Abstract:
I will present a Lagrangian action on fields, the critical points of which lead to solutions of the Einstein-Yang-Mills equations, in the spirit of Kaluza-Klein theories. The novelty is that the a priori fiber bundle structure hypothesis is not required: fields are defined on a "space-time" $Y$ of dimension $4+r$ without any a priori principal bundle structure, where $r$ is the dimension of the structure group. If the latter group is compact and simply connected, to each solution of the Euler-Lagrange equations it corresponds a 4-dimensional pseudo-Riemannian manifold $X$ (which can be interpreted as our usual space-time) in such a way that $Y$ acquires a principal bundle structure over $X$ equipped with a connection. Moreover the metric on $X$ and the connection on $Y$ are solutions of the Einstein-Yang-Mills system. If the structure group is $U(1)$ (the case which corresponds to the Einstein-Maxwell system) the situation is slightly degenerated and supplementary hypotheses are necessary.
Biography:
Professor Hélein got his PhD in 1989 at Ecole Polytechnique under the supervision of Jean Michel Coron, and became Professor at Ecole Normale Supérieure de Cachan two years later in 1991, he moved in 2003 to University of Paris 7, now called University of Paris Cité. He worked initially in harmonic maps between manifolds. Using Cartan's moving frame, he proved the milestone regularity result for weak harmonic maps on surface. His famous work with Bethuel-Brezis on the Ginzburg-Landau equation inspired a great deal of subsequent research. Since twenty years, his research interest moved to the mathematics physics for Hamiltonian formalism and integrable systems. He was awarded the Fermat Prize in 1999, and he was an invited speaker at ICM 1998. He was Director of the Federation of Mathematical Research of Paris Centre in 2009; elected member of Board of directors for Institute Henri-Poincaré from 2010 to 2014. He was also Scientific Director of National Network of Mathematics Libraries in France from 2012 to 2020, and has been involved at the service of mathematicians in many hard negotiations with the publishers of mathematics journals, as Springer or Elsevier.
Seminar by the NYU-ECNU Institute of Mathematical Sciences at NYU Shanghai
This event is open to the NYU Shanghai community and Math community.