Abstract:
Spin systems play a fundamental role in classical and quantum statistical mechanics. While the most basic example of the Ising model is relatively well understood, more complicated systems, and especially systems with continuous symmetry, still pose outstanding physical and mathematical challenges. One approach to the study of such spin systems proceeds by constructing an appropriate graphical representation, often taking the form of a system of random walks or loops. The course will introduce the loop representations associated to several spin systems, including the classical spin O(n) models (Ising, XY, Heisenberg and higher n models) and the quantum XY and Heisenberg models. (No knowledge of quantum mechanics will be assumed.) Many conjectures and open problems about the properties of the loops and their correlations will be discussed and some recent progress will be presented. Special attention will be given to the loop O(n) model, to the interchange (stirring) model and to (generalizations of) the Tόth and Aizenman-Nachtergaele loop representations, which have significant interest independently of their original spin system motivation.
Biography:
Ron Peled is a full professor at Tel Aviv University specializing in probability theory and statistical mechanics. He received his Ph.D. from the University of California, Berkeley, where his thesis won the mathematics department's Herbert Alexander prize and the statistics department's citation in probability, and continued to work at the Courant Institute of New York University before going to Israel. He is the recipient of a Clay liftoff fellowship, an Alon fellowship and a European Research Council (ERC) starting grant. In recent years he has worked on the rigorous study of phase transitions in statistical mechanics as well as on problems at the interface of probability, analysis and combinatorics.
Daniel Ueltschi is professor of mathematics at the University of Warwick (United Kingdom). He obtained his PhD in Theoretical Physics at the Ecole Polytechnique Federale of Lausanne (Switzerland) and subsequently held positions at Princeton, UC Davis, and Arizona, before moving to Warwick. His research deals with probabilistic approaches to condensed matter systems, and with various mathematical questions that arise from them. He is an expert of classical and quantum spin systems.
Schedule:
Course I: March 12, ECNU Zhongbei Campus
3:30-4:30 pm, Classroom 264, Geography Building
4:45-5:45 pm, Classroom 264, Geography Building
Course II: March 14, NYU Shanghai Pudong Campus
10:00 am-12:00 pm, Classroom 211
1:00-3:00 pm, Classroom 309
Course III: March 19, NYU Shanghai Pudong Campus
1:00-2:30 pm, Classroom 311
4:00-5:30 pm, Classroom 309
Course IV: March 21, NYU Shanghai Pudong Campus
10:00-11:30 am, Classroom 211
1:30-3:00 pm, Classroom 310
Math Mini-Course by the NYU-ECNU Institute of Mathematical Sciences at NYU Shanghai