SG-classes, singular symplectic geoemtry, and order preserving isomorphisms

Publikation: Qualifikations-/Studienabschlussarbeit › Dissertation

Autoren

  • Alessandro Pietro Contini

Organisationseinheiten

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Details

OriginalspracheEnglisch
QualifikationDoctor rerum naturalium
Gradverleihende Hochschule
Betreut von
  • Elmar Schrohe, Betreuer*in
Datum der Verleihung des Grades26 Juli 2023
ErscheinungsortHannover
PublikationsstatusVeröffentlicht - 2023

Abstract

Die geometrischen Kalküle von Pseudo-differenzial- und Fourier-Integraloperatoren beruhen auf den symplektischen Eigenschaften des Kotangentialbündels. Um neue Kalküle zu entwickeln, die an eine besondere Geometrie angepasst sind, ist es nötig, singulär-symplektische Mannigfaltigkeiten zu betrachten. Diese müssen zuerst verstanden werden, bevor man die zugehorigen Operatorkalküle konstruieren kann. In dieser Dissertation geben wir neue Einblicke in die singulär-symplektischen Strukturen, die aus asymptotisch-Euklidischen Mannigfaltigkeiten entstehen. Insbesondere rechnen wir aus, wie die Poisson-Klammer auf SG-Pseudo-Differenzialoperatoren wirkt, und definieren eine neue Klasse symplektischer Abbildungen, die an die geometrischen Besonderheiten angepasst sind. Wir betrachten außerdem die ordnungserhaltenden Isomorphismen der SG-Algebra und zeigen, dass unser Konzept von singulär-symplektischen Abbildungen natürlich in diesem Zusammenhang auftaucht. Wir benutzen es, um diese Isomorphismen als Konjugation mit einem SG-Fourier-Integraloperator zu charakterisieren.

Zitieren

SG-classes, singular symplectic geoemtry, and order preserving isomorphisms. / Contini, Alessandro Pietro.
Hannover, 2023. 68 S.

Publikation: Qualifikations-/Studienabschlussarbeit › Dissertation

Contini, AP 2023, 'SG-classes, singular symplectic geoemtry, and order preserving isomorphisms', Doctor rerum naturalium, Gottfried Wilhelm Leibniz Universität Hannover, Hannover. https://doi.org/10.15488/15187
Contini, A. P. (2023). SG-classes, singular symplectic geoemtry, and order preserving isomorphisms. [Dissertation, Gottfried Wilhelm Leibniz Universität Hannover]. https://doi.org/10.15488/15187
Contini AP. SG-classes, singular symplectic geoemtry, and order preserving isomorphisms. Hannover, 2023. 68 S. doi: 10.15488/15187
Contini, Alessandro Pietro. / SG-classes, singular symplectic geoemtry, and order preserving isomorphisms. Hannover, 2023. 68 S.
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T1 - SG-classes, singular symplectic geoemtry, and order preserving isomorphisms

AU - Contini, Alessandro Pietro

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N2 - The geometric theory of pseudo-differential and Fourier Integral Operators relies on the symplectic structure of cotangent bundles. If one is to study calculi with some specific feature adapted to a geometric situation, the corresponding notion of cotangent bundle needs to be adapted as well and leads to spaces with a singular symplectic structure. Analysing these singularities is a necessary step in order to construct the calculus itself. In this thesis we provide some new insights into the symplectic structures arising from asymptotically Euclidean manifolds. In particular, we study the action of the Poisson bracket on SG-pseudo-differential operators and define a new class of singular symplectomorphisms, taking into account the geometric picture. We then consider this notion in the context of the characterisation of order-preserving isomorphisms of the SG-algebra, and show that these are in fact given by conjugation with a Fourier Integral Operator of SG-type.

AB - The geometric theory of pseudo-differential and Fourier Integral Operators relies on the symplectic structure of cotangent bundles. If one is to study calculi with some specific feature adapted to a geometric situation, the corresponding notion of cotangent bundle needs to be adapted as well and leads to spaces with a singular symplectic structure. Analysing these singularities is a necessary step in order to construct the calculus itself. In this thesis we provide some new insights into the symplectic structures arising from asymptotically Euclidean manifolds. In particular, we study the action of the Poisson bracket on SG-pseudo-differential operators and define a new class of singular symplectomorphisms, taking into account the geometric picture. We then consider this notion in the context of the characterisation of order-preserving isomorphisms of the SG-algebra, and show that these are in fact given by conjugation with a Fourier Integral Operator of SG-type.

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