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.
In the last 30 years many new connections between representation theory (which is a branch of algebra) and geometry were found. The main goal of this project was to establish some new connections of this sort, which in addition also involve insight from recent works in mathematical physics. Specifically we study the so called symplectic resolutions - some special algebraic varieties (which are certain geometric objects) and show that they live in the center of a a triangle, whose vertices are formed by representation theory, modern algebraic geometry (e.g. the theory of derived categories of coherent sheaves) and mathematical physics (super-symmetric quantum field theory in dimensions 2 and 4). More specifically, it turns out that the same structures can be seen as qcoming from 1) Quantum cohomology (an object, initially defined by physicists, attached to an algebraic variety) of a symplectic resolution (specifically, the so called quantum differential equation) 2) Representation theory of certain algebras (similar to the famous double affine Hecke algebras) attached to a symplectic resolution (usually obtained by quantizing the symplectic structure) 3) Derived category of (equivariant) coherent sheaves on the resolution The conjectural relation between 1 and 3 comes from the famous mirror symmetry conjecture (proposed by physicists) and we were able to confirm it in the case of symplectic reolution. The relation between 2 and both 1 and 3 allows one get some very non-trivial new results in representation theory.
