Professor Dietmar Bisch, chair of the mathematics department at Vanderbilt University, has kindly written the following piece about the newly formed “Center for Noncommutative Geometry and Operator Algebras (NCGOA)” and its activities for the NCG blog.

FIFTH ANNUAL SPRING INSTITUTE ON NONCOMMUTATIVE GEOMETRY AND OPERATOR ALGEBRAS, Vanderbilt University, May 7 to May 16, 2007

The Fifth Annual Spring Institute on NCG & OA will begin this coming Monday, May 7, at Vanderbilt. The topic of this year’s meeting is index theory, Hopf cyclic cohomology and NCG. We expect about 100 participants from all over the world, including more than 60 students and postdocs. The following mini-courses will be featured at the meeting:

Alain Connes: Noncommutative geometry and motives

Nigel Higson, John Roe: Dirac operators and the Baum-Connes conjecture

Christian Kassel: An introduction to quantum groups

Atabey Kaygun, Masoud Khalkhali: From cyclic cohomology to Hopf cyclic cohomology

Giovanni Landi, Andrzej Sitarz: An Introduction to equivariant spectral triples

Ryszard Nest, Boris Tsygan: Algebraic index theorem

Michael Puschnigg: Cyclic cohomology and the Chern-Connes character

Moreover, there will be several research talks related to the topics of the spring institute. For more information, please consult the conference website.

Funding for the yearly NCGOA spring institute is provided by NSF and the College of Arts & Science at Vanderbilt University. The annual spring institute on NCGOA is organized every year by the Center for Noncommutative Geometry and Operator Algebras (NCGOA) at Vanderbilt under the direction of Alain Connes.

The Center consists of more than 20 graduate students, postdocs and senior mathematicians. The senior mathematicians in the area of NCG & OA atVanderbilt are Dietmar Bisch, Alain Connes, Bruce Hughes, Gennadi Kasparov, Jesse Peterson, Daoxing Xia, Guoliang Yu and Dechao Zheng.

Researchers affiliated with the Center cover a broad spectrum of scientific interests including K-theory of operator algebras, index theory, coarse geometry, von Neumann algebras, the theory of subfactors, operator theory, controlled topology, stratified spaces and quantum information theory. We are heavily involved in the training of students and postdocs and our work in this area is currently supported by an NSF Research Training Group grant.

Besides the annual spring institute, the Center’s activities include organization of seminars, lecture series and (small) workshops, and we are involved in running the annual East Coast Operator Algebras Symposium. Information about the previous NCGOA spring institutes is available on the web. The links are listed below. Each site contains links to lecture notes, articles or slides of talks which were sent to us by the speakers.

Just one word to say that I changed my mind and am lecturing on the metric aspect of NCG and in particular on the joint work with Chamseddine and Marcolli on the spectral action.

Hi,

I attended Yesterday’s letures and have several questions. During his

lecture Kassel said a quantum group is an object whose representations form a braided tensor category. I suppose this means not all Hopf algebras are quantum groups? Any one has an idea why is that? And exactly what is such an `object’?

I have another question. This morning Connes showed how gauge potentials will appear by considering self Morita equivalences of algebras. Masoud in his lecture also mentioned how the gauge invariance principle for Lagrangians leads to introduction of gague fields or connections as covariant derivatives. Is there any relation between the two mechanisms? I wonder also how Higgs fields appear in NCG?

Dear Chris,

It seems to me, in the context that we are talking now, the ways gauge fields appear in classical field theory and noncommutative geometry are rather different. Classically, if you try to promote a global symmetry, e.g. a U(1) symmetry psi (x) to e^{i phi}psi (x), to a local symmetry psi (x) to e^{i phi (x)}psi (x) and insist on invariance of your Lagrangian under this larger group of symmetries then you are forced to replace the partial derivatives by covariant derivatives in your Lagrangian by adding an extra term which is the gauge potential. This will also fix the transformation rule for gauge potentials. In NCG however, as Alain explained in his lecture, gauge bosons appear also when you try to push the metric structure (the spectral triple) from one algebra to a Morita equivalent one. In particular since NC algebras have non-trivial self Morita equivalences, you get the so called “inner fluctuations” of your metric by gauge bosons.

For your first question about braided categories and quantum groups I think it is a matter of your choice as to what should a quantum group be. From one point of view it is good enough to consider just any `dummy’ Hopf algebra a quantum group and vice-versa. It is true that the `classical’ quantum groups (quantum enveloping algebras and compact quantum groups) are closer to their classical counterparts in the sense that their category of representations has a braiding (if not a symmetry), defined through an specific R-matrix on your Hopf algebra. In this regard then quasi-triangular Hopf algebras are closer to groups and Lie algebras compared to a Hopf algebra in general.

This is great info to know.