Teremos no Auditório do Departamento de Matemática um Ciclo de Palestras em Geometria Diferencial com a Palestrante Professora Alice Barbara Tumpach - Université de Lille, que acontecerá nos dias :

12/09 quinta-feira às 10h30

18/09 quarta-feira às 10h30

25/09 quarta-feira às 10h30

LECTURES ON HAMILTONIANS SYSTEMS AND GEOMETRY
             BY ALICE BARBARA TUMPACH

Topic of the Lecture: Arnold’s approach [1] of understanding equations of hydrodynamic type as geodesics on a group of diffeomorphisms was the starting point of fruitful interactions between the field of partial differential equations and geometry [2, 3]. In a succession of three lectures, we will introduce Arnold’s view of hydrodynamics, and illustrate it with the geometric image of various hydrodynamic approximations: namely the Korteweg-de Vries (kdV) equation, the Camassa-Holm (CH) equation, and the Hunter-Saxton (HS) equation [3].

First Lecture: In the first lecture, we will use the example of matrix groups to define adjoint and co-adjoint actions of a Lie group on its Lie algebra and the dual of its Lie algebra. This will allow us to identity Lax equations as equations on (co-)adjoint orbits of a Lie group. This setting will than be generalized to the infinite dimensional case of the group of diffeomorphisms of the circle and its universal central extension, the Virasoro group, that we will introduce.

Second Lecture: In the second lecture, we will show that the Korteweg-de Vries equation, the Camassa-Holm equation, and the Hunter-Saxton equation can be regarded as equations of the geodesic flow related to different right-invariant metrics on the Virasoro group introduced in the first lecture (in the case of KdV and CH) or on an associated homogeneous space (in the case of HS).

Third Lecture: One main mechanism of integrability of evolution equations is the presence of two compatible Hamiltonian structures. It turns out that the Kortewegde Vries equation, the Camassa-Holm equation, and the Hunter Saxton equation, have the same symmetry group and similar bihamiltonian structures. The general setup of Hamiltonian formalism for Euler equations will be explained during the lecture, as well as the Magri scheme for generating conserved quantities in a bihamiltonian setup.

References
[1] Arnold, V.I., Sur la g´eome´etrie diffe´erentielle des groupes de Lie de dimension infinie et ses
applications `a l’hydrodynamique des fluides parfaits, Ann. Inst. Fourier (Grenoble) 16 (1966)
319–361;
[2] Arnold, V.I. and Khesin, B., Topological methods in hydrodynamics, V. Arnold, B. Khesin, Topological
Methods in Hydrodynamics, Springer, New York, 1998, xv+376pp.
[3] Khesin, B. and Misiolek, G., Euler equations on homogeneous spaces and Virasoro orbits, Advances
in Mathematics 176 (2003) 116–144.