Odd Dunkl Operators and nilHecke Algebras

I’ve always been interested in mathematics. It is the pinnacle of human logic and is unquestionably correct, leading to wonderful models of predicting weather and making transistors. I found math to be a beautiful art form with a personality; some equations are humble, some are lawless, and some are mysterious, teasing for further inquiry … I have conducted research in representation theory, the backbone of many mathematical ideas in algebra, topology, and particle physics. A major part of this field is the interplay between symmetries and the algebraic objects which control them. In the 1980’s, Charles Dunkl introduced certain operations involving both derivatives (rates of change) and certain reflections naturally associated to the symmetry of ordinary Euclidean space. These Dunkl operators have proven useful in both physics and mathematics, where they are used to study quantum many-body problems, conformal field theory, Lie theory, and harmonic analysis. In my project, I studied new Dunkl-type operators better adapted to a type of noncommutative space, which is a space in which the multiplication of quantities does not satisfy the familiar relation ab = ba …

• Mathematics