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Actualité des théorèmes de Noether

ou

The Noether Theorems, a hundred years later

Institut Henri Poincaré

Paris, le mercredi 23 janvier 2019




Mercredi 23 janvier, 9h-10h

Pierre Cartier

(Institut des Hautes Études Scientifiques)

"A la recherche du tenseur énergie impulsion en relativité générale : Noether nous y conduit


Comme on sait, l’article de Emmy Noether est une réponse à des questions posées par Klein et Hilbert sur le tenseur d’énergie-impulsion en relativité générale. Il faut y ajouter des contributions de Albert Einstein et de Hermann Weyl. Il semble que Emmy Noether n’ait pas poussé à bout les calculs pour la relativité générale. Nous proposons une formule explicite dans l’esprit de Emmy Noether.



Mercredi 23 janvier, 17h-18h

Maxim Kontsevich (Institut des Hautes Études Scientifiques)

On the origin of closed two-form in the calculus of variations


I will explain how the closed two-form on the space of solutions of Euler-Lagrange equation arises from the first cohomology with compact support of the space-time, using certain calculus of weighted de Rham differentials.
My approach is very robust, it is applicable to the discrete time case, and also is applicable to Noether type theorems.


Mercredi 23 janvier, 16h-17h

Elizabeth Mansfield

(University of Kent)

"Noether's Theorem, smooth and discrete


In this talk, I will illustrate progress, first in the understanding of the mathematical structure of Noether's conservation laws for a geometric group action, and second their adaptation to various discrete versions. One main theme has been to understand the mathematical structure of the laws in terms of invariants and an equivariant frame. Another main theme has been to embed the laws, a priori, into numerical schemes, so that we can claim that the scheme truly incorporates the physical symmetries of the underlying model. I will indicate how we may get around the famous 'no go' theorem by Ge and Marsden and achieve this last. Time permitting, I will show how Noether's Second Theorem may also be extended to difference systems.


Mercredi 23 janvier, 11h30-12h30

Peter J. Olver

(University of Minnessota)

"Emmy Noether and Symmetry”


Noether's celebrated two theorems connect variational symmetries with conservation laws and differential identities satisfied by the Euler--Lagrange equations.  In this talk I would like to concentrate on an underappreciated but nevertheless additional fundamental contribution of her foundational paper — the notion of a generalized symmetry that, a half century later, reappears in the modern theory of integrable systems.The talk will cover Noether's contributions to symmetry analysis, their rediscovery more than 40 years later, and subsequent developments in a variety of directions.


Mercredi 23 janvier, 14h30-15h30

Tudor Ratiu

(Shanghai Jiao Tong University et EPFL)

"Momentum maps for gauge and diffeomorphism groups 


This is a report on work with Tobias Diez on an extension of the classical momentum map to group valued momentum maps that can identify topological information. Modeled on thethe momentum map of Poisson Lie group actions, we introduce the concept of group valued momentum map and construct it explicitly for actions of the gauge and diffeomorphsim groups. These momentum maps have values in differential character groups.

I will concentrate on the example of hydrodynamics, where this momentum map produces Clebsch variables admitting integer valued non-zero helicity.



Mercredi 23 janvier, 10h30-11h30

David Rowe

(Johannes Gutenberg-Universität, Mainz)

Going to School with Fräulein Noether: Rudolf J. Humm in Göttingen“


During the war years, Emmy Noether taught courses in Göttingen, though officially these had to be offered under Hilbert’s name. As is well known, she was denied the chance to habilitate in Göttingen until 1919, which prompted Einstein’s remark that “the troops returning from the field would have been done no harm were they sent to school under Fräulein Noether.” A Swiss student, Rudolf J. Humm, came to Göttingen hoping to become an expert on relativity. He took Hilbert’s courses, but he also heard talks by Noether and learned about her work on energy conservation in general relativity. After briefly describing Humm’s interactions with Hilbert, Noether, as well as Einstein, this talk will focus on Noether’s unpublished paper on energy conservation, which Humm transcribed in 1918.


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L’article original de Noether (en allemand).