Subflexible symplectic manifolds, with Emmy Murphy, arXiv:1510.01867.
Abstract:
We introduce a class of Weinstein domains which are sublevel sets of flexible Weinstein manifolds but are not themselves flexible. These manifolds exhibit rather subtle behavior with respect to both holomorphic curve invariants and symplectic flexibility. We construct a large class of examples and prove that every flexible Weinstein manifold can be Weinstein homotoped to have a nonflexible sublevel set.
Squared Dehn twists and deformed symplectic invariants.
In preparation.
Abstract:
We give a complete characterization of those disk bundles over surfaces which embed as rationally convex strictly pseudoconvex domains in C^2. We recall some classical obstructions and prove some deeper ones related to symplectic and contact topology. We explain the close connection to Lagrangian surfaces with isolated singularities and develop techniques for constructing such surfaces. Our proof also gives a complete characterization of Lagrangian surfaces with open Whitney umbrellas, answering a question first posed by Givental in 1986.
A Geometric proof of a faithful linear-categorial surface mapping class group action,
arXiv:1108.3676v1.
Abstract: We give completely combinatorial proofs of the main results of [3] using polygons. Namely, we prove that the mapping class group of a surface with boundary acts faithfully on a finitely-generated linear category. Along the way we prove some foundational results regarding the relevant objects from bordered Heegaard Floer homology,
Stick index of knots and links in the cubic lattice, with Colin Adams, Michelle Chu, Thomas Crawford, Stephanie
Jensen, and Liyang Zhang. Journal of Knot Theory and its Ramifications, Vol. 21, No. 5 (2012).
Abstract: The cubic lattice stick index of a knot type is the least number of
sticks necessary to construct the knot type in the 3-dimensional cubic lattice.
We present the cubic lattice stick index of various knots and links, including
all (p, p + 1)-torus knots, and show how composing and taking satellites can be
used to obtain the cubic lattice stick index for a relatively large infinite class
of knots. We present several bounds relating cubic lattice stick index to other
known invariants. Finally, we define and consider lattice torsion index.
Abstract: We present a computer algorithm which confirms that an approximately
discete subgroup of PSL(2,C) is in fact discrete. The algorithm proceeds
by constructing the Dirichlet domain of the subgroup in H^{3} , and then checks
that the hypotheses of Poincares Theorem for Fundametal Polyhedra are satisfied.
In order to check that the hypotheses are exactly satisfied, we rely on
group theoretical properties resulting from certain geometric conditions of the
Dirichlet domain. We begin in Section 1 with a review of relevant background
material. In Section 2 we provide a formal statement of the problem, including
the restrictions we impose upon the subgroup of PSL(2,C). In Section 3 we
introduce the geometric conditions which must apply for our algorithm to hold,
and we prove that these are generically satisfied. We then show in Section 4
that these conditions are in fact sufficient to verify the hypotheses of Poincare's Theorem. In Section 5 we look at the field containing our matrix
entries.