## Summer School on Operator Algebras and Non-commutative Geometry

- Start Date: 06/14/2010
- End Date: 06/25/2010

Nate Brown (Penn State)

Heath Emerson (Victoria)

Marcelo Laca (Victoria)

Ralf Meyer (Goettingen)

Sergey Neshveyev (Oslo)

Andrew Toms (Purdue)

University of Victoria

This is an event organized by the PIMS CRG in Operator Algebras and Non-commutative Geometry. Interested potential participants please contact the organizing committee.

The Summer School will feature three 10-Lecture Series:

1. The structure of nuclear C*-algebras *(Brown and Toms)*

2. KK-theory and the Baum-Connes conjecture *(Emerson and Meyer)
*

3. C*-dynamical systems *(Laca and Neshveyev)*

SS_NCG_Program_wk1.pdf

SS_NCG_Program_wk2.pdf

DAY1_NOTES_SSNCG_2010.pdf

DAY2_NOTES_SSNCG_2010.pdf

DAY3_4_NOTES_SSNCG_2010.pdf

DAY5_NOTES_SSNCG_2010.pdf

Brown-VictoriaNotes.pdf

EM_ KK-theory_BC_Conjecture.pdf

TomsLectureNotes.pdf

Operator_Algebras_&_NCG_Report.pdf

Please see the summer school schedule in the **Downloads** section above.

Heath Emerson (University of Victoria)

Thierry Giordano (University of Ottawa)

Marcelo Laca (University of Victoria)

Ian Putnam (University of Victoria)

The summer school will be immediately followed by a conference on Selected Topics in Non-Commutative Geometry.

**REGISTRATION**

There will be a $50 registration fee payable at the beginning of the summer school.

**FUNDING**

Partial funding from PIMS and MSRI is available for graduate students and postdocs. Please contact Kristina McKinnon by email to pims@uvic.ca in this regard, stating your status (grad student, postdoc, etc.), affiliation, area of interest and the name of your advisor/supervisor. Please apply early, and in any case before March 15, as some of the funding is allocated on a first-come, first-serve basis, and is subject to that deadline.

**ACCOMMODATION:**

Accommodations have been blocked for participants and speakers. Individuals are responsible for securing their own accommodations. There are 25 dorm-type rooms available, and several cluster houses. Each cluster house has 4 bedrooms and is completely furnished. When booking your accommodations please quote title of event SSOANCG. These are on first-come first-serve basis. Please see UVic visitor's accommodation rates: http://housing.uvic.ca/visitor/visitoraccom.php. We also have several guest suites blocked at the Executive House Hotel downtown Victoria. They can be contacted at 1-800-663-7001, http://executivehouse.com

Campus Map: http://www.uvic.ca/home/about/campus-info/maps/

Activities in Victoria: http://www.tourismvictoria.com

**COURSE DESCRIPTION **

**The structure of nuclear C*-algebras. N. Brown (Penn State) and A. Toms (Purdue)**

Summary: This series of lectures will introduce students to nuclear C*-algebras and current developments in their classification. We will begin with elementary facts and hope to end near the current frontier of research. Topics will include K-theory, the Cuntz semigroup, decomposition rank, C*-dynamical systems and other things related to Elliott's classification program.

**KK-theory and the Baum-Connes conjecture. H. Emerson (Victoria) and R. Meyer (Goettingen)**

Summary: The Atiyah-Singer index theorem established an important link between topology and elliptic differential equations. One of the important ingredients was a homology theory for C*-algebras called K-theory. The Baum-Connes conjecture attempts to understand better how to calculate K-theory groups of particular C*-algebras. In the last 10 to 20 years it has sparked an extraordinary amount of activity and a lively interaction between C*-algebraists and specialists in other areas, like representation theory, geometric group theory, differential geometry, harmonic analysis and others. At the same time, K-theory is now used to classify C*-algebras. This workshop is intended as an introduction to the techniques used to define and study the Baum-Connes conjecture, especially Kasparov's equivariant KK-theory. In particular, we will start with a rapid introduction to K-theory for C*-algebras, which should complement the minicourse of Brown and Toms on classification of C*- algebras. The emphasis throughout will be on Kasparov's approach to the Baum-Connes conjecture, using duality, and on Baum and Connes' perspective on the index theorem, using geometric cycles to describe K-homology.

**C*-dynamical systems from number theory. M. Laca (Victoria) and S. Neshveyev (Oslo)**

Summary: The lecture series will start with the necessary background on algebraic number theory and on dynamical systems and their equilibrium states. We will then review key examples, some exhibiting uniqueness of equilibrium and some exhibiting non-uniqueness (i.e. phase transitions). Following this we will present and analyze in detail the remarkable system, due to Bost and Connes, exhibiting a phase transition with the spontaneous breaking, at low temperature, of a symmetry given by the Galois group of the maximal abelian extension of the rationals. The second half of the minicourse will aim to reach the state of the art in the subject and to shed light on its connection with explicit class field theory. It will include the phase transition of the generalization of the Bost-Connes system to imaginary quadratic fields due to Connes, Marcolli and Ramachandran and that of the GL2-system of Connes and Marcolli. We plan to develop some of the background, basic constructions and examples as guided exercises suitable for informal discussion.

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Group picture