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SUMMARY:Scalar Curvature\, Gauss-Bonnet Theorem and Einstein-Hilbert Acti
on for Noncommutative Tori
DTSTART;VALUE=DATE-TIME:20131217T143000Z
DTEND;VALUE=DATE-TIME:20131217T150000Z
DTSTAMP;VALUE=DATE-TIME:20200713T105718Z
UID:indico-event-393@indico.math.cnrs.fr
DESCRIPTION:Geometric spaces are described by spectral triples (A\, H\, D)
in non-commutative geometry. In this context\, A is an involutive noncomm
utative algebra represented by bounded operators on a Hilbert space H\, an
d D is an unbounded selfadjoint operator acting in H which plays the role
of the Dirac operator\, namely that it contains the metric information whi
le interacting with the algebra in a bounded manner. The local geometric i
nvariants such as the scalar curvature of (A\, H\, D) are extracted from t
he high frequency behavior of the spectrum of D and the action of A via sp
ecial values and residues of the meromorphic extension of zeta functions
ζa to the complex plane\, which are defined for a in A by\n\n ζa (s)
= Trace (a ⎜D⎜-s)\, ℜ(s) >> 0.\n\nFollowing the semina
l work of A. Connes and P. Tretkoff on the Gauss-Bonnet theorem for the ca
nonical translation invariant conformal structure on the noncommutative tw
o torus Tθ2\, there have been significant developments in understanding t
he local differential geometry of these C*-algebras equipped with curved m
etrics. In this talk\, I will review my joint works with M. Khalkhali\, in
which we extend this result to general translation invariant conformal st
ructures on Tθ2 and compute the scalar curvature. Our final formula for t
he curvature matches precisely with the independent result of A. Connes an
d H. Moscovici. I will also present our recent work on noncommutative four
tori\, in which we compute the scalar curvature and show that the metrics
with constant curvature are extrema of the analog of the Einstein-Hilbert
action.\n\nhttps://indico.math.cnrs.fr/event/393/
LOCATION:IHES Amphitéâtre Léon Motchane
URL:https://indico.math.cnrs.fr/event/393/
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