Using the theorems we have proven for finite dimensional integrals as motivation, I will define the Euclidean correlation functions for a quantum mechanical particle moving in an arbitrary smooth potential in terms of a sum over graphs and give a derivation of the Feynman rules for this simple system.

It is well known that there exist domains in C^n, n>1, such that all functions holomorphic therein extend holomorphically past the boundary. In this talk, we shall examine this phenomenon for certain refinements of the fundamental example of Hartogs. We shall look at a generalization of Hartogs' construction discovered by E .M. Chirka. Finally, we shall provide a partial answer to a related question raised by Chirka. There will be plenty of pictures, and very little familiarity with several complex variables will be required.

Our purpose in this general audience talk is twofold. First is to explain why solvable groups are quite necessary for studying geometry and representation theory. Second is to report on some recent progress of ours on the combinatorics of the orbits of a solvable subgroup of SL(n) acting on certain symmetric spaces and on their compactifications.

On a compact Lie group, there exists a 1-paramemter family of Dirac operators which interpolates the geometric Dirac operator (Levi-Civita), algebraic Dirac operator (cubic of Kostant), and the trivial Dirac operator (used in LQG). The spectral action of this family of operators is computed for $SU(2)$.

Sums of squares in rings have been studied by numerous authors in the past. Typical questions are: Which elements in a ring can be written as sums of squares? If an element in a ring can be represented as a sum of squares, how many squares are needed for such a representations. We study these questions for arbitrary commutative rings, in particular in the case where $-1$ can be written as a sum of $n$ squares for some positive integer $n$. Such rings are called rings of finite level at most $n$. We derive estimates in terms of $n$ for other invariants pertaining to sums of squares such as the sublevel and the Pythagoras number. We give some examples and pose some open questions. This is joint work with David Leep.