This summer school will familiarize students with the basic problems of the mathematical theory of Euclidean quantum fields. The lectures will introduce some of its prominent models and analyze them via the so called “stochastic quantization” methods, involving recently developed stochastic and PDE techniques. This is an area which is highly interdisciplinary combining ideas ranging from the theory of partial differential equations, to stochastic analysis, to mathematical physics. Our goal is to bring together students which are maybe familiar with some but not all of these subjects and teach them how to integrate these different tools to solve cutting-edge problems of Euclidean quantum field theory.

School Structure

The organizers plan to introduce gradually the basic tools and ideas needed in this area of research, which will cover at least the first 3/4 of the school.  The remaining time will be left to discuss other models and more recent developments. There will be two or three lectures per day, 1.5 hours each, plus one or two 1.5 hours long TA/problem session. 

Prerequisites

A basic knowledge of finite-dimensional probability and of Brownian motion (without stochastic calculus), of basic functional analysis and basic PDE theory: linear equations, Fourier transform, Sobolev spaces on R^d. This the ideal background but it is not strictly necessary: we are well aware that the participants will come from a wide range of knowledge and abilities, so we will work to bring them to a common ground. A few weeks before the school we plan to assign recommended readings that would help the participants get better prepared, for instance, reading material along the line of Section 2.2.1 and 2.3.1 of “Partial differential equations” by Evans, and Chapter 2 of “Stochastic Differential Equations: An Introduction with Applications” by Oksendal.

For eligibility and how to apply, see the Summer Graduate Schools homepage