Prof. Mukhin studies symmetries and structures arising in the context of conformal field theory, quantum field theory and exactly solvable models of statistical physics. He is often working on the border of several areas of mathematics, employing a combination of tools from representation theory, combinatorics, and analysis.
He has made a number of important contributions to the theory of algebraic Bethe ansatz, to the theory of finite-dimensional representations of affine quantum groups, to the combinatorics of representations of affine Lie algebras and W-algebras, to the theory of Jack and Macdonald polynomials, to the representation theory of quantum toroidal algebras, to the theory of quantum and classical Knizhnik-Zamolodchikov equations.
His major results include
the proofs of a number of fundamental conjectures concerning the q-characters of representations of quantum affine algebras (together with E. Frenkel of UC Berkeley)
the proofs of the several versions of the Shapiro-Shapiro conjecture, of the transversality of Real Schubert Calculus and of the simplicity of spectrum of the Gaudin model (together with V. Tarason of IUPUI and A. Varchenko of UNC Chapel Hill)
the pioneering study of Macdonald polynomials at negative rational values of the coupling constant (together with B. Feigin of Moscow Higher School of Economics, M. Jimbo of Rikkyo University, and T. Miwa of Kyoto University).
Prof. Mukhin has an impressive list of collaborators which includes researchers from many countries, both senior renown scientists and young mathematicians.
Prof. Mukhin has published over 90 scientific papers in refereed journals. Most of the papers can be downloaded from the archive at http://www.arxiv.org/find/math. The published versions can be found through MathSciNet. The journal references are updated at the personal website.
The few selected publications are: