Summer Graduate School

Models of random growth and of the separation of phases occurring when one substance is suspended in another often evince universal features, in which scaling exponents are shared among a broad class of such models. A foothold for understanding such features is often offered by studying a few special models that are exactly solvable, which means that exact formulas of algebraic or integrable origin are available. Showing that a broader range of models also have the features is a task that may rely on a range of robust probabilistic or geometric tools. The summer school will offer an introduction to random growth and phase separation, with an emphasis on tools that offer the prospect of proving universality for a wider class of models.

School Structure

There will be two lectures each morning, as well as afternoon problem sessions.

Prerequisites

The prerequisites for the summer school are a class in real analysis at the introductory graduate level, including the construction of Lebesgue measure, the Lebesgue integral and the Radon-Nikodym theorem, and point-set topology including compactness and continuity. This material corresponds to the first three chapters in Folland's `Real Analysis: Modern Techniques and Their Applications' (Second Edition).

In addition, students should have some familiarity with probability at an advanced undergraduate level: random variables, expectations, convergence of random variables, the law of large numbers, the central limit theorem and discrete-time martingales. This material is covered in the first nine chapters of John B. Walsh's `Knowing the Odds: An Introduction to Probability'.

Students should have taken a class that covers a significant span of this material, and may further prepare by reading such texts as Walsh's.

The school does not require formal prerequisites in partial differential equations. In practice, it may be helpful to prepare by reading about Hamilton-Jacobi PDE and conservation laws in Sections 3.4 and 3.4 of L.C. Evans' text `Partial Differential Equations'.

Application Procedure

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