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University of California – San Diego

1. Palmer, Joseph. Symplectic invariants and moduli spaces of integrable systems.

Degree: Mathematics, 2016, University of California – San Diego

In this dissertation I prove a number of results about the symplectic geometry of finite dimensional integrable Hamiltonian systems, especially those of semitoric type. Integrable systems are, roughly, dynamical systems with the maximal amount of conserved quantities. Though the study of integrable systems goes back hundreds of years, the earliest general result in this field is the action-angle theorem of Arnold in 1963, which was later extended to a global version by Duistermaat. The results of Atiyah, Guillemin-Sternberg, and Delzant in the 1980s classified toric integrable systems, which are those produced by effective Hamiltonian torus actions. Recently, Pelayo-Vu Ngoc classified semitoric integrable systems, which generalize toric systems in dimension four, in terms of five symplectic invariants. Using this classification, I construct a metric on the space of semitoric integrable systems. To study continuous paths in this space produced via symplectic semitoric blowups, I introduce an algebraic technique to study such systems by lifting matrix equations from the special linear group SL(2,Z) to its preimage in the universal cover of SL(2,R). With this method I determine the connected components of the space of semitoric integrable systems. Motivated by this algebraic technique, I introduce the notion of a semitoric helix; the natural combinatorial invariant of semitoric systems. By applying a refined version of the algebraic method to semitoric helixes I classify all possible minimal semitoric integrable systems, which are those that do not admit a symplectic semitoric blowdown. I also produce invariants of integrable systems designed to respect the natural symmetries of such systems, especially toric and semitoric ones. For any Lie group G, I construct a G-equivariant analogue of the Ekeland-Hofer symplectic capacities. I give examples when the capacity is an invariant of integrable systems, and I study the continuity of these capacities using the metric I defined on semitoric systems. Finally, as a first step towards constructing a meaningful metric on general integrable systems, I provide a framework to study convergence properties of families of maps between manifolds which have distinct domains by defining a metric on such a collection.

Subjects/Keywords: Mathematics; integrable systems; minimal models; semitoric systems; symplectic geometry; sympletic capacities; toric geometry

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6th Edition):

Palmer, J. (2016). Symplectic invariants and moduli spaces of integrable systems. (Thesis). University of California – San Diego. Retrieved from http://www.escholarship.org/uc/item/8fm2b234

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

Chicago Manual of Style (16th Edition):

Palmer, Joseph. “Symplectic invariants and moduli spaces of integrable systems.” 2016. Thesis, University of California – San Diego. Accessed September 29, 2020. http://www.escholarship.org/uc/item/8fm2b234.

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

MLA Handbook (7th Edition):

Palmer, Joseph. “Symplectic invariants and moduli spaces of integrable systems.” 2016. Web. 29 Sep 2020.

Vancouver:

Palmer J. Symplectic invariants and moduli spaces of integrable systems. [Internet] [Thesis]. University of California – San Diego; 2016. [cited 2020 Sep 29]. Available from: http://www.escholarship.org/uc/item/8fm2b234.

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

Council of Science Editors:

Palmer J. Symplectic invariants and moduli spaces of integrable systems. [Thesis]. University of California – San Diego; 2016. Available from: http://www.escholarship.org/uc/item/8fm2b234

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

2. Dijols, Sarah. Distinguished representations : the generalized injectivity conjecture and symplectic models for unitary groups : Autour des représentations distinguées : la conjecture d'injectivité généralisée et modèles symplectiques pour les groupes unitaires.

Degree: Docteur es, Mathématiques et informatique, 2018, Aix Marseille Université

Soit G un groupe connexe quasi-déployé défini sur un corps non-Archimédien de caractéristique nulle. On suppose que l'on se donne un sous-groupe parabolique standard de décomposition de Levi P=MU ainsi qu'une représentation irréductible tempérée τ de M. Soit ν un élement dans le dual de l'algèbre de Lie de la composante déployée de M; on le choisit dans la chambre de Weyl positive. La représentation induite IPGν) est appelée module standard. Quand la représentation τ est générique (pour un caractère non-dégénéré de U), i.e a un modèle de Whittaker, le module standard IPGν) est également générique.Casselman et Shahidi ont conjecturé que l'unique sous-quotient générique apparaissait nécessairement comme sous-représentation dans le module standard IPGν). Ceci a été démontrée dans le cas des groupes classiques SO(2n+1), Sp(2n), et SO(2n) quand P est un sous-groupe parabolique maximal de G, par Hanzer en 2010.Dans notre travail, nous formulons et étudions ce problème dans le contexte plus général d'un groupe connexe quasi-déployé tel que les composantes irréductibles de Σσ sont de type A,B,C ou D.Dans la deuxième partie de cette thèse (en commun avec D.Prasad), nous prouvons d'abord qu'il n'existe pas de representation cuspidale du groupe quasi-déployé \U2n(F) qui soit distinguée par son sous-groupe \Sp2n(F) pour F un corps local non-Archimédien. Nous prouvons ensuite le théorème équivalent pour un corps global: il n'existe pas de représentation cuspidale de \U2n(\Ak) qui ait une période symplectique non nulle pour k un corps de nombres ou corps de fonctions.

