Anatol Kirillov is working on algebraic analysis, Representation theory of quantum groups, Algebraic and tropical combinatorics, Fermionic formulae, dilogarithm identities for classical and quantum dilogarithms, quantum and classical Schubert calculus, quantum cohomology, Special functions and integrable systems.
  The most beautiful and deep results are
1) Construction of the rigged configuration bijection which is a refinement of the familiar Robinson-Schensted-Knuth correspondence.The rigged configuration bijection is a source of Fermionic formulae for characters and branching functions, as well as has many deep and unexpected applications to combinatorics of the symmetric group.
2) Quantum dilogarithm and dilogarithm identities.
3) The Bracket algebra and classical and quantum Schubert calculus.
4) Raising operators for the Macdonald polynomials.
5) Tropical combinatorics and discrete integrable systems.
6) Combinatorics around Painleve's equations.
For more information and details, seeURL: http://www.kurims.kyoto-u.ac.jp/~kirillov