Research Statement for Susama Agarwala

I am interested in geometric and algebraic questions motivated by high energy physics, particularly renormalization. I study renormalization using combinatorial Hopf algebra in the program established by Connes and Kreimer in [11]. The study of combinatorial Hopf algebras leads me to work on problems in many different fields, such as non-commutative geometry, number theory, and even control theory. I am also interested motives, particularly as they apply to multiple polylogarithms and field theories in configuration space. Combinatorial Hopf algebras have led me to study the process of renormalization certain types of field theories [2, 1, 5]. These algebras have applications in the study of multiple polylogarithms [18, 3]. They have an important role to play in non-commutative geometry [15]. In ongoing work, I am studying their relationship with gauge structures found in control theory. In more speculative work, I am investigating whether these Hopf algebras and their related structures may shed light on the Fundamental Lemma, at least in the case of GL2(k). Combinatorial Hopf algebras can be expressed as the Hopf algebra of rooted trees, Hrt. These are non-planar oriented trees with one marked vertex that all edges are oriented away from. The coproduct structure is defined by making a certain type of cuts on these trees. These structures first appeared in the proble