Let G be a quasi-split connected reductive group over a non-Archimedean local field F of characteristic zero. We assume we are given a standard parabolic subgroup P with Levi decomposition P=MU as well as an irreducible, tempered representation τ of M. Let now ν be an element in the dual of the real Lie algebra of the split component of M; we take it in the positive Weyl chamber. The induced representation IPGν) is called a standard module. When the representation τ is generic (for a non-degenerate character of U), i.e. has a Whittaker model, the standard module IPGν) is also generic. Casselman and Shahidi have conjectured that the unique irreducible generic subquotient of a standard module IPGν) is necessarily a subrepresentation. This conjecture known as the Generalized Injectivity Conjecture was proved for the classical groups SO(2n+1), Sp(2n), and SO(2n) for P a maximal parabolic subgroup, by Hanzer in 2010.In our work, we formulate and study this problem for any quasi-split connected reductive group such that the irreducible components of Σσ are of type A,B,C or D. In the second part of this thesis (joint work with D.Prasad), we prove that there are no cuspidal representations of the quasi-split unitary groups \U2n(F) distinguished by \Sp2n(F)…

Advisors/Committee Members: Heiermann, Volker (thesis director).

Subjects/Keywords: Conjecture d'injectivité généralisée; Modèles symplectiques; Modèles de Whittaker; Generalized Injectivity Conjecture; Symplectic models; Whittaker models; 510

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6th Edition):

Dijols, S. (2018). Distinguished representations : the generalized injectivity conjecture and symplectic models for unitary groups : Autour des représentations distinguées : la conjecture d'injectivité généralisée et modèles symplectiques pour les groupes unitaires. (Doctoral Dissertation). Aix Marseille Université. Retrieved from http://www.theses.fr/2018AIXM0194

Chicago Manual of Style (16th Edition):

Dijols, Sarah. “Distinguished representations : the generalized injectivity conjecture and symplectic models for unitary groups : Autour des représentations distinguées : la conjecture d'injectivité généralisée et modèles symplectiques pour les groupes unitaires.” 2018. Doctoral Dissertation, Aix Marseille Université. Accessed September 29, 2020. http://www.theses.fr/2018AIXM0194.

MLA Handbook (7th Edition):

Dijols, Sarah. “Distinguished representations : the generalized injectivity conjecture and symplectic models for unitary groups : Autour des représentations distinguées : la conjecture d'injectivité généralisée et modèles symplectiques pour les groupes unitaires.” 2018. Web. 29 Sep 2020.

Vancouver:

Dijols S. Distinguished representations : the generalized injectivity conjecture and symplectic models for unitary groups : Autour des représentations distinguées : la conjecture d'injectivité généralisée et modèles symplectiques pour les groupes unitaires. [Internet] [Doctoral dissertation]. Aix Marseille Université 2018. [cited 2020 Sep 29]. Available from: http://www.theses.fr/2018AIXM0194.

Council of Science Editors:

Dijols S. Distinguished representations : the generalized injectivity conjecture and symplectic models for unitary groups : Autour des représentations distinguées : la conjecture d'injectivité généralisée et modèles symplectiques pour les groupes unitaires. [Doctoral Dissertation]. Aix Marseille Université 2018. Available from: http://www.theses.fr/2018AIXM0194

3. Yamagishi, Ryo. On smoothness of minimal models of quotient singularities by finite subgroups of SLn(C) .

Degree: 2018, Kyoto University

Subjects/Keywords: quotient singularities; crepant resolutions; minimal models; Cox rings; symplectic reolutions

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6th Edition):

Yamagishi, R. (2018). On smoothness of minimal models of quotient singularities by finite subgroups of SLn(C) . (Thesis). Kyoto University. Retrieved from http://hdl.handle.net/2433/232219

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

Chicago Manual of Style (16th Edition):

Yamagishi, Ryo. “On smoothness of minimal models of quotient singularities by finite subgroups of SLn(C) .” 2018. Thesis, Kyoto University. Accessed September 29, 2020. http://hdl.handle.net/2433/232219.

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

MLA Handbook (7th Edition):

Yamagishi, Ryo. “On smoothness of minimal models of quotient singularities by finite subgroups of SLn(C) .” 2018. Web. 29 Sep 2020.

Vancouver:

Yamagishi R. On smoothness of minimal models of quotient singularities by finite subgroups of SLn(C) . [Internet] [Thesis]. Kyoto University; 2018. [cited 2020 Sep 29]. Available from: http://hdl.handle.net/2433/232219.

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

Council of Science Editors:

Yamagishi R. On smoothness of minimal models of quotient singularities by finite subgroups of SLn(C) . [Thesis]. Kyoto University; 2018. Available from: http://hdl.handle.net/2433/232219

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

.