L3

Let G be a compact semisimple Lie group. A classical paper of Atiyah and Bott (from 1982) studies the moduli space of flat G-bundles on a fixed Riemann surface S. Their approach completely determines the integral homology of this moduli space, using Morse theoretic methods. In the case where G is U(n), this moduli space is homotopy equivalent to the moduli space of holomorphic vector bundles on S which are "semi-stable". Previous work of Harder and Narasimhan determined the Betti numbers of this moduli space using the Weil conjectures. 20 years later, a Madsen and Weiss determined the homology of the moduli space of Riemann surfaces, in the limit where the genus of the surface goes to infinity.

My talk will combine these two spaces: I will describe the homology of the moduli space of Riemann surfaces S, equipped with a flat G-bundle E -> S, where we allow both the flat bundle and the surface to vary. I will start by reviewing parts of the Atiyah-Bott and Madsen-Weiss papers. Our main theorem will then be a rather easy consequence. This is joint work with Nitu Kitchloo and Ralph Cohen.

I will survey the recent work of Haskins and myself constructing new special Lagrangian cones in ${\mathbb C}^n$

for all $n\ge3$ by gluing methods. The link (intersection with the unit sphere ${\cal S}^{2n-1}$) of a special Lagrangian cone is a special Legendrian $(n-1)$-submanifold. I will start by reviewing the geometry of the building blocks used. They are rotationally invariant under the action of $SO(p)\times SO(q)$ ($p+q=n$) special Legendrian $(n-1)$-submanifolds of ${\cal S}^{2n-1}$. These we fuse (when $p=1$, $p=q$) to obtain more complicated topologies. The submanifolds obtained are perturbed to satisfy the special Legendrian condition (and their cones therefore the special Lagrangian condition) by solving the relevant PDE. This involves understanding the linearized operator and its small eigenvalues, and also ensuring appropriate decay for the solutions.

I'll discuss my ongoing attempt to modernise the theory of the image of J.
Some features
that I would like to have are as follows:

1) Most of the spectra involved in the story should be E_\infty (or strictly
commutative)
    ring spectra, and most of the maps involved should respect this structure.  New
    machinery for dealing with E_\infty rings should be used systematically.

2) As far as possible the constructions used should not depend on arbitrary choices
     or on gratuitous localisation.

3) The Bernoulli numbers should enter via their primary definition as coefficients of a
     certain power series.

4) The image of J spectrum should be defined as the Bousfield localisation of S^0 with
    respect to KO, and other properties or descriptions should be deduced from this one.

5) There should be a clear conceptual explanation for the parallel appearance of
    Bernoulli numbers in the homotopy groups of J, K(Z) and in spectra related to
    surgery theory.

We consider Markovian models on graphs with local dynamics. We show that, under suitable conditions, such Markov chains exhibit both rapid convergence to equilibrium and strong concentration of measure in the stationary distribution. We illustrate our results with applications to some known chains from computer science and statistical mechanics.