Pseudoholomorphic Invariants of Contact Manifolds

Symplectic and contact structures first arose in the study of classical mechanical systems, allowing one to describe the time evolution of complex systems such as planetary motion. In mathematics, certain geometric shapes with these additional structures are known as symplectic or contact manifolds. Solutions of classical mechanical systems can be interpreted in terms of flow lines of mathematical objects known as Hamiltonian or Reeb vector fields on a symplectic or contact manifold, respectively.

Understanding the evolution and distinguishing transformations of these systems led to the development of global invariants of symplectic and contact structures, which shed light on the interconnectedness of dynamics, geometry, and topology. The PI plans to build on her work in providing foundations and applications of contact invariants. The PI will continue and expand her efforts to increase the access and success of underrepresented students in pure and applied mathematics.

With the Rice Association for Women in Mathematics Chapter, she plans to organize a weekly Math Night at Rice University. She also plans to conduct (jointly with two other professors) two national studies to delineate forms of antiracism in academic advising in STEM fields and to examine the effect of academic advisors' practices on BIPOC (Black,Indigenous, and people of color) student psychological experiences and outcomes, and construct a set of best practices and effective behaviors in academic advising.

To train future generations, the PI will co-organize two international conferences with professional development programming for junior mathematicians and will offer research opportunities for undergraduates. She will continue to advise PhD students and mentor postdoctoral researchers.

The project concerns pseudoholomorphic curve based Floer theoretic invariants of contact and symplectic manifolds. These pseudoholomorphic curves are equivalence classes of solutions to a nonlinear Cauchy-Riemann equation which interpolate between closed periodic orbits of either a Hamiltonian or Reeb vector field. The PI will employ direct geometric methods to extend the transversality theory for the associated moduli spaces of pseudoholomorphic curves while isolating and accounting for “errant” phenomena, primarily through the use of obstruction bundle gluing methods.

The primary goals of this project are to provide foundations and refine structural aspects of nonequivariant, and (circle) equivariant contact homology, including the development of product structures and isomorphisms with symplectic homology. By exploring contributions from obstruction bundle gluing, she also plans to provide foundations for Legendrian contact homology in certain closed contact 3-manifolds.

The secondary goals of this project are to provide applications to dynamics, free loop spaces, and symplectic embedding problems, in part through developing computational methods for embedded contact homology of Seifert fiber spaces.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.