Recent results of the PI's and the co-PI's suggest a strong connection between the following mathematical objects and constructions: localization theory in representation theory in zero and positive characteristic; derived categories of coherent sheaves on algebraic symplectic varieties; small equivariant quantum cohomology; Casimir-type connections and their monodromy. The goal of the project is to gain a deeper and more detailed understanding of the links between these objects and develop new methods for enumerative algebraic geometry and representation theory based on those links.
Representation theory is a branch of mathematics based on the fact that surprisingly rich information about a mathematical or physical object is often hidden in the structure of its symmetries. Throughout some 100 years of its history, a major source of motivation and methods in representation theory has been the interaction with neighboring fields, such as the physics of elementary particles, number theory and geometry. The idea of the present project comes from a new connection of this sort, this time with recent constructions in algebraic geometry motivated by high energy physics. At present this connection has only been observed in particular, though impressive, examples. The aim of the project is to gain a better understanding of the nature of this connection and use this understanding to develop new methods for attacking current problems in several areas of mathematics.
Algebraic geometry is a branch of mathematics studying geometry of a system of solutions of algebraic (polynomial) equations. It developed rapidly throughout the later part of the 20th century, and is by now one of the most advanced and sophisticated areas of mathematical research. A new trend in this subject emerged in 1990's when ideas from quantum field theory (a branch of theoretical physics) led to new theorems in enumerative algebraic geometry. This led to creation of a new area within algebraic geometry known as mirror symmetry. Representation theory studies the algebraic structure of symmetries of a mathematical object. Throughout its history, interaction with both physics and algebraic geometry has been a major factor in the progress of representation theory. However, possible relation with mirror symmetry has not been explored until recently. The goal of the project, motivated by some striking parallels observed in examples, has been to develop applications of the concepts of mirror symmetry to representation theory and to use representation theoretic ideas to make progress in some outstanding questions of algebraic geometry. A number of major results were obtained. Quantum cohomology connections are a particular kind of differential equations arising in enumerative algebraic geometry motivated by physics. The PI's have conjectures that the structure of quantum connection controls some basic invariants of representations. Etingof formulated a precise conjecture based on that principle which was later established by Bezrukavnikov (jointly with I. Losev). Okounkov (jointly with D. Maulik) has used representation theoretic formalism to compute quantum connections for algebraic varieties known as quiver varieties. Bezrukavnikov has produced new results and conjectures on mirror symmetry and related geometric Langlands duality and their applications to representation theory. These and other results and new methods developed by the PI's in the framework of the project constitutes its intellectual merit. Several graduate students and postdocs were involved in closely related work, thus the project contributed to preparation of the next generation of experts. PI's have reported their accomplishments in several international and domestic conferences. The project motivated collaboration and exchange of ideas with colleagues in the U.K., France, Israel and Russia, thereby promoting international cooperation. This constitutes the broader impacts of the project.
