Applications of Equivariant Lifts in Algebraic and Symplectic Geometry

点击翻译按钮获取中文摘要

This is a project in symplectic geometry, a branch of pure mathematics. It focuses on Gromov-Witten theory. Gromov-Witten invariants are numbers that appropriately count curves in spaces satisfying certain conditions; just as the fact that there is exactly one straight line passing through any two given distinct points. These invariants have been a central object of study in several fields, from their origins in theoretical physics (string theory) to algebraic and symplectic geometries. Computing such numbers for general spaces is a difficult task. For spaces with symmetries, gauged Gromov Witten theory has proven to be a far-reaching tool used to systematically understand the behavior of Gromov-Witten invariants as well as to compute them. One of the major accomplishments of gauged Gromov-Witten theory is the discovery of several explicit formulas for the invariants, which have been recently used to tackle open questions. This research project aims to advance understanding in this area. Due to the multidisciplinary nature of this field, new results will find direct applications in other areas.One of the major accomplishments of gauged Gromov-Witten theory is the discovery of wall-crossing formulas, which in turn proved to be decisive in the proof of the so-called crepant conjecture. The present project undertakes work to further advance the research program in gauged Gromov-Witten theory for algebraic and symplectic quotients. The Principal Investigator and collaborators will work on an explicit computation of quantum K-theory of toric varieties and Grassmannians by means of the K-theoretic quantum Kirwan map, and extend wall crossing formulas to quantum K-theory. The Principal Investigator and collaborators will also study the moduli of parabolic gauged maps and the gauged Gromov-Witten theory for hypersurfaces. They will investigate relations to other constructions such as stable maps with p-fields and the Landau-Ginzburg A-model. The Principal Investigator will continue his joint work on a degeneration formula for Seidel elements and its relation to mirror symmetry and potential function of disc counting. He will work on a construction of invariants for toric orbifolds generalizing a Calabi quasi-morphism for projective spaces, and plans to use this to study non-displaceability. Finally, he will investigate the possibility of extending other equivariant tools into quantum cohomology and quantum K-theory.