Interactions Between Ring Theory and Representations of Algebras

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  • David Wehlau. Tamsen Whitehead McGinley. Tianyuan Xu. We focus the vast expertise in our Department around key themes that maximize our ability to provide mathematical solutions to global societal issues. Using the latest theories and applying up to date mathematical testing methods, we ensure we are assisting today's key challenges in areas such as health, engineering, finance and the physical world. Our researchers describe properties of geometric structures and develop the theory and applications of dynamical systems.

    We work at the interface of theory, computation and experimentation with the aim of understanding an array of complex continua. Our mathematicians work closely with life scientists to address key challenges in biology and medicine. We develop and analyse algorithms that compute numerical approximations and apply them to real-world problems. Space is limited, so please contact Ivan Corwin to inquire as to remaining space availability.

    There is some funding for junior researchers who wish to attend.

    Interactions between representation theory and commutative algebra, Barcelona

    To apply for funding contact Ivan Corwin with a brief description of your research interests and arrange a short recommendation letter from a senior researcher to be sent as well. Event poster. Titles and Abstracts Alexei Borodin From Macdonald processes to Hecke algebras and quantum integrable systems Abstract: The goal of the talk is to offer a brief informal introduction into some of the subjects represented at the workshop with an emphasis on connections between them.

    Alexander Povolotsky Integrable particle models with factorized steady states Abstract: Interacting particle models serve as a testing ground for the low-dimensional non-equilibrium statistical physics. Many of them share two important features, which make them amenable to exact analytic description. First one is the simplest structure of traslationally invariant stationary measure that has a form of product of one-site factors.

    This allows for complete characterization of the stationary correlations by the use of the toolbox of equilibrium statistical mechanics, developed for systems with noninteracting degrees of freedom. The second one, which makes the full dynamical problem also solvable, is the matrix of transition probabilities of the Markov process being the transfer-matrix of some quantum integrable system that, in particular, can be diagonalized by the Bethe anstaz.

    In the talk we show how to obtain very general interacting particle model, which possesses both the product stationary measure and the Bethe anstaz solvability. Then we give a brief review of its particular limiting cases interesting for physicists and mathematicians, some of which were studied before and some are new. Leonid Petrov Spectral theory for integrable interacting particle systems Abstract: I will present Plancherel isomorphism theorems for Bethe ansatz eigenfunctions of the q-Hahn Boson particle system introduced by Povolotsky in This is the most general zero-range stochastic particle system with product-form steady state which is solvable by the coordinate Bethe ansatz and depends on three parameters q,mu and nu.

    This in particular lead to explicit nested contour integral formulas for moments of the q-TASEP started from an arbitrary initial configuration. The Plancherel isomorphism theorems provide a different approach to earlier results of Babbitt, Thomas, and Gutkin, and also give a structural explanation of ASEP symmetrization formulas of Tracy and Widom. Vadim Gorin Random matrix asymptotics for the six-vertex model Abstract: The six-vertex or "square-ice" model is one of the most well-studied examples of exactly-solvable lattice models of statistical mechanics.

    The developments of the last 15 years suggest that the asymptotic behavior of this model should be governed by the probability distributions of random matrix origin. However, until recently the rigorous mathematical results in this direction were restricted to the so-called free fermion case, when the model can be analyzed via determinantal point processes.

    The stationary density correlations decay algebraically. We compute the longest relaxation time and the exact dynamical structure factor.

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    The density profile seen from the shock position is a hyperbolic tangent. This immediately entails the quantum integrability, the bispectral dual system, and the n-particle scattering operator for the chain in question. Based on joint work with Erdal Emsiz. Erdal Emsiz Discrete harmonic analysis on a Weyl alcove and the double affine Hecke algebra at critical level Abstract: I will speak about recent work on a unitary representation of the affine Hecke algebra given by discrete difference-reflection operators acting in a Hilbert space of complex functions on the weight lattice of a reduced crystallographic root system.

    Conferences and Meetings on Algebra

    I will indicate why the action of the center under this representation is diagonal on the basis of Macdonald spherical functions also referred to as generalized Hall-Littlewood polynomials associated with root systems. We use this representation to construct an explicit integrable discrete Laplacian on the discrete Weyl alcove corresponding to an element in the center. The Bethe Ansatz method is employed to show that our discrete Laplacian and its commuting integrals are diagonalized by a finite-dimensional basis of periodic Macdonald spherical functions.