E. Mukhin plans to continue his studies of the Algebraic Bethe Ansatz method in the relation to the non-homogeneous Gaudin, XXX, XXY and other models. The main problems to be addressed are: develop an approach to the Algebraic Bethe Ansatz based on populations of solutions of Bethe Ansatz equations; use the populations to solve the Bethe Ansatz equations explicitly; use populations to study the number of solutions of the Bethe Ansatz equations; prove the (modified) Bethe Ansatz Conjecture which claims that the Bethe vectors form a basis in the space of states for generic values of parameters, in the case of sl(N).
Diagonalization of Hamiltonians of many models of mathematical physics can be performed if the corresponding system of algebraic equations called Bethe Ansatz equations is solved. It was shown that the solutions of the Bethe Ansatz equations naturally form families which are called populations. In the case of the Gaudin model associated to sl(N), the populations are in one-to-one correspondence with the points of intersection of appropriate Schubert cycles and with the scalar differential operators of order N which have only polynomial solutions. Thus the sl(N) populations are given by N-dimesnional spaces of polynomials in one variable with prescribed singular points and exponents. E. Mukhin hopes that the study of the spaces of polynomials can make the Bethe Ansatz approach more tractable and resolve several old problems in this area.
