Göran Gustafsson Lectures in Mathematics

GÖRAN GUSTAFSSON Lectures in Mathematics

January 23 - 30 , 2012
KTH, Stockholm

Stanislav Smirnov
Geneva University and St.-Petersburg State University

Abstracts:

Lecture 1: Discrete complex analysis
Monday, January 23, 3.15-4.15 pm
Lecture hall F2, Lindstedtsvägen 26

It is well-known that discrete harmonic functions can be defined on any graph, e.g. by requiring the mean value property: the function at a vertex is the mean of the values at its neighbors. Such discrete functions share many properties of their continuous counterparts and have been very extensively studied. The theory of discrete analytic functions also has a long history, but is less developed. One starts with a planar graph, e.g. the square lattice, and then asks a function to satisfy some discretization of the Cauchy-Riemann equations. There are many possible discretizations, and some of them have deep connections to integrable systems. It turns out that much of the usual complex analysis can be transferred to the discrete setting, albeit with some difficulties. Sometimes discrete theories can be applied to the continuous case and even lead to easier proofs of several results, including the Riemann uniformization theorem. There are also exciting connections to combinatorics, probability, geometry and even computer science.

Coffee is served between 2.45 and 3.15 outside the lecture hall.

Lecture 2: Conformal invariance and universality in the Ising model
Wednesday, January 25, 3.15-4.15 pm
Lecture hall E1, Lindstedtsvägen 3

The celebrated Ising model (introduced by his advisor Lenz as a model of ferromagnetic behavior) is one of the simplest systems exhibiting an order-disorder transition. It is widely believed that in 2D it has a universal and conformally invariant scaling limit at criticality, which is used in deriving many of its properties. We will sketch the first proof of this fact for the critical Ising model on a wide family of graphs.

Based on the joint work with Dmitry Chelkak.

Lecture 3: Self-avoiding walks on hexagonal lattice
Monday, January 30, 3.15-4.15 pm
Lecture hall F2, Lindstedtsvägen 26

A famous chemist Flory proposed to consider self-avoiding (i.e. visiting every vertex at most once) walks on a lattice as a model for polymer chains. A number of predictions about their behavior was made by physicists, but very little is known mathematically. We will discuss our joint work with Hugo-Duminil-Copin, establishing Nienhuis' exponential asymptotics for the number of self-avoiding walks on the hexagonal lattice.

Sponsored by the Göran Gustafsson Foundation

2012-01-02