My talk at HUS

+ 14h00 – 16h00, 21/11/2008: Nguyen Dang Manh (Institution: Section of Analysis, Faculty of Mathematics-Mechanics-Informatics)

Title: Duality in Equivariant KK-theory and the structures of C*-algebras of homogeneous spaces. 

Place: 422 T1, Ha Noi University of Science.

Abstract: 

In this text, I will give a brief description of  my thesis including the main themes and results.

The first framework in my thesis is the study of a stabilization in the equivalent -theory. In 1980, Kasparov built equivariant -theory from the inequivariant one in order to solve a special case of the Novikov conjecture. This theory was developed rapidly by many mathematicians such as Cunzt, Higson, Skandalis, Meyer, Thomsen, etc. In 2000, Meyer constructed characteristics of elements of a -group by  elementary-essential Kasparov triples when he managed to build the equivariant version of Cuntz picture for equivariant -theory. The first part of my thesis is to study the Thomsen stabilization of them under the perturbation of degenerate ones.

The second part of my thesis is to study Thomsen duality in -theory and do our problem of finding its application. In 1981, Paschke obtained the isomorphism between  and the BDF-extension group  where  is a separable unital -algebra and  is the commutant  in the Calkin algebra of the image  of  under a trivial extension. In later years, this results were generalized by Vallette, Skandalis, Higson and Thomsen. In 2005, Thomsen showed that under minor assumptions (algebras are not necessary nuclear)  equivariant -groups of couples of algebras are isomorphic to -group of a -algebra. In my thesis, I will study this duality and apply this duality to investigate the relative -theory, a generalization of the relative -homology; in some certain cases, we have obtained the same results for the relative -theory as the relative -homology.

The last part of my thesis is to propose the concepts of noncommutative homogeneous spaces, discuss when we acquire actual homogeneous spaces, and research their structures. It is well-known that the noncommutative geometry programme proposed by Alain Connes is a very big open problem. In known cases, we have already built noncommutative versions of corresponding commutative objects, such as locally Hausdorff compact spaces--algebras, vector bundles-finite projective modules, groups-quantum groups, classical analysis-quantum analysis, -theory--theory, etc. Recently, Do Ngoc Diep has constructed many noncommutative objects successfully such as the noncommutative Chern characters for some certain groups, Riemann-Roch theorem and index theorem in NCG, Graded Cch cohomology in NCG, etc (cf. His publications in MathSciNet of AMS). Our problem of investigating the structures of noncommutative homogeneous spaces is a continuing project of the series. We have obtained descriptions in -theory level of these spaces.

(See also: http://bmgt.wordpress.com/seminar/)

Motivation of KK-theory

In this post, I will discuss why Kasparov had the idea of KK-theory and what the definition of Kasparov triples came from.

Complex K-theory is an extraordinary cohomology theory on compact Hausdorff spaces. An natural interest is to find the corresponding homology theory, i.e. K-homology. From the fact that by Spanier-Whitehead duality We have already built K-homology group of a finite complex.

In 1968, Atiyah proposed the problem: Can K-homology of compact Hausdorff spaces be defined by operator theory ?.

He gave a suggestion about a generalized Fredholm operator for some compact Hausdorff space :

is the set of all triples , where

(in progess)

INTERACTION BETWEEN MATHEMATICS AND PHYSICS

There will be a so nice colloquy of Prof. Pierre Cartier:

Colloquium

Title : INTERACTION BETWEEN MATHEMATICS AND PHYSICS (probability ,combinatorics ,
algebraic geometry)
Speaker: Professor Pierre Cartier, IHES, Research Director of CNRS
Date: Fri. Nov. 14, 2008, 9:30-11:00
Place: Inst, of Math., room 301, 18 Hoàng Quốc Việt

ABSTRACT
I would like to show how deep mathematical techniques played an increasingly important role in the problems of theoretical physics . Statistical Physics relies extensively on methods from probabilitytheory . I mention especially Gaussian processes , Poisson processes , random media , martingales (averaging . Renormalization theory inQuantum Field Theory (elementary particles ) have been completely revolutionised by the intoduction of Hopf algebras by Connes and Kreimer . Lastly , the very new methods of motives in algebraic geometry have been applied by S. Bloch and his collaborators to calculate integrals connected to Feynman diagrams . I shall finish by describing the so-called COSMIC GALOIS GROUP which I suggested some years ago .

