Abstract: 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 $SU(2)$.

Abstract: 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.