The mirror to the quintic in P^4 is a family X_p of Calabi-Yau 3-folds over a thrice-punctured sphere. As p moves in a loop around each of the three punctures, we can parallel transport classes in H^3(X_p), and observe the monodromy. H^3(X_p) is four-dimensional, and the monodromy can be expressed by matrices in Sp(4,Z). These matrices generate a subgroup which is dense in Sp(4,Z), but it was not known whether or not it was of finite index. We showed that the subgroup is isomorphic to the free product Z/5 * Z, from which it follows that it cannot be of finite index. The mirror quintic family is one of 14 similar families of CY 3-folds; our methods establish similar results for 7 of the 14 families. For the other 7, it has recently been shown that the monodromy is of finite index, so our result is best possible. This talk is based on joint work with Chris Brav, arXiv:1210.0523.

Connes and others have developed a notion of non-commutative geometry (NCG) that generalizes Riemannian geometry, and provides a framework in which the standard model of particle physics, coupled to Einstein gravity, may be concisely and elegantly recast. I will explain how this formalism may be reformulated in a way that naturally generalizes from non-commutative to non-associative geometry. In the process, several of the standard axioms of NCG are conceptually reinterpreted. This reformulation also suggests a new constraint on the class of NCGs used to describe the standard model of particle physics. Remarkably, this new condition resolves a long-standing puzzle about the embedding of the standard model in NCG, by precisely eliminating from the action formula the collection of seven unwanted terms that previously had to be removed by an extra (empirically-motivated, ad hoc) assumption.

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 $\mathbb{C}^N$.

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.

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.