Grant: Discovery Grants Program - Individual

I am most interested in problems involving interplay between several branches of mathematics. There is an abundance of such problems involving Hopf algebras; these are algebraic structures arising in numerous parts of mathematics, as well as in physics. Examples of Hopf algebras include group algebras, universal envelopes of Lie algebras, algebras of representative functions on Lie groups, coordinate algebras of algebraic groups, and quantum groups. I plan to study algebraic structure of Hopf algebras in order to classify them, construct new interesting examples, and apply them to problems in algebra, analysis, and combinatorics. Discrete structures often give rise to natural Hopf algebras in which multiplication encodes the assembly and comultiplication encodes the disassembly of combinatorial objects under investigation. These Hopf algebras often enable us to see deep connections between algebra, analysis, and combinatorics. A point in case is recent joint work in which we constructed a combinatorial Hopf algebra in order to study a free analogue of joint distributions in probability.