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Concept

Géométrie symplectique

Résumé

La géométrie symplectique est un domaine de la recherche mathématique, s'intéressant à l'origine à une formulation mathématique naturelle de la mécanique classique et développé avec une notion d'entrelacement entre la géométrie différentielle et les systèmes dynamiques, avec des applications en géométrie algébrique, en géométrie riemannienne et en géométrie de contact. Formellement, elle consiste en l'étude des 2-formes différentielles fermées non dégénérées — appelées formes symplectiques — sur les variétés différentielles. Histoire Étymologie L'adjectif symplectique fut introduit par Hermann Weyl (1885 – 1955) pour désigner le groupe symplectique Sp(2n), le groupe des automorphismes linéaires réels de C^n conjuguant la multiplication par i à elle-même. Ce groupe était appelé « groupe du linéaire complexe » et une confusion était possible avec le groupe des automorphismes linéaires complexes. Hermann Weyl justifie son choix : Plus exa

Source officielle

https://fr.wikipedia.org/wiki/G%C3%A9om%C3%A9trie_symplectique

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Cours associés (2)

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Séances de cours associées (8)

Séances de cours associées

Chargement

A b-symplectic slice theorem

Anna Kiesenhofer

In this article, motivated by the study of symplectic structures on manifolds with boundary and the systematic study of b-symplectic manifolds started in Guillemin, Miranda, and Pires Adv. Math. 264 (2014), 864-896, we prove a slice theorem for Lie group actions on b-symplectic manifolds.

WILEY2022

Chargement

The aim of this dissertation is to solve numerically the following problem, denoted by P : given a Riemannian manifold and two points a and b belonging to that manifold, find a tangent vector T at a, such that expa(T) = b, assuming that T exists. This problem is set under an optimal control formulation, which requires the definition of an objective function and a space of control, the choice of a method for the calculation of the descent direction of that function in the space of control and the use of an optimization algorithm to find its minimum, which corresponds to the solution of the original problem by construction. Several techniques are necessary to be put together, coming from the fields of geometry, numerical analysis and optimization. The first part will concern a recalling of the mathematical context in which this formulation takes place. The general principles of optimal control will also be given. In the second part, we will present an intrinsic formulation of the optimal control problem associated to P, based on Jacobi fields, which will play the role of the so called adjoint state. This derivation leads to necessary optimality conditions. We will illustrate explicitly that formulation by treating the specific case of Riemannian manifolds with constant sectional curvature. Then, we will derive the optimal control problem in coordinates, not only to check the intrinsic formulation but also to reveal how it is hidden behind the expressions in coordinates. Their use reveals some quantities whose interpretation may be given this way. Moreover, we will show that more possibilities exist to chose the cost function and the control space in coordinates. In a second step, an alternative approach will consider the Hamiltonian formulation of geodesics. This is an incursion into symplectic geometry. We will then reformulate the Riemannian optimal control problem in its Hamiltonian version. In the third part, the numerical methods used for solving P will be presented. The discretization imposes the definition of new discrete optimal control problems. The technique shows that the discrete adjoint state equation strongly depends on the numerical scheme used to solve the direct problem. We will give a collection of numerical computations in the specific case of parametric piece of surfaces, where the surface can be defined by one or several Bézier patches, each one corresponding to a chart, which is representative of a Riemannian manifold. We will compare the different numerical approaches. The last but one part will be devoted to the interesting application of wooden roof building, where the structure is made of wooden boards, with geodesic trajectories on the designed piece of surface. The Geos (Geodesic solver) software has been developed for that purpose. After having introduced some specific numerical methods used in the code, we present the Geos application interface (AI) developed as a tool for the conception of such a roof. We then show an existing wooden structure built according to that mean. Finally, we will summarize the results of our research and discuss future possible prospects.

EPFL2007

Chargement

Motivic and p-adic Localization Phenomena

Dimitri Stelio Wyss

In this thesis we compute motivic classes of hypertoric varieties, Nakajima quiver varieties and open de Rham spaces in a certain localization of the Grothendieck ring of varieties. Furthermore we study the

p

-adic pushforward of the Haar measure under a hypertoric moment map

\mu

. This leads to an explicit formula for the Igusa zeta function

\FI_\mu(s)

\mu

, and in particular to a small set of candidate poles for

\FI_\mu(s)

. We also study various properties of the residue at the largest pole of

\FI_\mu(s)

. Finally, if

\mu

is constructed out of a quiver

\Gamma

we give a conjectural description of this residue in terms of indecomposable representations of

\Gamma

over finite depth rings. The connections between these different results is the method of proof. At the heart of each theorem lies a motivic or

p

-adic volume computation, which is only possible due to some surprising cancellations. These cancellations are reminiscent of a result in classical symplectic geometry by Duistermaat and Heckman on the localization of the Liouville measure, hence the title of the thesis.

EPFL2017

Chargement

EPFL2007

Motivic and p-adic Localization Phenomena

Dimitri Stelio Wyss

p

-adic pushforward of the Haar measure under a hypertoric moment map

\mu

. This leads to an explicit formula for the Igusa zeta function

\FI_\mu(s)

\mu

, and in particular to a small set of candidate poles for

\FI_\mu(s)

. We also study various properties of the residue at the largest pole of

\FI_\mu(s)

. Finally, if

\mu

is constructed out of a quiver

\Gamma

we give a conjectural description of this residue in terms of indecomposable representations of

\Gamma

over finite depth rings. The connections between these different results is the method of proof. At the heart of each theorem lies a motivic or

p

EPFL2017