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Methods Of Modern Mathematical Physics

This paper introduces five notions, including π-algebras, π-ideals, Hopf π-algebras, π-modules and Hopf π-modules, verifies the fundamental isomorphism theorem of π-algebras and studies some algebraic properties of Hopf π-algebras as well.

Hopf π- Algebras

Part I. Examples and Fundamental Concepts Introduction 1. First examples 2. Equivalence, classification, and invariants 3. Principle classes of asymptotic invariants 4. Statistical behavior of the orbits and introduction to ergodic theory 5. Smooth invariant measures and more examples Part II. Local Analysis and Orbit Growth 6. Local hyperbolic theory and its applications 7. Transversality and genericity 8. Orbit growth arising from topology 9. Variational aspects of dynamics Part III. Low-Dimensional Phenomena 10. Introduction: What is low dimensional dynamics 11. Homeomorphisms of the circle 12. Circle diffeomorphisms 13. Twist maps 14. Flows on surfaces and related dynamical systems 15. Continuous maps of the interval 16. Smooth maps of the interval Part IV. Hyperbolic Dynamical Systems 17. Survey of examples 18. Topological properties of hyperbolic sets 19. Metric structure of hyperbolic sets 20. Equilibrium states and smooth invariant measures Part V. Sopplement and Appendix 21. Dynamical systems with nonuniformly hyperbolic behavior Anatole Katok and Leonardo Mendoza.

Introduction to the Modern Theory of Dynamical Systems: EQUILIBRIUM STATES AND SMOOTH INVARIANT MEASURES

Advances in mathematics

The theory of pseudo differential operators, discussed in § 1, is well suited for investigating various problems connected with elliptic differential equations. However, this theory fails to be adequate for studying equations of hyperbolic type, and one is then forced to examine a wider class of operators, the so-called Fourier integral operators (Egorov [1975], Hormander [1968, 1971, 1983, 1985], Kumano-go [1982], Shubin [1978], Taylor [1981], Treves [1980]).

Fourier Integral Operators

The wavefront set provides a precise description of the singularities of a distribution. Because of its ability to control the product of distributions, the wavefront set was a key element of recent progress in renormalized quantum field theory in curved spacetime, quantum gravity, the discussion of time machines or quantum energy inequalitites. However, the wavefront set is a somewhat subtle concept whose standard definition is not easy to grasp. This paper is a step by step introduction to the wavefront set, with examples and motivation. Many different definitions and new interpretations of the wavefront set are presented. Some of them involve a Radon transform.

A smooth introduction to the wavefront set

The wavefront set of a distribution describes not only the points where the distribution is singular, but also the 'directions' of the singularities. Because of its ability to control the product of distributions, the wavefront set was a key element of recent progress in renormalized quantum field theory in curved spacetime, quantum gravity, the discussion of time machines or quantum energy inequalitites. However, the wavefront set is a somewhat subtle concept whose standard definition is not easy to grasp. This paper is a step-by-step introduction to the wavefront set, with examples and motivation. Many different definitions and new interpretations of the wavefront set are presented. Some of them involve a Radon transform.

https://iopscience.iop.org/article/10.1088/1751-8113/47/44/443001/pdf

Le sujet de la these est la construction d'une theorie perturbative des champs quantiques en interaction sur un espace-temps courbe, suivant un point de vue concu par Stueckelberg et Bogoliubov et developpe par Epstein-Glaser sur l'espace de Minkowski plat. En 2000 un progres important fut realise par Brunetti et Fredenhagen qui reussirent a etendre la theorie d'Epstein-Glaser en exploitant le point de vue developpe par Radzikowski pour definir les etats quantiques sur un espace-temps courbe en terme d'ensembles de front d'onde. Ces resultats furent ulterieurement generalises par Fredenhagen, Brunetti, Hollands, Wald, Rejzner, etc. Aux theories de Yang-Mills et de la gravitation. Cependant, meme pour des theories sans invariance de jauge, de nombreux details mathematiques sont restes inexplores et parfois sans verification. Dans cette these, on construit d'une facon totalement rigoureuse cette theorie dans le cas des champs sans invariance de jauge. Dans mon travail, j' ai revisite completement cette theorie, resolvant au passage plusieurs questions laissees en suspens, incorporant de nombreux resultats nouveaux autour de ce programme et, le cas echeant, apportant des details beaucoup plus precis sur les contre-termes dans le processus de renormalisation, une comprehension plus approfondie des ambiguites et une description geometrique des ensembles de front d'onde. L'ensemble de la these utilise un large eventail de techniques mathematiques : de la geometrie differentielle et pseudo riemannienne, des techniques d'analyse micro-locale et de geometrie symplectique pour les fronts d'onde, de l'analyse fonctionnelle, des resultats fins de la theorie des distributions,

https://tel.archives-ouvertes.fr/tel-00918759/document

Renormalization of quantum field theory on curved space-times, a causal approach

We study the twisted Ruelle zeta function ζX (s) for smooth Anosov vector fields X acting on flat vector bundles over smooth compact manifolds. In dimension 3, we prove Fried conjecture, relating Reidemeister torsion and ζX (0). In higher dimensions, we show more generally that ζX (0) is locally constant with respect to the vector field X under a spectral condition. As a consequence, we also show Fried conjecture for Anosov flows near the geodesic flow on the unit tangent bundle of hyperbolic 3-manifolds. This gives the first examples of non-analytic Anosov flows and geodesic flows in variable negative curvature where Fried conjecture holds true.

https://hal.archives-ouvertes.fr/hal-01828967/document

The Fried conjecture in small dimensions

We study the twisted Ruelle zeta function $\zeta_X(s)$ for smooth Anosov vector fields $X$ acting on flat vector bundles over smooth compact manifolds. In dimension $3$, we prove Fried conjecture, relating Reidemeister torsion and $\zeta_X(0)$. In higher dimensions, we show more generally that $\zeta_X(0)$ is locally constant with respect to the vector field $X$ under a spectral condition. As a consequence, we also show Fried conjecture for Anosov flows near the geodesic flow on the unit tangent bundle of hyperbolic $3$-manifolds. This gives the first examples of non-analytic Anosov flows and geodesic flows in variable negative curvature where Fried conjecture holds true.

Nguyen Viet Dang

Papers

A smooth introduction to the wavefront set

A smooth introduction to the wavefront set

Renormalization of quantum field theory on curved space-times, a causal approach

The Fried conjecture in small dimensions

Fried conjecture in small dimensions