Abstract: Let $G$ be a domain in $\mathbb{R}^N$ and let $w$ be a point in $G$. This talk is concerned with harmonic functions on $G$ with the property that their homogeneous polynomial expansion about $w$ are "universal" in the sense that they can approximate all plausible functions in the complement of $G$. We will discuss topological conditions under which such functions exist, and the role played by the choice of the point $w$. These results can be generalised for the corresponding class of universal holomorphic functions on certain domains of $\mathbb{C}^N$.

Abstract: On any manifold with Spin$^c$ structure, one can construct a corresponding Dirac operator, which is a first-order elliptic differential operator whose principal symbol can be expressed in terms of Clifford multiplication. Dirac operators are Fredholm on compact manifolds, but not on noncompact manifolds.

I'll give a construction due to Atiyah that deforms the symbol of a $G$-invariant Dirac operator into a transversally elliptic symbol whose equivariant index is well-defined, by "pushing" the characteristic set of the symbol off the zero section using an invariant vector field. This construction is essentially topological in nature, and has been used by Paradan, Ma and Zhang, and others in the study of the "quantization commutes with reduction" problem in symplectic geometry.

I will say a few words about this problem, and will end with a discussion of a construction due to Maxim Braverman of a generalized Dirac operator whose analytic index coincides with the topological index of the "pushed symbol" of Atiyah.

Abstract: The theory of projective modules has, from its inception, taken as inspiration for theorems and techniques ideas from the topological theory of vector bundles on (nice) topological spaces. I will explain another chapter in this story: ideas from classical homotopy theory can transplanted to algebraic geometry via the Morel-Voevodsky $\mathbb{A}^{1}$-homotopy category to deduce new results about classification and splitting problems for projective modules over smooth affine algebras. Some of the results I discuss are the product of joint work with Jean Fasel.