Program
Mini-course I: Geometry
Lecturer: Eleonora Di Nezza
Hours: 6
Title: Kähler geometry & Monge-Ampère equations
Abstract: In this mini-course, I will introduce fundamental concepts and definitions in Kähler geometry. We will then explore the notion of Kähler-Einstein metrics, and I will demonstrate how the search for these canonical metrics is equivalent to solving complex Monge-Ampère equations. Following that, I will present a variational approach, which involves minimizing an appropriate functional, known as the Mabuchi functional, whose critical points correspond to Kähler-Einstein metrics.
Mini-course II: Probability
Lecturer: Frank den Hollander & Marco Zamparo
Hours: 6
Title: Large deviation theory and selected applications
Abstract: In this mini-course we will illustrate the basic principles of large deviation theory and describe some applications in statistical physics. Large deviation theory is a natural framework for the description of large (= rare) fluctuations. The first three hours are devoted to describing key definitions and key theorems (= basic theory, examples, importance sampling). The last three hours are devoted to applications in statistical physics and the theory of stochastic processes, demonstrating that large deviation principles can be used to characterise phase transitions and the arrow of time (= breaking of ensemble equivalence, phase transitions and rate functions, entropy production in Markov chains).
Advanced Course I
Lecturer: Rolf Andreasson, Robert Berman, Jakob Hultgren
Hours: 8
Title: Probabilistic contruction of Kähler-Einstein metrics
Abstract: The goal of this course is to explain a surprising connection between Kähler geometry and probability theory, namely that Kähler-Einstein metrics can be constructed as many particle limits of certain canonical point processes on complex manifolds. More precisely, we will connect the Mabuchi functional in the variational approach to Kähler-Einstein metrics explained by Eleonora Di Nezza in Mini Course I to the rate function in a large deviation principle, a concept explained by Frank den Hollander and Marco Zamparo in Mini Course II. We will illustrate these phenomena by looking closely at the 1D case, and, if time permits, the toric case where some of the concepts can be understood in terms of optimal transport theory.
Advanced Course II
Lecturer: Alexander Drewitz, Bingxiao Liu, George Marinescu
Hours: 7
Title: Random holomorphic sections and their zeros
Abstract: This mini-course explores the construction of random holomorphic sections on complex manifolds and examines their zeros using Bergman kernels. Random holomorphic sections generalize the concept of random polynomials and Gaussian analytic functions of one complex variable. The geometric setting involves high tensor powers of a positive holomorphic line bundle over a K\”{a}hler manifold, to which we associate a sequence of Gaussian holomorphic sections. After analyzing Bergman kernel expansions, we will discuss key results on equidistribution, concentration of measures (also called large deviation estimates), and the central limit theorem for the zeros of these Gaussian holomorphic sections as the tensor powers go to infinity. In addition, we will review recent advances and developments in the field.
Seminar
Speaker: Alessandra Cipriani
Hours: 1
Title: The discrete Gaussian free field on a compact manifold
Abstract: In this talk we aim at defining the discrete Gaussian free field (DGFF) on a compact Riemannian manifold. The DGFF, also known as harmonic crystal in the physics literature, is a central model of random surfaces (random height functions). In probability, it has been widely used in connections to several other models: random walks, domino tilings, random matrices, SLE curves — just to name a few. We will review the DGFF in the Euclidean setup, when it is defined on graphs (generally, lattices). Secondly, we will explain the difficulties one encounters in extending the definition to a manifold. Finally, we will construct a suitable random graph that replaces the square lattice in Euclidean space, and prove that the scaling limit of the DGFF is given by an object called “the manifold continuum Gaussian free field”.
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