Time and Location:

• The lectures take place on Fridays from 1-2:30pm in Room 122 of the Math. Sc. building.

• The first lecture is on October 1st. The last lecture will be on January 15th. There will be no lecture on December 24th.

Contact Information:

• Office: Room A02 (GCOE Annex, in front of the Mathematical Sciences Building).

• E-mail: ponge [at] ms [dot] u-tokyo [dot] ac [dot] jp.

• Phone: 03-5465-7261.

Final Report:

• Details about the final report will be given sometime in December.

Course Outline:

• A main aim noncommutative geometry 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 overview of some of the tools and methods of noncommutative geometry and present some important geometric applications.

• In the first part of the semester (Oct. 1 though Nov. 5) we shall 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).

• The 2nd part of the course (Nov. 12 through Dec. 17) will focus on noncommutative geometry and local index theory. More specifically, we shall see how noncommutative geometry provides us with a purely algebraic framework in which the local index formula holds ultimately.

• The last two lectures (Jan. 8 & 15) will present an important geometric application of the local index formula in noncommutative geometry. Namely, the transverse index theorem in diffeomorphism-invariant geometry of Connes and Moscovici.

Contents:

• Chapter 1: Spectrum and duality between spaces and algebras. Lecture notes (pdf).

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

• Chapter 3: Characteristic values and Operator Ideals. Lecture notes (pdf).

• Chapter 4: Quantized calculus. Lecture notes (pdf).

• Chapter 5: Pseudodifferential operators. Lecture notes (pdf). Appendices on densities and Sobolev spaces (pdf).

• Chapter 6: The Noncommutative residue. Lecture notes (pdf).

• Chapter 7: Quantized calculus and lower dimensional volumes. Lecture notes (pdf). Link to my paper on lower dimensional volumes (pdf).

• Chapter 8: The Atiyah-Singer index theorem. Notes on Clifford algebras, Clifford modules and Spin groups (pdf). Notes on Spin structures, Dirac operators and Fredholm operators (pdf). Notes on the local index formula of Atiyah-Singer (pdf). Appendix on characteristic classes (pdf).

• Chapter 9: K-theory. Lecture notes (pdf).

• Chapter 10: Cyclic cohomology. Lecture notes (pdf).

• Chapter 11: The local index formula in noncommutative geometry. Lecture notes (pdf).

• Chapter 12: The transverse index theorem in diffeomorphism-invariant geometry. Lecture notes (pdf).

Main References:

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

2. Connes, A.; Moscovici, H.: The local index formula in noncommutative geometry. Geometric and Functional Analysis 5 (1995), 174-243. (Available online here.)

3. Connes, A.; Moscovici, H.: Hopf algebras, cyclic cohomology and the transverse index theorem. Communications in Mathematical Physics 198 (1998), 199--246. (Available online here.)

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

5. Higson, N.: The residue index theorem of Connes and Moscovici. Surveys in Noncommutative Geometry, 71-126, Clay Mathematics Proceedings 6, AMS, Providence, 2006. (Available online here.)

6. Ponge, R.: Noncommutative geometry and lower dimensional volumes in Riemannian geometry. Letters in Mathematical Physics 83 (2008) 19-32. (Available online here.)

7. Skandalis, G.: Noncommutative geometry, the transverse signature operator, and Hopf algebras (translated from French by R. Ponge and N. Wright). Operator algebras and noncommutative geometry II, Encyclopaedia of Mathematical Sciences, 121, pp. 115-134. Springer Verlag, Berlin, 2004.