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The Center of Excellence of Algebraic Methods and     Application                                                            

 Isfahan University of Technology                         

 

      

Summer School  on Noncommutative Geometry

       

Time:
Workshop:    June 7-9, 2008    (18th Khordad to 20th Khordad, 1387)
Seminar:       June 11-12, 2008 (22nd Khordad to 23th Khordad, 1387).

Location:

Department of Mathematical Science,

Isfahan University of Technology, Isfahan, Iran

   

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  ThechnologyIran)

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: 

Noncommutative topology of operator algebras

D. Kucerovsky

 New Brunswick

Canada

Elements of noncommutative geometry

G. Landi

Trieste

Italy

Spectral Triples and Local Index Formula

A. Pal

Delhi Centre

India

Cyclic Cohomology, Hopf Cyclic Cohomology

B. Rangipour

 New Brunswick

Canada

 

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