Abstract: We investigate the question: for which functions $f(x_1,...,x_n),~g(x_1,...,x_n)$ the asymptotic expansion of the integral $\int g(x_1,...,x_n) e^{\frac{f(x_1,...,x_n)+x_1y_1+...+x_ny_n}{\hbar}}dx_1...dx_n$ consists only of the first term. We reveal a hidden projective invariance of the problem which establishes its relation with geometry of projective hypersurfaces of the form $\{(1:x_1:...:x_n:f)\}$. We also construct various examples, in particular we prove that Kummer surface in $\P^3$ gives a solution to our problem. This is a joint work with Maxim Kontsevich.

Abstract: In this talk I will introduce the notation of a matrix ensemble with a focus on the Gaussian Unitary Ensemble (GUE) as an example. I will introduce its basic properties in connection with map enumeration. In particular, I will outline a proof of the Genus Expansion Formula for moments of the GUE. Time permitting we will discuss the famous Harer-Zagier formula.