Algebraic Geometry of Hitchin Integrable Systems and Beyond

This research project focuses on algebraic geometry and moduli spaces. Algebraic geometry is the study of varieties, which are in turn the sets of solutions of polynomial equations. Moduli spaces are parameter spaces of varieties, which concern the behavior of varieties as the defining polynomials vary. In the last few decades, fundamental connections have been found relating moduli spaces in algebraic geometry to other fields including representation theory, topology, and quantum field theory in mathematical physics. This project aims to study several classes of moduli spaces which lie at the crossroads of central areas in mathematics and physics. The investigator will develop new tools concerning these varieties, attack long standing questions, and explore new connections. These projects will increase communication between the communities of enumerative geometry, topology, Hodge theory, and mathematical physics. The new developments will generate more activities and offer questions for graduate students and postdocs who are interested in these areas. Graduate students will be supported by this award. The research of the investigator will center around three projects: (1) to study the P=W phenomenon and the topological mirror symmetry for general reductive groups. This will bridge more systematically symmetries of groups in representation theory and symmetries of moduli spaces in algebraic geometry; moreover, a local version of P=W will be explored concerning several conjectures relating algebraic geometry of singularities to knot invariants in topology; (2) to study Hodge theory of Lagrangian fibrations. This will connect the general theory of perverse sheaves and Hodge modules to concrete and interesting examples of integrable systems and symplectic varieties; (3) to study perverse sheaves in enumerative geometry. This concerns relating Gromov-Witten and Donaldson-Thomas invariants to the more mysterious work of Gopakumar and Vafa. This direction will provide new perspectives in understanding the connections between algebraic geometry and quantum physics. To achieve these goals, the investigator together with his collaborators and students, will develop a set of tools including support theorems associated with the decomposition theorem, vanishing cycles techniques, localization methods, techniques in algebraic geometry of positive characteristics, and symmetries in hyper-Kähler geometries. 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.