Integrable systems and random matrices
Team members
Research topics
- The study of eigenvalues of random matrices.
- Determinantal point processes, repulsive particle systems, random growth models
- Asymptotic analysis of orthogonal polynomials and Hankel, Toeplitz and Fredholm determinants.
- Asymptotic analysis via Riemann-Hilbert problems.
- Integrable differential equations, such as Painlev茅 equations and nonlinear wave equations (like the Korteweg-de Vries equation which describes shallow water waves).
Representative publications
- M. Cafasso and T. Claeys, A Riemann-Hilbert approach to the lower tail of the KPZ equation (arxiv:1910.02493)
- C. Charlier and T. Claeys, Large gap asymptotics for Airy kernel determinants with discontinuities, Comm. Math. Phys. (2019), https://doi.org/10.1007/s00220-019-03538-w
- T. Claeys and I. Krasovsky, Toeplitz determinants with merging singularities, Duke Math. Journal 164, no. 15 (2015), 2897-2987
- M. Adler, Mark, K. Johansson, Kurt, and P. van Moerbeke, Double Aztec diamonds and the tacnode process, Adv. Math. 252 (2014), 518鈥�571
- T. Claeys, A. Its, and I. Krasovsky, Emergence of a singularity for Toeplitz determinants and Painlev茅 V, Duke Math. Journal 160, no. 2 (2011), 207-262
- P. van Moerbeke, Random and integrable models in mathematics and physics. Random matrices, random processes and integrable systems, 3鈥�130, CRM Ser. Math. Phys., Springer, New York, 2011
Useful links
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Geometry and mathematical physics
Team members
Research topics
My research concerns the following fields: the theory of symmetric spaces, harmonic analysis, non-commutative geometry and mathematical physics. I am particularly interested in the interactions between curvature and the theory of deformations in a non-formal setting.
My current work has three main directions. First, I develop geometric methods essentially coming from symmetric spaces and from representation theory with the aim of obtaining non-commutative spaces in the sense of A. Connes by non-formal quantization in a framework of operator algebras. An important tool is a generalization to solvable Lie groups of the deformation method of M.A. Rieffel for the actions of $ {\ mathbb R} ^ d $.
Second, in the context of semisimple symplectic symmetric spaces, I study certain applications in harmonic analysis of quantization by covariant deformations. Third, through collaborations with physicists, I consider, in the context of $ C ^ \ star $ - algebras, the problem posed by the definition of non-commutative D-branes in a string theory evolving in a curved space-time, such as a locally anti-Sitter causal black hole.
P. Bieliavsky
Representative publications
- P. Bieliavsky; Semisimple symplectic symmetric spaces, Geom. Dedicata 73 (1998), no. 3, 245-273.
- P. Bieliavsky; Symmetric spaces and star representations, Advances in Geometry, Progr. Math. 172, Birkhauser (Boston), 1999, 71-82.
- P. Bieliavsky, Strict quantization of solvable symmetric spaces, Journal of Symplectic Geometry 1 (2002), no. 2, 269-320. (math.QA/0010004.)
- P. Bieliavsky, Y. Maeda, Convergent star product algebras on "$ax+b$", Lett. Math. Phys. 62 (2002), no. 3, 233-243.
- P. Bieliavsky, M. Massar, Oscillatory integral formulae for left-invariant star products on a class of Lie groups, Lett. Math. Phys. 58 (2001), no. 2, 115-128.
- P. Bieliavsky, M. Rooman, Ph. Spindel, Regular Poisson structures on massive non-rotating BTZ black holes, Nuclear Phys. B 645 (2002), no. 1-2, 349-364.
- P.Bieliavsky, M.Pevzner, Symmetric spaces and star representations III. The Poincar\'e disk, Noncommutative Harmonic Analysis, Progress in Mathematics, 220, Birkh盲user Boston, P. Delorme, M. Vergne eds (2004). (math.RT/0209206).
Statistical models and integrability
Team members
Research topics
- Study of exactly solvable quantum spin chains, in particular the Heisenberg XXZ model. These models provide simple descriptions of quantum magnetism and their rich mathematical structure allows exact correlation functions to be computed.
- Combinatorial statistical physics: relationships between vertex models on two-dimensional networks and the enumeration of combinatorial objects such as alternating-sign matrices and plane partitions.
- Exact supersymmetry in spin chains and related mathematical structures such as (co-)homology.
- Conformal logarithmic theories: general aspects (including that of undecomposable Virasoro modules) and application to statistical models on a network (sandpile model, dimers, loop models).
- Study of the algebraic structures underlying the integrability and conformal properties of critical network models.
- Universal properties (arctic curves, correlations) in the tiling of domains by simple tiles, such as rectangles or diamonds.
Representative publications
- C. Hagendorf, A. Morin-Duchesne, Symmetry classes of alternating sign matrices in the nineteen-vertex model. J. Stat. Mech. (2016) 053111
- A. Morin-Duchesne, J. Rasmussen, P. Ruelle, Y. Saint-Aubin, On the reality of spectra of Uq(sl2)-invariant XXZ Hamiltonians, J. Stat. Mech. (2016) 053105
- A. Morin-Duchesne, J. Rasmussen, P. Ruelle, Integrability and conformal data of the dimer model, J. Phys. A: Math. Theor. 49 (2016) 174002
- C. Hagendorf, The nineteen-vertex model and alternating sign matrices. J. Stat. Mech. (2015) P01017
- A. Morin-Duchesne, J. Rasmussen, P. Ruelle, Dimer representations of the Temperley-Lieb algebra, Nucl. Phys. B 890 (2015) 363--387
- C. Hagendorf, T. Fokkema, L. Huijse, Bethe-ansatz solvability and supersymmetry of the M(2) model of single fermions and pairs. J. Phys. A: Math. Theor. 47 (2014) 485201
- J.G. Brankov, V.S. Poghosyan, V.B. Priezzhev, P. Ruelle, Transfer matrix for spanning trees, webs and colored forests, J. Stat. Mech. (2014) P09031
- C. Hagendorf, Spin chains with dynamical lattice supersymmetry. J. Stat. Phys. 150 (2013) 609-657
- P. Ruelle, Logarithmic conformal invariance in the Abelian sandpile model, J. Phys. A: Math. Theor. 46 (2013) 494014
- C. Hagendorf, P. Fendley, The eight-vertex model and lattice supersymmetry. J. Stat. Phys. 146 (2012), 1122-1155
- N.Sh. Izmailian, P. Ruelle, C.-K. Hu, Finite-size corrections for logarithmic representations in critical dense polymers, Phys. Lett. B711 (2012) 71--75