Diagrammatic and geometric techniques in representation theory

点击翻译按钮获取中文摘要

My research is based on the interface between representation theory, algebraic geometry, low dimensional topology and mathematical physics. It centres around both the foundational theory and applications of studying a special class of non-commutative algebras. These algebras arise from the notion of deformation quantization: a commutative algebra can deform to an interesting non-commutative one, and one can study the representation theory of that algebra by considering the geometry of its classical limit. This approach is fundamental in the study of quantum mechanics (the Heisenberg uncertainty principle expresses the failure of commutativity), but also has a fruitful history in the study of Lie algebras. ******The relevant class of algebraic varieties I consider are "symplectic singularities." Every symplectic singularity has an associated non-commutative algebra, which we call its "universal enveloping algebra," constructed using deformation quantization. These include the universal enveloping algebras of Lie algebras as a special case; one basic principle of my research is to study how statements about Lie algebras can be modified to hold in the general case. ******Mathematical physics appears as a source of these algebras: certain special quantum field theories give us many examples of these algebras. The program I propose in this grant is to understand how the geometry of these singularities, the representation theory of their deformations, and the associated quantum field theories relate to each other, and can be applied in other areas such as combinatorics and topology. ******My most important work over the past decade has been dedicated to the idea that these varieties appear in "dual pairs" that arise in physics. This cast the whole field in a new light, and understanding how properties of these varieties are related under duality has stimulated study of many different aspects of symplectic singularities. The connection between the representation theory and geometry of dual varieties is subtle, but this duality can be seen as a "geometrification'' and "categorification" of many dualities in mathematics, such as Schur-Weyl, rank-level and Gale duality. ******Moving forward, the biggest question facing me is to understand better how these insights can be applied in mathematical physics, and conversely, how the ideas of physics can brought to bear on the mathematical questions of the proposal. ********