A central question in symplectic geometry is to determine which symplectic manifolds are symplectomorphic. We provide novel tools to answer this question in a new context: We use time-independent incomplete Hamiltonian flows to excise interesting closed subsets of positive codimension from symplectic manifolds. In this talk I will focus on our reduction of this symplectic question to a differential-topological question. In tomorrow's colloquium I will discuss this differential-topological question, but the two talks will be independent of each other. This is joint work with Xiudi Tang, available on the arXiv.

Let $M$ be an open subset of $\mathbb{R}^n$, or, more generally, a manifold. A vector field on $M$ integrates to a flow, which -- unless the vector field is complete -- is not everywhere defined for all times. Its time-one map is a diffeomorphism between open subsets of $M$. We say that this diffeomorphism excises a closed subset $Z$ of $M$ if its domain is the complement of $Z$ in $M$ and its image is all of $M$.

If a closed subset $Z$ of $M$ is diffeomorphic to the epigraph of a lower-semicontinuous function, we build a vector field on $M$ whose time-one flow excises $Z$ from $M$. Examples of such $Z$ include the ray $[0,\infty)$, what we call a "Cantor brush", and a "box with a tail". We do not know if this sufficient condition for excisability is necessary.

This is part of a joint project with Xiudi Tang about excisability in symplectic geometry. In the Geometry and Topology seminar on Wednesday I will discuss other parts of this work, but the two talks will be independent of each other.