Geometry and mathematical physics

Team members

Pierre BIELIAVSKY
Thibaut GROUY

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, Birkhäuser Boston, P. Delorme, M. Vergne eds (2004). (math.RT/0209206).