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.
Modern theoretical physics operates with mathematical objects of immense sophistication, which is only to be expected of a theory that aims to describe the incredible complexity of our physical world. These include the study of extended objects, such as strings, propagating in some target manifolds. Mathematically, this links with study of families of algebraic curves in algebraic manifolds. Several years ago, a set of striking, bold, and challenging conjectures were put forward that concern algebraic curves in a large and very important class of target manifold, the so-called equivariant symplectic resolutions. Equivariant symplectic resolutions include manifolds that a very familiar from geometric representation theory, as well as so-called Nakajima varieties, which play a key role in the study of supersymmetric gauge theories. Obtaining a mathematical proof of these conjectures, as well as the development of the right mathematical framework for them, became the goals that we set for ourselves at the beginning of the projects. The introduction to http://arxiv.org/abs/1211.1287 contains an extended discussion of these conjectures aimed at a mathematical audience. We are very happy to report that most of these challenging mathematical problems were successfully resolved and thus extended the frontiers of our knowledge by a great deal. These newly developed techniques and ideas are now being applied and extended in many new and exciting directions.
