the symplectic form is locally only defined up to positive scale. For example, for the symplectization R x Y of a contact manifold Y, translation in the R direction are symplectomorphisms up to scale, and hence the quotient (R/Z) x Y is naturally an LCS manifold. The importation of symplectic techniques into LCS geometry is somewhat subtle because of this ambiguity of scale.
In 2016, Hutchings introduced a knot filtration on the embedded contact homology (ECH) chain complex in order to estimate the linking of periodic orbits of the Reeb vector field, with an eye towards applications to dynamics on the disk. Since then, the knot filtration has been computed for certain lens spaces by myself, and the "action-linking" relationship has been studied for generic contact forms on general three-manifolds by Bechara Senior-Hryniewicz-Salomao.
I will survey recent progress toward Khovanov homology for links in general 3-manifolds based on categorification of $q$-series invariants labeled by Spin$^c$ structures. Much of the talk will focus on the $q$-series invariants themselves. In particular, I hope to explain how to compute them for a general 3-manifold and to describe some of their properties, e.g. relation to other invariants labeled by Spin or Spin$^c$ structures, such as Turaev torsion, Rokhlin invariants, and the "correction terms'' of the Heegaard Floer theory.
I will discuss recent work with K. Honda and Y. Huang on proving the Giroux correspondence between contact structures and open book decompositions. Though our work extends to all dimensions (with appropriate adjectives), this talk will focus on the 3-dimensional proof. I will first recall Giroux’s argument for existence of supporting open book decompositions, formulating it in the language adapted to our proof. The rest of the talk will be spent describing the proof of the stabilization correspondence.
The study of contact and symplectic manifolds has relied heavily on understanding Legendrian and Lagrangian submanifolds in them -- both for constructing the manifolds using these submanifolds, and for computing invariants of the ambient space in terms of invariants of the submanifolds. This thesis explores the construction of Legendrian submanifolds in high dimensional contact manifolds (greater than 3) in two directions.
Knots in contact manifolds are interesting objects to study. In this talk, I will focus on knots in overtwisted manifolds. There are two types of knots in an overtwisted manifold, one with overtwisted complement (known as loose) and one with tight complement (known as non-loose). Not very surprisingly, non-loose knots behave very mysteriously. They are interesting in their own right as we still do not understand them well.
This week, we'll continue discussing the rational blowdown and use it to construct small exotic 4-manifolds. We will see how we can view the rational blowdown as a "monodromy substitution." Finally, if time allows, we will discuss knot surgery on 4-manifolds.
Contact 3-manifolds arise organically as boundaries of symplectic 4-manifolds, so it’s natural to ask: Given a contact 3-manifold Y, does there exist a symplectic 4-manifold X filling Y in a compatible way? Stein fillability is one such notion of compatibility that can be explored via open books: representations of a 3-manifold by means of a surface with boundary and its self-diffeomorphism, called a monodromy.
