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Bastien Jean

On Von Neumann dimension and the Atiyah L2 index theorem
Jeudi, 22 Septembre, 2022 - 17:00
Résumé : 
The aim of this talk is to explain the $L^2$-index theorem of Atiyah. Consider a Riemannian manifold $\tilde{M}$ endowed with a free and proper action of a discrete group $\Gamma$ with compact quotient $M:=\tilde{M} /\Gamma$, we are interested in the study of an elliptic differential operator $\tilde{D}$ between hermitian vector bundle on $\tilde M$ obtained as the lifting of a differential operator $D$ on $M$. If $\Gamma$ is finite, the usual index theorem gives us $index(\tilde{D}) = |\Gamma|\cdot index(D)$. The Atiyah index theorem is a generalisation including the case of infinite covering, however in this setting the index of $\tilde D$ is not well defined due to the non-compactness of $\tilde M$ and to solve this problem we will first need to introduce the notion of $\Gamma$-dimension defined by Von Neumann.
Institution de l'orateur : 
Institut Fourier
Thème de recherche : 
Salle : 
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