From a general perspective, problems we deal with in our research lie at the intersection of two major pathways in mathematics of the past two centuries. One of them is the in-depth pursuit of the intricate properties of algebraic curves - in the form inherited from works of Gauss, Abel, Jacobi, Riemann, Klein and Poincare. The other is the broad conceptual landscaping of mathematical physics dictated by the progress of classical and quantum mechanics, and often associated with the names of Hamilton, Maxwell, Gibbs, Poincare, Hilbert, Einstein and Weyl. It is string theory that in the search for the ultimate laws of nature places algebraic curves at the center of modern fundamental physics, generating new mathematical questions and pointing out plausible answers with an amazing pace and persistence. Some of the problems we work on are motivated by such questions, some others hopefully provide answers that string theorist did not really anticipate. The award will be used to support at least three PhD students. The PI's work will also influence the climate in STEM education through his involvement with K-12 and higher level educational projects, such as Math Olympiads, math circles, book publishing, and expository writing.
The specific goals of the project build upon the success of the theory of the so-called K-theoretic Gromov-Witten invariants, developed by the PI among other researchers, and studying topological invariants of phase manifolds of Hamiltonian systems based on properties of vector bundles over moduli spaces of holomorphic curves in these manifolds. From a formal homotopy theory viewpoint, K-theoretic invariants, as well as their cohomological predecessors, should be specializations of much more general cobordism-valued invariants. An approach, referred here as "formal", disregards the subtle stacky or orbifold properties of the moduli spaces, and should be contrasted with the "genuine" theory fully capturing these subtleties. In this project, the PI and his collaborators will move beyond cohomological and K-theoretic Gromov-Witten invariants, and explore the possibility of defining and computing genuine, as opposed to formal, Gromov-Witten invariants with values in complex cobordisms. One direction is to include quantum "chi-y"-theory (based on the Hirzebruch genus), with expected connections with, and applications to representation theory and quiver varieties. Another direction is to explore the specialization to the theory (where y=-1) based on the topological Euler characteristics of moduli spaces of stable maps. Yet another one is to examine the prospects of Gromov-Witten theory based on elliptic cohomology. The general question on the background is: Which extraordinary cohomology theories can home all compact complex orbifolds (as opposed to manifolds)?
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.
