The following are some samples of questions that I enjoy thinking about.
Q: How do basic shapes of space change when allowed to evolve under symplectic isotopies?
My dissertation work was related to a problem that is commonly referred to as the ''symplectic camel''. This theorem is based on the image that a camel cannot squeeze through the eye of a needle. Mathematically, it says that if one starts with a standard inclusion of a 4 dimensional ball, then one cannot pass from one half space of 4-space to another disjoint half space joined by a small passageway in the dividing hyperplane.
Q: When are two open subsets of standard symplectic manifolds symplectically equivalent?
Work on the symplectic camel led me to construct invariants for open subsets of euclidean space by studying the space of symplectic embeddings of balls. Namely, for different shapes of spaces, one can prove that the topology of associated embedding spaces changes. As a postdoc, I worked on a much finer invariant for open subsets of euclidean space. These invariants come through symplectic homology. In particular, I developed a notion of symplectic homology that is defined through a finite dimensional technique of generating functions that parallels the symplectic homology theory defined through pseudo-holomorphic curves.
Q: How do symplectic embeddings interact?
The symplectic camel and Gromov's non-squeezing theorem are famous results about embeddings of a single ball. I enjoy thinking about how numerous balls symplectically interact. As a postdoc, I did work finding optimal symplectic packings of a number of symplectic manifolds. Recently, my student Jean Mastrangeli has written a dissertation where she packs open subsets in cotangent bundles of tori. The following picture represents packing a 4-dimensional subset of the cotangent bundle of the 2-dimensional torus with 6 4-dimensional objects.
Q: What are invariants of legendrian links?
When working with symplectic manifolds, balls are a natural object to study under symplectic isotopies. When working in contact manifolds (the odd dimensional analog of symplectic manifolds), studying the movement of knots or links under contact isotopies is natural. In particular, a challenge is to find legendrian knots/links that are topologically equivalent but are not equivalent with respect to a contact isotopy. Recently, I have found such examples of links in a standard contact manifold which is diffeomorphic to the open solid torus. They are pictured below.
Recently, I have been found ways to use the technique of generating
function to associate polynomials to two component links inside a
manifold diffeomorphic to the open solid torus. These polynomials carry
the same information as the polynomials obtained from
linearized contact homology which is defined through the theory of holomorphic
Q: What shapes of space can be achieved with restricted legendrian submanifolds?
In recent years, I have been fascinated with the notion of legendrian tangles. I have been working with a class of tangles that satisfy a minimality condition and been studying how this condition puts restrictions on the types of tangles that can be realized. I find myself learning more and more knot theory as time goes by. I have constructed legendrian invariants that can distinguish topologically equivalent tangles.