We introduce Feynman categories and show that they naturally define bi-algebras. In good circumstances these bi-algebras have Hopf quotients. Corresponding to several levels of sophistication and decoration (both terms have technical definitions), we recover the Hopf algebras of Goncharov and Brown from number theory, a Hopf algebra of Baues used in the analysis of double loop spaces and the various Hopf algebras of Connes-Kreimer used in QFT as examples of the general theory. Co-actions also appear naturally in this context as we will explain.

We will first define the moduli space of algebraic-group-valued representations of finitely presented groups. Then we will briefly describe how non-commutative rings influence the structure of the coordinate ring of these moduli spaces. Lastly, we will illustrate this general relationship by constructing minimal generating sets of the coordinate rings of these moduli spaces in some specific examples.

In this talk we will discuss an ongoing work on random field models. First we will review a work by Parisi and Sourlas. They conjectured that the infrared fixed point of such random field models should be described by a supersymmetric conformal field theory (CFT). From this they argued that the disordered CFT admits a description in terms of a CFT in two less spacetime dimensions but without the disorder. We will explain how the dimensional reduction is realized. Finally we will discuss when and how the RG flow of the random field theory reaches the SUSY fixed point.

Using Dwork’s trace formula, we express the exponential sum associated to a Laurent polynomial as the trace of a chain map on a twisted de Rham complex for the torus over the p-adic field. Treating the coefficients of the polynomial as parameters, we obtain the p-adic Gelfand-Kapranov-Zelevinsky (GKZ) system, which is a complex of $D^{dagger}$-modules with Frobenius structure.