Principal Investigator: Alexander Givental
The project will pursue various problems of Gromov-Witten theory, that is, the theory of topological invariants of phase spaces of Hamiltonian systems. Our research focuses on the axiomatic structure of Gromov-Witten invariants, their generalizations, their relationships with integrable systems and singularity theory, and methods of their computation, including those associated with Riemann-Roch theorems and the mirror conjecture. A central goal of this project is to resolve or advance a decade-old open problem of expressing Gromov-Witten invariants of Kahler manifolds defined as holomorphic Euler characteristics of complex vector bundles in terms of cohomological invariants of these bundles. Such expression would establish a true "quantum" analogue of the Hirzebruch-Riemann-Roch theorem. As a technical tool, the orbifold version of the classical Hirzebruch-Riemann-Roch theorem will be applied to Kontsevich's moduli spaces of stable maps. Applications of the theory to finite difference equations and integrable systems, representation theory of quantum groups, and the mirror symmetry phenomenon are expected.
From a more general perspective, problems we deal with in our research lie on the crossroad of two major pathways in mathematics of the last 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 the modern landscape of fundamental physics, and generates new mathematical questions and points 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 theory did not really anticipate.
The classical Riemann-Roch formula is considered the central general result in the 19th century theory of complex apgebraic curves, which goes back to Gauss, Abel, Jacobi, and Riemann. It enters to the modern mathematical paradigm in at least two different pathways. In the middle of the 2oth century, in the work of Hirzebruch and Atiyah-Singer, the formula was generalized fromcurves to higher dimensions, becoming the universal and efficient link between algebraic geometry and topology. On the other hand, the string theory raised the demand for the theory of algebraic curves, a.k.a. Riemann surfaces, a.k.a. the strings' world sheets. The present project lies on the intersection of the two pathways. Higher-dimensional complex algebraic spaces (a.k.a. symplectic manifolds, a.k.a. phase spaces of conservative dynamical systems of classical mechanics) become populated with moving strings, and the space of a string's optimaltrajectories becomes the object of study. The Riemann-Roch theory on such spaces is the main achievement of the project. It expresses certain relevant integers (which can be vaguely interpreted as counting the quantum degrees of freedom of the string) in terms of geometry of the space of its classical trajectories. Aa a piece of current research, the project is tightly related to many other areas of mathematical physics (such as, for instance, hypergeometric functions, representation theory, combinatorics, classical and quantum dynamical systems), supplies these areas with new problems, provides solutions to some old ones, and draws methods and motivations from them. It also presents an excellent training field for beginning researchers. More specifically, one graduate student, Valentin Tonita, completed an award-winning PhD thesis at UC Berkeley,and won a competitive research postdoc position at the IMPU (Japan). Yet in a larger perspective, the outcome of this project is just another tread on the stair leading from the ideas of the 19th century classics toward future applications of these ideas to our fuller understanding of the universe.
