Time and Location:

• The lectures take place on Thursdays from 10:40am-12:10pm in Room 118 of the Math. Sc. building (first lecture on April 16th).

Lectures Notes:

• Lecture notes on Chapter 1: Spectrum and Duality between Spaces and Algebras (pdf).

• Lecture notes on Chapter 2: Operators on a Hilbert Space (pdf).

• Lecture notes on Chapter 3: Characteristic Values and Compact Operators (pdf). (Updated on June 26th; latest version includes a proof of the formula for the trace of a kernel operator).

• Lecture notes on Chapter 4: Quantized Calculus (pdf). (Posted on June 26th).

• Lecture notes on Chapter 5: Pseudodifferential Operators (pdf). (Posted on June 27th; an updated version will be posted by September 7th).

• Lecture notes on Chapter 6: The Noncommutative Residue. (To be posted by September 9th).

• Lecture notes on Chapter 7: Quantized Calculus on a Manifold . (To be posted by September 12th).

• Lecture notes on Appendix A: Densities on a Manifold (pdf). (Posted on June 27th).

• Lecture notes on Appendix B: Operator Ideals and Symmetric Norms (pdf). (Posted on June 27th).

Final Report's Description:

• The project should consist of summaries of all the chapters of the course, including the chapters that will be covered during the make-up lectures in September (but there is no need to make summaries of the appendices posted on the course's website).

• For each chapter, the summary should highlight the main definitions and results of the chapter. It should be about 2-5 pages depending on the length and the importance of the chapter (but it's OK if it is longer).

• The summary of Chapter 3 (Quantized Calculus) should contain a sketch of the construction of the Dixmier trace.

• The summary of Chapter 7 (Quantized Calculus for a Smooth Manifold) should contain a sketch of the proof of the equality between the noncommutative residue and the Dixmier trace.

• The report should be written in English and sent by e-mail to me by September 25th.

• The students should mention in their reports their student ID numbers and the kana or romanji transliterations of their names (if applicable).

Objectives of the Course:

• Noncommutative geometry is a new subject launched by Alain Connes in the 80s. A main aim is to translate the tools of differential geometry into the operator theoretic language of quantum mechanics. More precisely, using the duality between spaces and algebras, we want to replace the study of noncommutative spaces, which hardly make sense, by that of noncommutative algebras playing formally the roles of the algebras of functions on "ghost" noncommutative spaces.

• The aim of this course is to provide an introduction to some tools and methods of noncommutative geometry. In the Summer semester we will see how the basic tools of calculus can be translated into the language of noncommutative geometry. As an application we will see how to give sense to "lower dimensional volumes" for Riemannian manifolds (e.g. we see will how to define the area of any Riemannian manifold dimension >2).

Contents:

• Chapter 1: Spectrum and Duality between Spaces and Algebras (2 lectures).

• Chapter 2: Operators on a Hilbert Space (0 lecture; lecture notes will be provided).

• Chapter 3: Characteristic Values and Compact Operators (2 lectures)

• Chapter 4: Quantized Calculus (2 lectures).

• Chapter 5: Pseudodifferential Operators (3 lectures).

• Chapter 6: The Noncommutative Residue (1 lecture).

• Chapter 7: Quantized Calculus for a Smooth Manifold (2 lectures).

Main References:

• Connes, A.: Noncommutative Geometry. Academic Press, San Diego, 1994. (Available online here .)

• Gracia-Bondía, J.M.; Várilly, J.C.; Figueroa, H.: Elements of Noncommutative Geometry. Birkhäuser Boston, Boston, 2001.

• Additional references will be provided along the way.