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ICMAT Laboratory "Marius Junge"

The Marius Junge Laboratory is a joint project with the teams lead by Javier Parcet (ICMAT) on noncommutative harmonic analysis and David Pérez-García (UCM) on quantum information theory.

Quantum mechanics dramatically changed the laws of physics. The attempt to provide a rigorous mathematical foundation rapidly attracted the attention of John von Neumann, who was the first noticing that classical/relativistic notions of measure and geometry did not reflect anymore all our knowledge of the physical world. This led him to propose a “quantization of mathematics” which had a deep transformative effect on mathematics as a discipline. We can now find noncommutative forms of classical theories which allow the basic objects to obey nonclassical commutation relations. Noncommutative geometry and topology, quantum probability or noncommutative measure theory illustrate it. More recent examples are topics such as quantum groups, operator spaces and noncommutative Lp spaces.
The aim of the project is to accept the quantum mechanical perspective in the context of harmonic analysis and information theory. We are interested in extending classical estimates for Fourier multipliers to group von Neumann algebras and other noncommutative scenarios. Among other topics, this includes Hörmander-Mihlin multiplier type theorems, Calderón-Zygmund and Littlewood-Paley methods, hypercontractivity and log-Sobolev inequalities, transference techniques… Additionally, some of these problems lead us to new viewpoints in classical harmonic analysis. Regarding quantum information, we will be focusing on quantum channels, Bell inequalities and entanglement theory and quantum games via operator spaces.