The Undergraduate Summer School provides opportunities for talented undergraduate students to enhance their interest in mathematics. This program is open to undergraduates at all levels, from first-year students to those who have just completed their undergraduate education. Based on the backgrounds of the accepted students, they will be divided into two groups - introductory and advanced. There will be several activities organized for these groups, with some specifically intended for either the introductory or advanced group. There will be time for study groups and individual projects guided by advisors, as well as other activities.

2007 Course Descriptions: (updated May 15, 2007)

1. Introductory Course: A non-technical introduction to recent advances in discrete probability and mathematical physics; Marek Biskup, University of California Los Angeles

The course will focus on a number of model systems that have been oflarge interest to both probabilists and mathematical physicists in the last twodecades. Specific examplesinclude random walks, branching processes,percolation, Ising and Potts models, spanning trees, dominoes, etc. These models have theadvantage of a relatively simple formulation as well as the availability of interesting resultsthat can be obtained by relatively non-technical means.

The questions we hope to addressare, e.g., what forces the simple random walk return to its starting point in dimensions 1 and 2 and why this fails in dimensions three and higher, why is the mean offspring equal to 1 critical for survival of afamily line, how many infinite clusters can one have in bondpercolation on the square lattice, etc. Time and interest permitting, we may casually wander into some more advanced topics, e.g., why do large clusters in critical percolation look the same regardless of the orientation of the lattice axes, what makes the Ising ferromagnet have a phasetransition at low temperatures, how can one count the number of spanning trees of a given graph, etc.

2. Advanced Course: An introduction to Brownian motion and its applications; Omer Angel, University of Toronto

Brownian motion is the most basic stochastic process, fundamentally connected to many areas of mathematics as well as physics, finance and more. After over a century of research, some aspects of Brownian motion remains at the forefront of research in probability theory.

The course will begin with an introduction to Brownian motion, as well as to some of the relations between discrete and continuous random structures. We start by defining B.M. as a random function and constructing it explicitly. Some of the topics we may cover are: Properties of B.M. paths (continuity, non-differentiability and Holder continuity); the relation between B.M. and random walks, and the Skorokhod embedding theorem, structure of the zero set of B.M., which is a naturally occurring fractal; We will then consider B.M. in higher dimensions, and study the symmetries of B.M., including conformal invariance as well as the relation to harmonic analysis and to differential equations. A further important application we will consider is the continuum random tree (CRT), which describes the large scale structure of typical trees. We will begin with combinatorial encodings of trees, and study the relation between the CRT and B.M., Some extra topics that may be visited include polar sets, intersection exponents of B.M. as wells as (very) short introductions to SLE and to the Gaussian free field.

For both courses:

Prerequisites include basic curriculum in calculus, linear algebra, and analysis. Some exposure -- e.g. through a basic introductory course -- to probability will definitely be useful though we will try to cover the necessary facts as they arrive. Some basic facts concerning complex analysis and (partial) differential equations will be described as needed, though prior familiarity with them will help students get maximal benefit from the course.

Both the Introductory Course and the Advanced Course will run pretty much in parallel (the actual lectures will be scheduled back to back) and there will be common lecture notes. Several of the topics discussed in the Introductory Course in a discrete setting will be mirrored in the Advanced Course in a continuous setting, and the relation between the discrete and continuous is one of the main themes of the whole program. The lecturers feel the best strategy for most participants is to attend both courses, and gradually focus attention on the level which they find appropriate.

PCMI's Undergraduate Summer School is supported in part by the National Security Agency and in part by the National Science Foundation grant # DMS- 0437137.