Research

Research Interests:

I am an algebraist with strong combinatorial and mathematical physical tastes. My early work is on the Hopf algebraic approach to renormalization, as developed by Connes and Kreimer. My results in this area of work have translated the computationally intensive problem of understanding energy scale flows on Quantum Field theories into a compact statement about connections on Lie groups.

Since then, I have moved in two distinct directions. I have exploited the Hopf algebras underlying a class of functions, called the multiple polylogarithms, to understand their symmetries. Multiple polylogarithms are direct generalizations of Multiple Zeta Values (MZVs), which, in turn, are generalizations of the Riemann Zeta function. Therefore, understanding relations between multiple polylogarithms leads to discovering relations between MZVs, which addresses Zagier's conjecture about the dimension of the graded components of the algebra of MZVs. Along these same lines, I have identified a correspondence between a subcategory of Mixed Tate Motives and a family of oriented graphs. The beauty of this project is that it reduces the very complicated task of understanding and working with Mixed Tate motives to the problem of understanding and working with a simple family of graphs.

Multiple polylogarithms appear in the amplitudes of a Yang Mills theory called SYM N=4. In a completely separate line of study, I associate the diagrams and integrals that appear in this theory with positive Grassmannians using purely combinatorial techniques such as matroids.

For a less technical (general pure mathematics audience) exposition of some of my research interests, click here.

For a more technical (and more complete) exposition of my research interests, click here.