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.
Symmetry abounds in nature. From the atoms in a crystal to the planets in the solar system, our universe is often arranged in symmetrical patterns. Sitting at the interface of geometry, algebra and analysis, representation theory is the mathematical study of those patterns. One of the main results of this project is the precise formulation of a conjectural link relating groups, the mathematical entities which describe the symmetries of macroscopic physical systems, and quantum groups, which describe the symmetries of certain (microscopic) physical systems arising in statistical mechanics. Such a link is obtained through the study of the multivaluedness of solutions of a system of differential equations constructed by the PI and called the Casimir equations. In collaboration with a graduate student, funded through this proposal as a research assistant, the PI developed algebraic tools that reduce the mathematical proof of this link to the verification of its simplest instance, and made significant progress towards that verification. In a different direction, and in collaboration with Tom Bridgeland (Oxford), the PI uncovered a novel dictionary linking wall–crossing, the jumps that algebraic varieties undergo when the equations defining them are changed, and Stokes phenomena, which describe the changes in asymptotics of ordinary differential equations near a singular point. The PI has been advising three graduate students on topics related to the subject of this proposal, and trained several more through the running of intensive seminars and graduate courses.
