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Professor A. Borodin will give a series of ten lectures on Determinantal point processes and representation theory. In the last decade mathematics has seen a remarkable synthesis of the ideas from statistics and probability theory, mathematical physics, algebra and geometry. Determinantal random point processes first appeared in random matrix theory in the 1960's. They describe the behaviour of eigenvalues of a random Hermitian matrix. Methods of orthogonal polynomials and asymptotic analysis are used to study important characteristics such as correlation functions. The representation theory of finite and compact groups is a well understood area. A breakthrough in the representation theory of infinite versions of classical groups was achieved by A. Borodin and G. Olshanski earlier in this century. The ideas and methods of probability theory were crucial in the study of harmonic analysis on the infinite symmetric and unitary groups. The course of ten lectures will be devoted to the new fast developing area on the intersection of probability theory and algebra which involves the above and other interactions. Despite the novelty and the variety of methods the lecture course is accessible to postgraduate students or other mathematicians interested in the topic. The following topics will be covered at different level of details: 1. The Young graph and classification of characters of the infinite symmetric group. 2. The Gelfand-Tsetlin graph and classification of characters of the infinite-dimensional unitary group. 3. Other examples of branching graphs. Connections to random matrices and Macdonald polynomials. 4. Non-ergodic Gibbs measures and the problem of noncommutative harmonic analysis. 5. Determinantal point processes. Determinantal structure of the ergodic measures on the Gelfand-Tsetlin graph and of the Schur measures. 6. Examples of asymptotic analysis of determinantal processes. Applications to random matrices and last passage percolation. 7. Markov dynamics on Gibbs measures. Applications to one-dimensional exclusion processes and random directed polymers. There will also be supplementary lectures by Professors I. Corwin (Microsoft
Research and MIT), P. Ferrari ( If you intend to come please fill in the
registration form by 18 February 2012. Funding is available to support
attendance of participants with the first preference given to Organisers: Dr Misha
Feigin ( Enquiries: LMSlectures2012@gmail.com |
Website a.kirpichnikova@gmail.com (Anna Kirpichnikova)