SR2

This is joint work with Uri Onn. We use motivic integration to get the growth rate of the sequence consisting of the number of conjugacy classes in quotients of G(O) by congruence subgroups, where $G$ is suitable algebraic group over the rationals and $O$ the ring of integers of a number field.

The proof uses tools from the work of Nir Avni on representation growth of arithmetic groups and results of Cluckers and Loeser on motivic rationality and motivic specialization.

We will give an introduction to the theory of d-manifolds, a new class of geometric objects recently/currently invented by Joyce (see http://people.maths.ox.ac.uk/joyce/dmanifolds.html). We will start from scratch, by recalling the definition of a 2-category and talking a bit about $C^\infty$-rings, $C^\infty$-schemes and d-spaces before giving the definition of what a d-manifold should be. We will then discuss some properties of d-manifolds, and say some words about d-manifold bordism and its applications.

Chern-Simons theory is topological gauge theory in three dimensions that contains an interesting class of operators called Wilson lines/loops, which have connections with both physics and pure mathematics. In particular, it has been shown that computations with Wilson operators in Chern-Simons theory reproduce knot invariants, and are also related to Gauss linking invariants. We will discuss the complex generalizations of these ideas, which are known as holomorphic Chern-Simons theory, Wilson operators, and linking, in the setting of Calabi-Yau three-folds. This will (hopefully) include a definition of all three of these holomorphic analogues as well as an investigation into how these ideas can be translated into simple homological algebra, allowing us to propose the existence of "homological Feynman rules" for computing things like Wilson operators in a holomorphic Chern-Simons theory. If time permits I may say something about physics too.

We will explain Bridgelands results on the stabiltiy manifold of a K3 surface. As an application we will define the stringy Kaehler moduli space of a K3 surface and comment on the mirror symmetry picture.

More perspectives on spectra.

This is the first in a series of $\geq 2$ talks about Stable Homotopy Theory. We will motivate the definition of spectra by the Brown Representability Theorem, which allows us to interpret a spectrum as a generalized cohomology theory. Along the way we recall basic notions from homotopy theory, such as suspension, loop spaces and smash products.