Summary of my research
A copy of my research statement can be found
I study symplectic topology, in particular the symplectic topology of Stein
manifolds. Stein manifolds are properly embedded complex submanifolds
of complex Euclidean space. They have a symplectic form coming from this
embedding which is a biholomorphic invariant.
My favourite Stein manifolds are smooth affine varieties.
The tool I use to study these manifolds is a homology theory called symplectic homology.
Recently I have been interested in finding out which algebraic properties
of a smooth affine variety can be recovered from its symplectic structure.
I have also been thinking about using a homology theory
call wrapped Floer homology to study smooth affine varieties.
Exotic Stein manifolds.
In the following papers we study Stein manifolds that are diffeomorphic
but not symplectomorphic to complex Euclidean space. These are called exotic
We construct infinitely many pairwise non-symplectomorphic
exotic Stein manifolds.
We use symplectic homology to distinguish them. The hard part is showing
that we can get some finite invariant from symplectic homology.
My thesis (777k)
Lefschetz fibrations and
If I have two different Hamiltonians then I can ask if some
point p gets sent to the same point q under both flows.
We will call such a point a leafwise intersection point if the second
Hamiltonian flows for time 1 only.
In this paper we construct many Hamiltonians H such that
for any other generic Hamiltonian K there are infinitely many leafwise intersections
sitting on a chosen energy level of H.
This chosen energy level is a contact hypersurface constructed
using Stein manifolds.
embeddings and infinitely many leaf-wise intersections
(joint work with Peter Albers)
Symplectic homology and Lefschetz fibrations.
Lefschetz fibrations are certain fibrations on Stein manifolds
with symplectic fibers. These have a monodromy map which is a symplectomorphism
on the fiber. There is a Floer homology group associated to this monodromy
map and we relate this to symplectic homology. This calculation
can be used to construct many symplectomorphisms with fixed points.
A spectral sequence for
Stein manifolds and smooth affine varieties.
Many Stein manifolds are not symplectomorphic to smooth affine varieties
for topological reasons. For instance there are many Stein manifolds
with infinite topology, but smooth affine varieties are homotopic to finite CW complexes.
Cotangent bundles turn out to be symplectomorphic to Stein manifolds.
Some manifolds such as spheres have cotangent bundles symplectomorphic to smooth affine varieties.
In this paper we show that many cotangent
bundles cannot be symplectomorphic to smooth affine varieties.
The growth rate of symplectic homology and affine varieties
Undecidability of Stein manifolds.
If we have two diffeomorphic Stein manifolds thats are described in an explicit way
then one can ask if there is some algorithm to tell us if they are symplectomorphic
or not. In this paper we show that we cannot find such an algorithm.
If we have a group presentation then in general there is no algorithm
to tell us if it represents the trivial group or not.
In this paper we build an exotic Stein manifold SP associated to each group presentation
These Stein manifolds have the property that if P_1 and P_2 are presentations
representing the trivial group then SP_1 and SP_2 are symplectomoporphic.
Also if P_1 is the trivial presentation and
and if P_3 represents a nontrivial group then SP_1 cannot be symplectomorphic to SP_3.
So if there was an algorithm telling us if SP was symplectomorphic to
the Stein manifold associated to the trivial presentation then
we would also get an algorithm telling us when a particular group
presentation is trivial or not which is impossible.
Computability and the growth rate of symplectic
Infinitely many Reeb orbits.
If we have some compact Riemannian manifold Q then we can construct
its unit cotangent bundle. This is the set of cotangent vectors
of length one. Such a set is a hypersurface in the unit cotangent bundle.
There are other hypersurfaces called fiberwise starshaped hypersurfaces
which are naturally contact manifolds.
There is a non-trivial vector field on this called its Reeb vector field.
The flow of this vector field is a generalization of the geodesic flow.
A flowline which is closed is called a Reeb orbit.
A celebrated result of
Gromoll and Meyer in 1969 showed that Q had infinitely many geodesics
under certain topological conditions.
We generalize this result by showing that each fiberwise starshaped
hypersurface has infinitely many Reeb orbits.
Local Floer homology and
infinitely many simple Reeb orbits