CAREER: Floer theories and Reeb dynamics of contact manifolds

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This award is funded in whole or in part under the American Rescue Plan Act of 2021 (Public Law 117-2). The mathematical structures known as contact and symplectic manifolds have their origins in the study of classical mechanical systems from physics, which allow one to describe systems such as planetary motion and wave propagation. The equations of motion can be described in terms of mathematical objects known as flow lines of Hamiltonian and Reeb vector fields. Understanding the dynamics of these vector fields led to the development of global Floer theoretic invariants of symplectic and contact manifolds. By developing foundations and applications of these invariants, the PI's work will illuminate the interconnectedness of dynamics, embeddings, knot theory, geometry, and topology. The PI will expand her efforts to increase the access and success of underrepresented students in mathematics and academia more broadly. She is conducting, jointly with others, 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) PhD student psychological experiences and outcomes. Based on the results of the studies, she will 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 early-career mathematicians, including a panel discussion on academic jobs for graduate students. She will also develop and implement summer research programs for undergraduates that will include graduate students and a postdoctoral researcher.The project concerns Floer theoretic invariants and Reeb dynamics of contact manifolds. The dynamics of Reeb vector fields are subtly related to the underlying contact structure as well as the topology and geometry of the underlying manifold. Reeb vector fields realize distance minimizing flow and arise as the restriction of Hamiltonian vector fields to contact type hypersurfaces and boundaries of symplectic manifolds. Reeb dynamics additionally govern embeddings between symplectic manifolds with contact type boundary as well as the topology of symplectic fillings of contact manifolds. The PI's research has a particular emphasis on the study of closed periodic Reeb orbits (circular flow lines) by way of the development of various Floer theories, so as to capture different dynamical, geometric, and topological phenomenon. She will also provide applications to surface dynamics and the study of contact and symplectic manifolds. The project includes community building, mentorship, recruitment, and retention programming for undergraduate and graduate students in mathematics, as well as the training of students in communication skills.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.