Prof. Dr. Michał Wrochna – Freiburg Institute for Advanced Studies

Prof. Dr. Michał Wrochna

Cergy Paris University
Mathematics

External Senior Fellow (Marie S. Curie FCFP)
September 2021 – August 2022

Last Update: 31.08.2022

Curriculum Vitae

Michał Wrochna is Professor of Mathematics in Cergy Paris Université (former Université de Cergy-Pontoise). Previously he was Maître de Conférences in Grenoble at the Institut Fourier, and he held fellowships and shorter visiting positions at Université Paris-Sud, Stanford University, the University of Cambridge and the Institut des Hautes Études Scientifiques in Bures-sur-Yvette. His principal area of research is mathematical physics and he is particularly interested in applications of methods from partial differential equations, microlocal or asymptotic analysis and spectral theory, often in geometric contexts. Presently his main research topic is Quantum Field Theory on curved spacetimes, and he has worked on problems including Hadamard states, renormalisation, AdS/CFT correspondence, QFT in external potentials and gauge theories. More generally, he is interested in various problems where there is a relationship between classical and quantum dynamics, or where local and global aspects are tied together in an intricate way.

Selected Publications

The Unruh state for massless fermions on Kerr spacetime and its Hadamard property (with Christian Gérard and Dietrich Häfner), Annales Scientifiques de l’École Normale Supérieure, 2020
The massive Feynman propagator on asymptotically Minkowski spacetimes (with Christian Gérard), American Journal of Mathematics 141 (6), 1501–1546 (2019)
Quantum fields from global propagators on asymptotically Minkowski and extended de Sitter spacetimes (with András Vasy), Annales Henri Poincaré, 19 (5), 1529–1586 (2018)
Analytic Hadamard states, Calderón projectors and Wick rotation near analytic Cauchy surfaces (with Christian Gérard), Communications in Mathematical Physics 366 (1), (2019)
A mechanism for holography for non-interacting fields on anti-de Sitter spacetimes (with Wojciech Dybalski), Classical and Quantum Gravity 36 (8), (2019)

FRIAS Project

Non-elliptic spectral theory and emerging quantum geometries

The spectral theory of elliptic differential operators, their Fredholm theory and their relationship with Riemannian geometry are widely studied topics in mathematical analysis. The main goal of this project is to develop an equally rich global theory of hyperbolic operators, including the wave and Dirac operators on Lorentzian manifolds. Open problems concern the existence of fractional powers, the relation of trace densities to the curvature, the validity of geometric index formulae, and the nature of the Dirichlet-to-Neumann map on anti-de Sitter spaces. The primary application will be to gain insight into the interaction of quantum fields with spacetime geometry, with the ultimate goal of establishing dynamical equations that couple quantum degrees of freedom to geometry.

To deal with non-elliptic problems, the crucial idea in this project is to combine methods from microlocal and global or spectral analysis in the form of powerful propagation estimates, unifying propagation of singularity theorems and the positive commutator methods from Mourre theory.