Deformation theory from the point of view of fibered categories

We give an exposition of the formal aspects of deformation theory in the language of fibered categories, instead of the more traditional one of functors. The main concepts are that of tangent space to a deformation problem, obstruction theory, versal and universal formal deformations. We include proofs of two key results: a versione of Schlessinger's Theorem in this context, and the Ran--Kawamata vanishing theorem for obstructions. We accompany this with a detailed analysis of three important cases: smooth varieties, local complete intersection subschemes and coherent sheaves.Comment: corrected several typos and made some minor improvements to the expositio

Lorentzian approach to noncommutative geometry

This thesis concerns the research on a Lorentzian generalization of Alain Connes' noncommutative geometry. In the first chapter, we present an introduction to noncommutative geometry within the context of unification theories. The second chapter is dedicated to the basic elements of noncommutative geometry as the noncommutative integral, the Riemannian distance function and spectral triples. In the last chapter, we investigate the problem of the generalization to Lorentzian manifolds. We present a first step of generalization of the distance function with the use of a global timelike eikonal condition. Then we set the first axioms of a temporal Lorentzian spectral triple as a generalization of a pseudo-Riemannian spectral triple together with a notion of global time in noncommutative geometry.Comment: PhD thesis, 200 pages, 9 figures, University of Namur FUNDP, Belgium, August 201

Real Lie Algebras of Differential Operators and Quasi-Exactly Solvable Potentials

We first establish some general results connecting real and complex Lie algebras of first-order differential operators. These are applied to completely classify all finite-dimensional real Lie algebras of first-order differential operators in $R^2$. Furthermore, we find all algebras which are quasi-exactly solvable, along with the associated finite-dimensional modules of analytic functions. The resulting real Lie algebras are used to construct new quasi-exactly solvable Schroedinger operators on $R^2$.Comment: 33 pages, plain TeX. To apper in Phil. Trans. London Math. Soc. Please typeset only the file rf.te