The Kardar-Parisi-Zhang (KPZ) equation is a non-linear stochastic partial differential equation whose statistical properties are believed to describe a large class of mathematical models including interacting particle systems, random growth models, directed polymers, and branching diffusion processes. The purpose of this project is to make significant progress towards developing a theory of the exact solvability of the KPZ equation. In particular, this theory should lead to exact formulas for the probability distributions of the solution to the KPZ equation when started with various important types of initial data. This project will involve a number of fields of mathematics and this direction has already produced results of independent interest to these fields, which include: Macdonald symmetric function theory, tropical combinatorics and Whittaker functions, and certain quantum integrable systems.
Since its discovery two hundred years ago the Gaussian distribution (bell curve) has come to represent one of mathematics greatest societal and scientific contributions ? a robust theory explaining and analyzing much of the randomness inherent in the world. Physical and mathematical systems accurately described in terms of Gaussian statistics are said to be in the Gaussian universality class. This class, however, is not all encompassing. For example, classical extreme value statistics or Poisson statistics better capture the randomness and severity of events ranging from natural disasters to emergency room visits. More recently, significant research efforts have been focused on understanding systems which are not well-described in terms of any of the classically developed statistics. The failure of these systems to conform to classical descriptions is generally due to a non-linear relationship between natural observables and underlying sources of random inputs and noise. A variety of models for complex systems such as growth processes, polymer chains, mass transport, traffic flow, queueing theory, driven gases, and turbulence have been actively studied for over forty years in mathematics, physics, material science, chemistry and biology. All of these systems fail to conform with classical Gaussian statistics, as has been observed through experimental evidence involving turbulent liquid crystals, crystal growth on a thin film, facet boundaries, bacteria colony growth, paper wetting, crack formation, and burning fronts. Surprisingly, despite their differences, all of the systems fall into a new statistical universality class whose properties are described in terms of a single model called the Kardar-Parisi-Zhang (KPZ) equation. The purpose of this project is to develop a statistical understand of the KPZ equation and its universality class.
