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Noncommutative
geometry is one of the most rapidly growing areas of mathematics which
has applications in many branches of mathematics and physics. One
of the goals of NCG
is to use algebras, especially noncommutative algebras and their
derivatives, to study
"spaces" which appear generally, but not
exclusively, in the classical geometry,
number theory, and physics. Among its tools, K-theory and
cyclic cohomology are more well
known and applicable. In this workshop
we overview general noncommutative
geometry, K-Theory, cyclic cohomology, Hopf cyclic
cohomology, spectral triples
and the Connes-Moscovici's
local index formula. There are four mini courses designed to be
accessible by graduate students and young researchers,
simultaneously the workshop is
rich enough to attract interested experienced researchers. The last
two days of the
workshop/conference are assigned for contributed talks.
Organizer:
Mahmood Behboodi ( Isfahan University of Thechnology, Iran)
Bahram Rangipour (University of New Brunswick, Canada)
Scientific Committee:
Professor G. Landi (Universita` di Trieste, Trieste,
Italy)
Professor H. Moscovici (Ohio State University, OH,
USA).
Speakers of the mini-courses:
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Noncommutative topology of operator
algebras
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D. Kucerovsky
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New Brunswick
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Canada
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Elements of noncommutative geometry
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G. Landi
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Trieste
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Italy
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Spectral
Triples and Local Index Formula
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A.
Pal
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Delhi Centre
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India
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Cyclic Cohomology, Hopf Cyclic
Cohomology
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B. Rangipour
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New Brunswick
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Canada
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Abstract of mini courses :
1-Noncommutative topology of operator algebras:
Abstract:
K-theory, originally
the study of vector bundles in topology, has become a powerful tool in
the subject of operator algebras, and has led to profound applications
throughout mathematics. We give an introduction to K-theory and its deep
bivariant version, Kasparov's KK-theory. Kasparov's theory has some
entirely new features with no exact counterpart in K-theory,
particularly the Kasparov product, and we explain how to calculate
the Kasparov product in favorable cases.
2-Elements of noncommutative geometry:
Abstract:
We give an
introduction to noncommutative geometry and its use. In the presented
approach, a geometric space is given a spectral description as a triple
(A, H, D) consisting of a *-algebra A represented on a Hilbert
space H together with an unbounded self-adjoint operator D
interacting with the algebra in a bounded manner. The aim is to
carry geometrical concepts over to a new class of spaces for which the
algebra of functions A is noncommutative in general. We supplement
the general theory with examples which include toric
noncommutative spaces and spaces coming from quantum groups.
3-Spectral triples and local index formula in NGG:
Abstract:
We will start
with a general motivation of the notion of a spectral triple using
commutative manifolds followed by a description of the notion and a
few examples, commutative as well as nocommutative. We then
describe in more detail what one does with a spectral triple. In
particular, how it relates to K-homology, cyclic cohomology and the
index theorem, gives rise to the notion of dimension in the
noncommutative set up, gives rise to a pseudodifferential calculus
which in turn help formulate a local index theorem. Second half of
the course will be a detailed study of Connes' local index
theorem for SU_q(2). For this, we start with a brief introduction
to the notion of a compact quantum group and its homogeneous
spaces, followed by a brief sketch of construction of good spectral
triples on SU_q(2). We then go into a detailed computation of the local index
formula following Connes' paper on SU_q(2).
4-Hopf algebras in noncommutative geometry and their Hopf-cyclic
chomology:
Abstract:
The
computation of the local index formula by Connes and Moscovici shed light
on the
role of Hopf algebras in noncommutative geometry as the true source of
symmetry. In our
lectures we survey Hopf algebras and show how they appear in
noncommutative geometry and
discuss why they are important. We also review cyclic cohomology of
algebras and Hopf
algebras and explain the relation between them. In the last part we
illustrate the
theory by computing some examples.
Call for Papers
and Contact Information:
Papers for presentation at the conference
should be sent
electronically to the following organizers.
Mahmood Behboodi,
Department of Mathematical Sciences, Isfahan
University of Thechnology, Isfshan,
Iran, Post Code: 8415683111, Phone: (+98) 311 391 3612 , Fax:(+98)311912602,
Email: mbehbood@cc.iut.ac.ir
Bahram Rangipour,
Department of Mathematics and
Statistics,University of New Brunswick, Fredericton, New Brunswick,
CANADA E3B 5A3,
Phone: 506-458-7370,
Fax: (506)
453-4705,
Email : bahram@unb.ca
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