... interviewed President Truman during a train stop in Cumberland, Maryland in 1949.

Accomplished in many areas of mathematics, Nelson is especially well known for his successful application of probability to quantum field theory, work for which he received the American Mathematical Society's Steele Prize for Seminal Contribution to Research in 1995. The AMS recognised two papers published in 1966 and 1973, respectively, that "showed for the first time how to use the powerful tools of probability theory to attack the hard analytic questions of constructive quantum field theory," the award citation said. The latter paper "fired one of the first shots in what became known as the Euclidean revolution", according to the AMS.

I would like to thank my advisor, Edward Nelson, for suggesting this problem, providing invaluable advice, and giving much needed encouragement. I would also like to thank Bob Anderson for reading the original draft carefully and pointing out a number of errors. I would like to thank the National Science Foundation for graduate support.

Let $\mathbb{Z}^{d} = \{(z_{1}, z_{2}, ... , z_{d}): z_{i} \in \mathbb{Z}\}$ be the integer lattice. A k-step self-avoiding path is a sequence of points $[x_{0}, x_{1}, ... , x_{k}]$ in $\mathbb{Z}^{d}$ with $x_{0} = 0, ||x_{i} - x_{i-1}|| = 1$ for $i = 1, ..., k$, and $x_{i} ≠ x_{j}$ for $i ≠ j$. The study of self-avoiding paths originated in chemical physics as an attempt to model polymer chains. Two major questions were asked. First, if $\gamma_{d}(k)$ is the number of self-avoiding paths of length k, then how does $\gamma_{d}(k)$ grow with k? Second, what is the mean square distance of the self-avoiding walk from the origin, or more generally, what is the limiting distribution as k approaches infinity of the walk after k steps? While much numerical evidence has been gathered to answer these questions, especially in dimensions two and three, few sharp analytical results have been found.

No satisfactory results about this have been proven. Even our knowledge of the behaviour of $\gamma_{d}(n)$ for large n is still very limited. The author obtains results for a different probability measure on the self-avoiding walks. His measure is induced by taking an infinite simple random walk on $\mathbb{Z}^{d}$ and "erasing of loops" from its path until a self-avoiding path is obtained. For $d ≥ 5$, the author proves an invariance principle for his self-avoiding walks .... He also obtains some asymptotic results about his loop erasing process for $d = 4$. Even though the final result is given in classical terms the author has phrased his proof in terms of nonstandard analysis.

This paper describes joint work with Oded Schramm and Wendelin Werner establishing the values of the planar Brownian intersection exponents from which one derives the Hausdorff dimension of certain exceptional sets of planar Brownian motion. In particular, we proof a conjecture of Mandelbrot that the dimension of the frontier is $\large\frac{4}{3}\normalsize$. The proof uses a universality principle for conformally invariant measures and a new process, the stochastic Loewner evolution (SLE), introduced by Schramm. These ideas can be used to study other planar lattice models from statistical physics at criticality. I discuss applications to critical percolation on the triangular lattice, loop-erased random walk, and self-avoiding walk.

One of the main goals in statistical physics is to understand macroscopic behaviour of a system given the interactions which are mainly microscopic but may exhibit long range correlations. Such models often depend on a parameter and at a critical value of the parameter the collective interaction switches from being microscopic to macroscopic. Critical phenomena is the study of such systems at or near this critical value.

There is a wide class of models (percolation, self-avoiding walk, Ising and Potts model, loop-erased random walk and spanning trees, ...) whose behaviour is very dependent on the spatial dimension. There exists a critical dimension above which the behaviour is relatively simple (although it is not always trivial to prove this is true!), but below the critical dimension there is "non mean-field" behaviour with nontrivial critical exponents for long-range correlations and fractal structures arising.
...
Given the explosive nature of the field, I will not try to give an overview. I have decided to give a personal perspective and to focus on several loop measures and related models, loop-erased random walk (related to uniform spanning trees) and the Gaussian free field. I start by introducing one of the main characters, discrete loop measures, and show how they are related to some well known objects, spanning trees and determinant of the Laplacian. It also generates one of the random fractals, the loop-erased random walk, and we then discuss what it means to take a scaling limit. This leads to a review of two of the main players in the field: the conformally invariant Brownian loop measure and the Schramm-Loewner evolution (SLE). I discuss a number of properties of SLE and focus on the most recent part to finish the characterisation, the natural or fractal parametrisation of the curve.

The Society for Industrial and Applied Mathematics' George Polya Prize was awarded to Gregory F Lawler of Cornell University, Oded Schramm of Microsoft Corporation and Wendelin Werner of Université Paris-Sud at SIAM's Annual Meeting in Boston, 10-14 July 2006. The prize was established in 1969 and is given every two years, alternatively in two categories. One is a notable application of combinatorial theory. The other is for a notable contribution in another area of interest to George Polya such as approximation theory, complex analysis, number theory, orthogonal polynomials, probability theory, or mathematical discovery and learning. In 2006, the George Polya Prize is given for a notable contribution in another area of interest to George Polya.

Lawler, Schramm and Werner received the prize for their groundbreaking work on the development and application of stochastic Loewner evolution (SLE). Of particular note is the rigorous establishment of the existence and conformal invariance of critical scaling limits of a number of 2D lattice models arising in statistical physics.

Lawler was honoured "for his comprehensive and pioneering research on erased loops and random walks," and Le Gall was selected "for his profound and elegant works on stochastic processes." According to the prize citation, "the work undertaken by these two mathematicians on random processes and probability, which [has] been recognised by multiple prizes, became the stepping stone for many consequent breakthroughs."

Gregory Lawler has made trailblazing contributions to the development of probability theory. He obtained outstanding results regarding a number of properties of Brownian motion, such as cover times, intersection exponents, and dimensions of various subsets. Studying random curves, Lawler introduced a now-classical model, the Loop-Erased Random Walk (LERW), and established many of its properties. While simple to define, it turned out to be of a fundamental nature, and was shown to be related to uniform spanning trees and dimer tilings. This work formed much of the foundation for a great number of spectacular breakthroughs, which followed Oded Schramm's introduction of the SLE curves. Lawler, Schramm, and Werner calculated Brownian intersection exponents, proved Mandelbrot's conjecture that the Brownian frontier has Hausdorff dimension $\large\frac{4}{3}\normalsize$, and established that the LERW has a conformally invariant scaling limit. These results, in turn, paved the way for further exciting progress by Lawler and others.

Besides his research, Lawler has been involved in the tournament bridge world in investigations of cheating among top competitors, especially the use of statistics to verify allegations. When not doing math, he plays guitar and is in charge of music at the Beverly Unitarian Church in Chicago. In summers he plays on the mathematics department softball team at University of Chicago.

I am a maths professor at the University of Chicago who played more bridge when I was younger (became Life Master when I was 22 in the late 1970s) but have played only infrequently in the last twenty years.