Quantum Harmonic Analysis

Principal Investigator: Javier Parcet



Inspired by Heisenberg’s matrix mechanics, John von Neumann proposed a quantization of mathematics beyond the classical/relativistic notions of measure and geometry. Along the years, his program gave rise to groundbreaking theories. Quantum harmonic analysis over quantized measure spaces (von Neumann algebras) remains a huge challenge with a tremedous potential impact. This long standing gap is partly explained by a limited use so far of powerful Lp techniques and the absence of points after quantization. My recent results around noncommutative Calderón-Zygmund theory and Fourier multipliers on group von Neumann algebras open a door towards more ambitious goals. The primary line of this project is to analyze the Lp convergence of Fourier series/integrals with frequencies given in a locally compact nonabelian group. Fefferman’s ball multiplier theorem, Kakeya maximal estimates or Bochner-Riesz type multipliers are very much unexplored in group algebras.

Beyond their intrinsic interest, this is also motivated by its relation to Connes’ 1980 SLn(Z) problem via an unexpected connection with nonsmooth methods from Euclidean harmonic analysis. I propose two major goals. The first one is to find a local smoothness criterium for Lp multipliers over high rank semisimple Lie groups. This would be the first positive result in this direction after decades of negative results from Haagerup, Cowling, Lafforgue or de la Salle among others. The second goal is to prove the CBAP for L4(LG) with G = SL(3,R) and other rank 2 semisimple Lie groups. This is in line with the most ambitiuos aspects of my Research Proposal for the ERC Consolidator Grant. Interdisciplinarity is absolutely essential in this project. It requires to combine harmonic analysis, operator algebra, geometric group theory, noncommutative geometry and quantum probability.