Masoud Khalkhali, Curvature in Noncommutative Geometry
Δημοσιεύτηκε στις 8 Οκτ 2014
In
this talk Masoud Khalkhali, of University of Western Ontario, discusses
various topics related to the notion of curvature for noncommutative
tori, including scalar curvature, the Gauss-Bonnet theorem, and the
Einstein-Hilbert action.
References relevant to this talk include:
http://arxiv.org/abs/1110.3511
http://arxiv.org/abs/1301.6135
http://arxiv.org/abs/1111.1358
http://arxiv.org/abs/1005.4947
http://arxiv.org/abs/1410.0475
http://arxiv.org/abs/1307.5367
This
talk was given at the Perimeter Institute in May of 2014. This talk, as
well as many, MANY others can be downloaded at the PI online video
library -- follow the link below:
http://www.perimeterinstitute.ca/vide...
A pdf for this talk can be found at the following link:
http://pirsa.org/pdf/loadpdf.php?pirs...
Khalkhali's abstract for this talk is as follows:
"After
the seminal work of Connes and Tretkoff on the Gauss-Bonnet theorem for
the noncommutative 2-torus and its extension by Fathizadeh and myself,
there have been significant developments in understanding the local
differential geometry of these noncommutative spaces equipped with
curved metrics. In this talk, I will review a series of joint works with
Farzad Fathizadeh in which we compute the scalar curvature for curved
noncommutative tori and prove the analogue of Weyl's law and Connes'
trace theorem. Our final formula for the curvature matches precisely
with the one computed independently by A. Connes and H. Moscovici. I
will then report on our recent work on the computation of scalar
curvature for noncommutative 4-tori (which involves intricacies due to
violation of the Kähler condition). We show that metrics with constant
curvature are extrema of the analogue of the Einstein–Hilbert action. A
purely noncommutative feature in these works is the appearance of the
modular automorphism from Tomita–Takesaki theory of KMS states in the
final formulas for the curvature."
this talk Masoud Khalkhali, of University of Western Ontario, discusses
various topics related to the notion of curvature for noncommutative
tori, including scalar curvature, the Gauss-Bonnet theorem, and the
Einstein-Hilbert action.
References relevant to this talk include:
http://arxiv.org/abs/1110.3511
http://arxiv.org/abs/1301.6135
http://arxiv.org/abs/1111.1358
http://arxiv.org/abs/1005.4947
http://arxiv.org/abs/1410.0475
http://arxiv.org/abs/1307.5367
This
talk was given at the Perimeter Institute in May of 2014. This talk, as
well as many, MANY others can be downloaded at the PI online video
library -- follow the link below:
http://www.perimeterinstitute.ca/vide...
A pdf for this talk can be found at the following link:
http://pirsa.org/pdf/loadpdf.php?pirs...
Khalkhali's abstract for this talk is as follows:
"After
the seminal work of Connes and Tretkoff on the Gauss-Bonnet theorem for
the noncommutative 2-torus and its extension by Fathizadeh and myself,
there have been significant developments in understanding the local
differential geometry of these noncommutative spaces equipped with
curved metrics. In this talk, I will review a series of joint works with
Farzad Fathizadeh in which we compute the scalar curvature for curved
noncommutative tori and prove the analogue of Weyl's law and Connes'
trace theorem. Our final formula for the curvature matches precisely
with the one computed independently by A. Connes and H. Moscovici. I
will then report on our recent work on the computation of scalar
curvature for noncommutative 4-tori (which involves intricacies due to
violation of the Kähler condition). We show that metrics with constant
curvature are extrema of the analogue of the Einstein–Hilbert action. A
purely noncommutative feature in these works is the appearance of the
modular automorphism from Tomita–Takesaki theory of KMS states in the
final formulas for the curvature."
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