In this talk I will introduce the moduli space of curves, and a class of vector bundles “of conformal blocks” on the moduli space of curves. I’ll give a nonspecialist definition of these bundles, which have connections to algebraic geometry, representation theory and mathematical physics. I’ll talk about how by studying the bundles we can learn about the moduli space of curves, and vice versa, focusing on just a few recent results, and open problems.
Let $X$ be a compact manifold with the boundary $Y$ and $R(k)$ be a
Dirichlet-to-Neumann operator: $R (k):f \to \partial_n u |_Y$ where u solves
$$
(Delta+k^2) u=0, \ u|_Y=f.
$$
We establish asymptotics as $k\to \infty$ of the number of eigenvalues of
$k^{-1}R (k)$ between $a$ and $b$.
We will discuss tools, used to solve this problem: sharp semiclassical spectral
asymptotics and Birman-Schwinger principle.
This is a joint work with Andrew Hassell, Australian National University.
