Research

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.

Education

• BS and MS: Moscow State University, mechmat (1992, with Honors), Russia
• PhD: University of North Carolina at Chapel Hill (1998), USA
• PostDoc: MSRI (1999), University of California at Berkeley (2000-2001)

Publications & Professional Activity

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:

• E.E. Demidov, Yu.I Manin, E. Mukhin, D.V. Zhdanovich, Non-standard quan-
  tum deformations of GL(N) and constant solutions of the Yang-Baxter equation,
  Common trends in mathematics and quantum theories, Kyoto, 1990. Progr.
  Theor. Phys. Suppl. 102 (1991), 203-218
• E. Frenkel, E. Mukhin, Combinatorics of q-characters of finite-dimensional representations of quantum affine algebras, arXiv:math/9911112, Commun. Math. Phys., 216 (2001), 23--57
• E. Mukhin, V. Tarasov, A. Varchenko, The B. and M. Shapiro conjecture in real algebraic geometry and the Bethe ansatz, arXiv:math/0512299, Annals of Mathematics (2) 170 (2009), no. 2, 863-881
• E. Mukhin, V. Tarasov, A. Varchenko, Schubert calculus and representations of the general linear group, arXiv:07114079, J. Amer. Math. Soc. 22 (2009), no. 4, 909-940
• E. Mukhin, V. Tarasov, A. Varchenko, Bethe algebra of Gaudin model, Calogero-Moser space and Cherednik algebra, arXiv:09065185 (2009), 1-24, accepted to IMRN
• E. Mukhin, C. A. S. Young, Extended T-systems, arXiv:1104.3094, Selecta Math. (N.S.) 18 (2012), no. 3, 591–-631
• B. Feigin, M. Jimbo, T. Miwa, E. Mukhin, Symmetric polynomials vanishing on
  the shifted diagonals and Macdonald polynomials, math.QA/0209042, Int. Math. Res. Not. (2003), no. 18, 1015-1034
•  A. Molev, E. Mukhin,  Invariants of the vacuum module associated with the Lie superalgebra gl(1|1),  arXiv:1502.03511, J. Phys. A, 48 (2015), no.31, 314001, 1-20
•  E. Mukhin, V. Tarasov,  Lower bounds for numbers of real solutions in problems of Schubert calculus,  arXiv:1404.7194, Acta Mathematica, 217 (2016), no.1, 177-193