Nicolas Ginoux

Wave equations on Lorentzian manifolds and quantization

This book provides a detailed introduction to linear wave equations on Lorentzian manifolds (for vector-bundle valued fields). After a collection of preliminary material in the first chapter one finds in the second chapter the construction of local fundamental solutions together with their Hadamard expansion. The third chapter establishes the existence and uniqueness of global fundamental solutions on globally hyperbolic spacetimes and discusses Greens operators and well-posedness of the Cauchy problem. The last chapter is devoted to field quantization in the sense of algebraic quantum field theory. The necessary basics on C*-algebras and CCR-representations are developed in full detail. The text provides a self-contained introduction to these topics addressed to graduate students in mathematics and physics. At the same time it is intended as a reference for researchers in global analysis, general relativity, and quantum field theory.

Classical and Quantum Fields on Lorentzian Manifolds

We construct bosonic and fermionic locally covariant quantum field theories on curved backgrounds for large classes of fields. We investigate the quantum field and n-point functions induced by suitable states.

CCR- versus CAR-quantization on curved spacetimes

We provide a systematic construction of bosonic and fermionic locally covariant quantum field theories on curved backgrounds for large classes of free fields. It turns out that bosonic quantization is possible under much more general assumptions than fermionic quantization.

A spectral estimate for the Dirac operator on Riemannian flows

We give a new upper bound for the smallest eigenvalues of the Dirac operator on a Riemannian flow carrying transversal Killing spinors. We derive an estimate on both Sasakian and 3-dimensional manifolds, and partially classify those satisfying the limiting case. Finally, we compare our estimate with a lower bound in terms of a natural tensor depending on the eigenspinor.

Optimization of tire noise by solving an Integer Linear Program (ILP)

One important aim in tire industry when finalizing a tire design is the modeling of the noise characteristics as received by the passengers of the car. In previous works, the problem was studied using heuristic algorithms to minimize the noise by looking for a sequence under constraints. These constraints are imposed by tire industry. We present a new technique to compute the noise. We also propose an integer linear program based on that technique in order to solve this problem and find an optimal sequence. Our study shows that the integer linear programming approach shows significant improvement of the found tire designs, however it has to be improved further to meet the calculation time restrictions for real world problem size.

A new upper bound for the Dirac operator on hypersurfaces

We prove a new upper bound for the first eigenvalue of the Dirac operator of a compact hypersurface in any Riemannian spin manifold carrying a non-trivial twistor spinor without zeros on the hypersurface. The upper bound is expressed as the first eigenvalue of a drifting Schrodinger operator on the hypersurface. Moreover, using a recent approach developed by O. Hijazi and S. Montiel, we completely characterize the equality case when the ambient manifold is the standard hyperbolic space.