Bethe algebras

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The Bethe algebras are important commutative subalgebras found in many infinite-dimensional algebras such as universal enveloping algebras of polynomial current algebras, Yangians, affine quantum groups, etc. Recent progress in the study of the Bethe algebras led to the proofs of the B. and M. Shapiro conjecture, of the tranversality conjecture, of the reality of Schubert Calculus, of the simplicity of the spectrum of the gl(N) Gaudin model, of the simplicity of the Heisenberg XXX chain, of the correspondence of the Bethe vectors and Fuchsian monodromy-free operators as part of the geometric Langlands program and others. The PI expects that the ideas used to obtain these results can be systematized and further developed to a procedure which can be applied to many other important cases. In particular, the PI intends to study the images of the Bethe subalgebras in natural representations of the larger algebras by identifying it with a ring of regular functions on a suitable algebraic variety and then combining the information from the representation theory and algebraic geometry.PI intends to establish and study new close relations between algebraic geometry and theory of integrable models. Such a connection is expected to produce a variety of new results in both disciplines.