Integrable Systems in the XXI century :
symplectic, algebraic and spectral theory
This program aims to bring together specialists in algebraic aspects of the theory of classical and quantum completely integrable systems, with specialists in semiclassical analysis and geometric aspects of the theory of completely integrable systems.
The program will make an emphasis on the latest research developments. A key component of the program is that it will serve as a platform to discuss the main problems in the field, and layout new directions for the participants to follow. We will achieve these by having a mixture of "research activities" and "down time" for the participants to talk to each other and think on their own about the research problems pertaining the program. The program will start off with a workshop consisting a mini-course and a series of research talks. There will be an active visitor and seminar program under which junior and senior researchers will work at the Bernoulli Center for varying periods.
PROGRAM TOPICS: the Program's topics to be covered in the program events described above will vary depending on the speakers and audience interests, and may include (but are not limited to)
(A) SYMPLECTIC AND SPECTRAL GEOMETRY OF INTEGRABLE SYSTEMS: symplectic invariants and classifications of integrable systems in low dimensions, constructions of integrable systems based on Poisson Lie groups and dynamical Poisson Lie groups, semiclassical and algebraic quantizations, semiclassical spectral and inverse spectral problems for integrable systems, involving Toeplitz and pseudodifferential operators.
(B) ALGEBRAIC AND GEOMETRIC STRUCTURES OF INTEGRABLE SYSTEMS: The focus of this part of the program will vary depending on the participants in residence at the time. We particularly want to focus on dynamical R-matrices in representation theory and the theory of integrable systems and special functions. This will include the following specific topics. The classical and quantum dynamical Yang-Baxter equations. Conceptual meaning: Poisson and quantum groupoids (dynamical quantum groups), module categories. Schiffmann's generalization of the Belavin-Drinfeld classification. Quantization results for classical dynamical r-matrices for semisimple Lie algebras. Dynamical R-matrices and matrix-valued Macdonald-Ruijsenaars systems and functions. Dynamical Weyl groups, q-deformed Casimir connections. Orthogonality and Macdonald-Mehta-Cherednik identities for matrix valued Macdonald functions. Vertex-IRF transformations. q-deformed Kazhdan-Lusztig functors.
(C) HAMILTONIAN STRUCTURES IN TOPOLOGICAL FIELD THEORIES: relations between various previously unconnected domains, namely the integrable systems and loop groups from one side and cluster varieties on the other. Cluster varieties are mainly used to give combinatorial description of two classes of spaces - namely the Teichmuller spaces and simple Lie groups. Both classes intersect in the notion of higher Teichmuller spaces - the spaces of discrete subgroups of split real Lie groups. Though study of cluster varieties nowadays is a very popular subject, there remains a class of questions in the domain, where almost nothing has been done so far.
PLANED SEMINARS AND WORKING GROUPS (subject to changes)
- Member's seminar, Mondays 3-4pm
- Working group 1: "Spectral theory of integrable systems", meeting Wed 2-4pm
- Working group 2: "Algebraic and geometric structures of integrable systems" meeting Fr 2-4pm
- Working group 3: "Hamiltonian structures in topological field theories" meeting Th 2-4pm
- Alvaro Pelayo (Institute for Advanced Study)
- Nicolai Reshetikhin (University of California, Berkeley)
- San Vũ Ngọc (IUF, Université de Rennes 1)
July 1, 2013 - December 31, 2013
CENTRE INTERFACULTAIRE BERNOULLI (CIB)
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Johannes Sjöstrand (Université de Bourgogne, France)
Thuesday, December 10, 2013; 5:15-6pm
Eigenvalue distribution for non-self-adjoint differential operators, with and without analyticity
- Manuel de Leon (ICMAT, Madrid)
July 24, 2013
The Hamilton-Jacobi theory: a geometric review
Craig Tracy (University of California Davis, USA)
Thursday July 4, 5:15-6pm
Integrable Models in Statistical Physics and Their Universality