(See the formal announcement, also here)

Introduction to Lie Algebras and Lie Groups

Fall 2008

International Master Class

Institute of Mathematics

Vietnam Academy of Science and Technology

---

This course will cover the basic theory of Lie groups and Lie algebras. The prequisites include knowledge of linear algebra and group theory as covered by Algebra courses and basic notions of differential geometry (manifolds, vector fields,… etc).

	TIME and PLACE	13:30 – 16:00, Monday, Wednesday and Thursday at Lecture hall 301A, Building A5. The first lecture will be held on Wednesday, October 29, 2008.
	INSTRUCTOR	Professor Pierre Cartier, IHES The best way to contact Professor P. Cartier is during the lecture or at his office (room 110, building A5)
	CONTENTS
		Introduction: Global and infinitesimal symmetris Lie algebras: Basic definitions, enveloping algebra, Hopf lgebras, classical Lie algebras, Cartan subalgebras (roots and weights) Lie groups: Classical Lie groups, Lie algebra of a Lie group, algebraic groups, maximal torus and Bruhat decomposition Basic results about linear representations A glimpse into modern developments: Quantum groups , Lie groupoids
	TEXTBOOKS
		A. Kirillov Jr., Introduction to Lie Groups and Lie Algebras, Cambridge University Press, 2002 N. Bourbaki, Lie groups and Lie algebras Chapter 1-3 ISBN 3-540-64242-0, Chapters 4-6 ISBN 3-540-42650-7, Chapters 7-9 ISBN 3-540-43405-4 J. P. Serre, Lie Algebras and Lie Groups: 1964 Lectures given at Harvard University, LNM 1500, Springer R. Carter et al., Lectures on Lie Groups and Lie Algebras, LMS Student Texts Series, 1995 J. E. Humphreys, Introduction to Lie Algebras and Representaion theory, Springer 1978 Note: Almost all of these textbooks are available at the library of the Institute of Mathematics. Some of them are available electronically also. (The official statement is here)

What is “algebraic KK-theory” ?

Fact: Grothendieck had ideas about K-groups in 1950s.  In 1959  Atiyah and Hirzebruch brought his ideas to topology and created topological K-theory as well as operator K-theory which is K-theory for the category of C*-algebras in few years later. At the same time, Quillen also created algebraic K-theory. In 1980, Kasparov generalized operator K-theory to KK-theory. The key of Kasparov’s idea is Fredholm modules which is generalized object of Fredholm operators, also elliptic operators. They are all analytic objects. Then roughly speaking, KK-theory was built on analytic techniques.

Question: What is the same as KK-theory in algebra which generalizes algebraic K-theory ?

What is the picture of the website’s header about?

Have you ever wondered why I used a so alien picture for the header of my site ?

It is from an article of New York Times:

In Medieval Architecture, Signs of Advanced Math

K. Dudley and M. Elliff

What is a quantum group?

In order to crack the motivation and content of the Noncommutative Geometry (NCG) Program, the crucial thing that one must know is how NCG pass through quantum apparatus from the corresponding classical one.

Renew my site

After a longly inactive time of this site as though being abandoned, I decide to renew my site by eliminating some old topics and making some new topics and anticipate dozens of my research-consistent topics to fulfill my ambition. Admittedly, this seems very difficult to be come true and actually a trophy (for me). I will convince myself to believe that I will not be distracted and do the things in steps.All people of various majors are welcome.