My research in math focused on the study of periodic orbits of a Reeb vector field on the 3-sphere. Reeb vector fields
arise in a fairly natural context in Hamiltonian mechanics: to give a large class of unambiguous, concrete, and
extremely natural examples, any Hamiltonian of the
classical form "kinetic energy" + "potential energy" gives rise to a Reeb vector field on the "energy hypersurface" energy =
constant. So, for example, any geodesic flow is a special instance of the flow of a Reeb vector field.
Using cylindrical contact homology I study some instances of the question of whether the existence of a certain orbit set (perhaps
with some more topological or local information about the orbit set) implies the existence of other closed orbits. This is
illustrated with some examples on the three sphere. As an application,
I show how to obtain a
version of a result of Angenent on closed geodesics in the 2-sphere (
see Annals of Mathematics, 2005 ) as a corollary.
We estimate from below the number of leafwise intersections of a bounding restricted contact type hypersurface in an exact
symplectic manifold, assuming the Hofer norm of the Hamiltonian perturbation is less than a certain quantity (the least period among contractible Reeb orbits on the
hypersurface). The example of a disc in the plane demonstrates that some bound on the Hamiltonian is necessary for such a general statement. The estimate on
leafwise intersections is weaker than the estimate coming from computing local Rabinowitz Floer Homology (work of Albers and Frauenfelder), but there are no hypotheses
either on the Hamiltonian being generic or on so-called "periodic leafwise intersections" (ones for which the leaf is
closed).
Preprints, and such...
A Poincare-Birkhoff Theorem for Reeb flows on S^3 (with Umberto Hryniewicz and Pedro
Salomão)- We extend the results of the previous article when the components are degenerate or hyperbolic. We concentrate on
the Hopf link in the 3-sphere, though the techniques can be applied more broadly.
Simply linked orbits in the tight 3-sphere - (with Umberto Hryniewicz and Pedro Salomão) (Working draft) -We
give some conditions on an unknotted, self-linking -1 periodic orbit P which guarantee that there must be another periodic orbit Q which is linked
with P (i.e. Q has linking number 1 with P). We also give some information on the knot type of the orbit Q: it must be (at least) concordant to an unknot.
Finite Energy Foliations and billiards in the plane, in preparation with Peter Albers and Umberto Hryniewicz
The first error
occurs already in the title: I discuss only cylindrical contact
homology, not actually the more general linearized contact homology (-:
N.B. The results of this thesis can be found (and are expanded upon) in the above papers/preprints, so I suggest looking
there instead. I leave this up here anyway for now.
In this thesis, I do the preliminary work on defining a cylindrical
contact homology on the complement of a link of Reeb orbits. We prove
that if the link consists of elliptic Reeb orbits, the contact form is
non-degenerate, and that no closed Reeb orbit on the complement of the
link is contractible (within the complement), then the usual count of
holomorphic cylinders indeed defines a differential for a chain
complex. We discuss briefly possible invariance properties and
techniques for proving these properties: however, this contains only a
sketch of what is involved and not full proofs.
I have found a few mistakes since (for example, the possibility of
branched covers of trivial cylinders is ignored in the proof of the
central theorem) - the appropriate corrections, and natural extensions, can be found in the next article (Contact Homology of orbit complements and implied existence).
Miscellaneous
Here is a link to a talk I gave on some of the work in Contact Homology of orbit
complements and implied existence.
Slides for a talk (with some moving pictures).
A simpler version of the above (still pictures).
Two videos of a trajectory of the 3-body problem (made using matlab/octave and code from here):
this is the first video (2.6 MB, in rotating coordinates) and this is the second video (3.7 MB, in inertial
coordinates) (sorry about the size. Also, I think the videos are not of the same trajectory, even though they look kind of similar. They are both, however, just above
the first critical energy, evident because the orbit visits both primaries.).
An example of simple harmonic motion, courtesy of the wikimedia commons:
