PI: Nicolai Reshetikhin

Quantum field theory (QFT) is one of the most fundamental concepts in modern theoretical physics. It is also known to be one of the biggest mathematical challenges. Semiclassical (perturbative) description of QFT is one of the most studied directionswhich is also one of the most widely used in physics. Non-pertirbative approaches branch off to string theory, geometry, statistical mechanics and probability theory. The problem of mathematical justification of path integrals lead to the development of constructive field theory. One of the important problems in this direction is to connect analytic techniques of constructive field theory with geometric methods developed over last two decades. Perhaps a biggest challenge remains quantization of gauge theories. Here many new recent developments clarified boundary structure of local QFT's with gauge symmetry. An important class of integrable two-dimensional non-inear QFT's can be constructed non-perturbatively, due to infinite dimensional hidden symmetry. Many of such theories are relativistically invariant, have interesting physical applications. On the mathematical side they are intrinsically related to infinite dimensional Lie algebras, Yangians, quantized universal enveloping algebras and related algebraic structures. Classical counterparts of such theories are soliton equations with rich related geometric structure. They are also related to solvable models in statistical mechanics and to integrable probability, the area at the interface of probability theory, representation theory and statistical mechanics. Quantum field theories with conformal symmetry, conformal field theories (CFT) are known to be a universal tool to describe two dimensional critical phenomena. They also describe boundary structures to an important class of three-dimensional topological quantum field theories (TQFT). Finite dimensional integrable systems provide remarkable class of dynamical systems in its own rank. First examples of such system appeared more than one hundred years ago. Classical integrable systems are deeply rooted in algebraic geometry, in the geometry of Poisson Lie groups and related geometric structures. For quantum integrable systems mathematical base is in representation theory. Currently, among other directions, the group is focused on selected problems in CFT and TQFT, on aspects of isomonodrtomy deformations and Stokes phenomena, on superintegrable systems such as spin Calogero-Moser type systems and on topics in discrete time integrable systems. More details about research of individual members can be found in their individual pages. Because of the interdisciplinary nature of the group's research direction we work closely with other groups, such as representation theory, statistics, string theory and quantum information.