MATH 5016: Topics in Probability
Description: Advanced topics in probability theory, theory of random processes, mathematical statistics, and related fields. With a change of content, this course is repeatable to a maximum of twelve credits.
MATH 5016 - Section 1: Rough Paths Theory
Description: The rough paths theory was discovered by Terry Lyons in the 1990's. The theory allows to solve differential equations driven by rough signals. The theory is deterministic but perfectly applies to the study of differential equations driven by rough random signals such as the Brownian motion or even potentially rougher signals such as the fractional Brownian motion. Rough paths theory is nowadays an active domain of research and its application to non linear partial differential equations led to the regularity structures theory, for which M. Hairer was awarded the Fields medal in 2014.
MATH 5016 - Section 2: Limit Shapes and KPZ Universality Class
Description: Since the days of Boltzmann, it has been well accepted that natural phenomena, when described using tools of statistical mechanics, are governed by various "laws of large numbers." For practitioners of the field this usually means that certain empirical means converge to constants when the limit of a large system is taken. However, evidence has been amassed that such laws apply also to geometric features of these systems and, in particular, to many naturally-defined shapes. Earlier examples where such convergence could be proved include certain interacting particle systems, invasion percolation models and spin systems in equilibrium statistical mechanics. The last decade has seen a true explosion of "limit-shape" results. New tools of combinatorics, random matrices and representation theory have given us new models for which limit shapes can be determined and further studied: dimer models, polymer models, sorting networks, ASEP (asymmetric exclusion processes), sandpile models, bootstrap percolation models, polynuclear growth models, etc. The goal of the course is to attempt to confront this "ZOO" of combinatorial examples with older foundational work and develop a better understanding of the general limit shape phenomenon.
Sections: Fall 2017 on Storrs Campus
|11211||5016||001||Lecture||TuTh 12:30:00 PM-01:45:00 PM||MONT314||Baudoin, Fabrice|
|14480||5016||002||Lecture||TuTh 02:00:00 PM-03:15:00 PM||MONT313||Li, Zhongyang|