My research interests range from algebraic combinatorics, representation theory to mathematical physics. I have been interested in combinatorial properties of quantum algebras. Quantum algebras have their roots in two dimensional solvable lattice models in statistical mechanics. Kashiwara showed that at zero temperature the quantum algebras exhibit beautiful combinatorial properties. Mathematically these are formulated in the crystal base theory. With Kirillov and Shimozono I have established a relation between crystal base theory and other combinatorial objects, called rigged configurations, which also arise from physics. A generalization of this result provides new fermionic formulas of the physical partition sums which, as follows by work of the Stony Brook group, encapture the statistics of the particle in the physics model (actually these statistics are similar to those found in the fractional quantum Hall effect; see here). In terms of q-series these yield generalizations of the famous Rogers-Ramanujan identities.
I am also part of the Focused Research Group on Affine Schubert Calculus: Combinatorial, geometric, physical, and computational aspects funded by the NSF. More information can be found at the FRG website.
I am especially fond of understanding relations between
different areas in mathematics and mathematical physics.
Often this helps to unravel hard problems in one area by
exploiting the techniques of another area. I am also a developer of the open-source
computer package MuPAD-Combinat
and more recently Sage-Combinat.
In particular, I have implemented crystal bases and Kirillov-Reshetikhin crystals.
Some of my papers
with Jason Bandlow and Nicolas Thiery
preprint arXiv:0806.3131v1 [math.CO]
with Florent Hivert and Nicolas Thiery
preprint arXiv:0804.3781v1 [math.RT]
with Thomas Lam and Mark Shimozono
preprint arXiv:0710.2720[math.CO]
with Jean-Christophe Novelli
RIMS Kokyuroku Bessatsu B8 (2008) 71-83
( arXiv:0706.2996[math.CO] )
with Masato Okado
Representation Theory 12 (2008) 186-207
( arXiv:0706.2224[math.QA] )
J. Algebra
319 (2008) 2938-2962
( arXiv:0704.2046[math.QA] )
with Ghislain Fourier and Mark Shimozono
J. Algebra
309 (2007) 386-404
( math.QA/0605451 )
MSJ Memoirs 17 (2007) 75-104 (published by the Mathematical Society of Japan)
( math.QA/0512161 )
Formal Power Series and Algebraic Combinatorics, San Diego, California 2006
( pdf file )
with Lipika Deka
J. Combinatorial Theory, Series A
113 (2006) 1435-1461
( math.CO/0509194 )
International Mathematics Research Notices, Volume 2006,
Article ID 97376, Pages 1-27
( math.QA/0508107 )
with Lipika Deka
Commun. Math. Phys. 260 (2005) 711-725
( math-ph/0412084 )
with Mark Shimozono
J. Algebra 295 (2006) 562-610
( math.QA/0412376 )
with Philip Sternberg
in: A. Kuniba, M. Okado (Eds.), RIMS Kokyuroku 1429
"Combinatorial Aspect of Integrable Systems", to appear.
( postscript file )
with Philip Sternberg
J. Alg. Combin. 23 (2006) 317-354
( math.QA/0408113 )
J. Algebra 285 (2005) 292-334
( math.QA/0406248 )
in: B. Lulek, T. Lulek and A. Wal (Eds), "Symmetry and Structural Properties
of Condensed Matter", Vol. 7, World Scientific, Singapore 2003; pp. 201-224
( math-ph/0210014 )
with Masato Okado and Mark Shimozono
Commun. Math. Phys.
238 (2003) 187-209
( math.QA/0209082 )
with Masato Okado and Mark Shimozono
"Algebraic Combinatorics and Quantum Groups", Edited by N. Jing, World Scientific (2003), 85-124
( math.QA/0203163 )
with Masato Okado and Mark Shimozono
J. Algebra 267 (2003) 212-245
( math.QA/0111288 )
in: MathPhys Odyssey 2001, M. Kashiwara and T. Miwa (eds.),
Birkhaeuser Boston, Cambridge, MA, 2002, pp. 437-454
( math.CO/0107214 )
with Masato Okado and Mark Shimozono
Representation Theory
7 (2003) 101-163
( math.QA/0105017 )
with Masato Okado and Mark Shimozono
Contemp. Math. 291 (2001) 29-53
( math.QA/0104268 )
with Mark Shimozono and Dennis E. White
Advances in Applied Mathematics 30 (2003) 258-272
( postscript file )
with Mark Shimozono
Commun. Math. Phys. 220 (2001) 105-164
( math.QA/0001114 )
with Mark Shimozono
in: Formal Power Series and Algebraic Combinatorics,
D. Krob, A.A. Mikhalev, A.V. Mikhalev (eds.),
12th International conference, FPSAC'00, Moscow, Russia, June 2000,
Proceedings, Springer Verlag Berlin/Heidelberg 2000, pp. 367-378.
( postscript file )
with S. Ole Warnaar
in: Physical Combinatorics, M. Kashiwara and T. Miwa (eds.),
Birkhaeuser Boston, Cambridge, MA, 2000, pp. 163-183.
( math.QA/9909044 )
with S. Ole Warnaar
Contemp. Math. 297 (2002) 227-255
( math.QA/9906092 )
with Anatol N. Kirillov and Mark Shimozono
Selecta Mathematica (N.S.) 8 (2002) 67-135
( math.CO/9901037 )
with Anatol N. Kirillov and Mark Shimozono
Seminaire Lotharingien de
Combinatoire 42 [B42j] (1999)
( postscript file )
with Mark Shimozono
Advanced Studies in Pure Mathematics 28,
Combinatorial Methods in Representation Theory (2000) 305-325
( math.QA/9812106 )
with George E. Andrews and S. Ole Warnaar
J. Amer. Math. Soc. 12 (1999) 677-702
( math.QA/9807125 )
with S. Ole Warnaar
Commun. Math. Phys. 202 (1999) 359-401
( math.QA/9802111 )
with Alexander Berkovich, Barry M. McCoy, and S. Ole Warnaar
Nucl. Phys. B499 [PM] (1997) 621-649
( hep-th/9702026 )
with S. Ole Warnaar
The Ramanujan Journal 2 (1998) 459-494
( q-alg/9701007 )
with Alexander Berkovich and Barry M. McCoy
Commun. Math. Phys. 191 (1998) 325-395
( q-alg/9607020 )
with S. Ole Warnaar
The Ramanujan Journal 2 (1998) 327-349
( q-alg/9607014 )
with S. Ole Warnaar
Int. J. of Mod. Phys. B11 (1997) 189-195
( q-alg/9604015 )
with Alexander Berkovich and Barry M. McCoy
Physica A 228 (1996) 33-62
( hep-th/9512182 )
Nucl. Phys. B467 (1996) 247-271
( hep-th/9510168 )
Nucl. Phys. B459 (1996) 393-436
( hep-th/9508050 )
with Peter van Nieuwenhuizen
Phys. Rev. D50 (1994) 967-970
( hep-th/9408125 )