Time and Location:

• The lectures take place on Tuesdays and Thusrsdays from 2pm to 3:15pm in 20-116.

Contact Information:

• Office: 27-314.

• E-mail: ponge [dot] snu [at] gmail [dot] com.

• Phone: 02-887-4694.

Evaluation:

• Evaluation will be based on homework. There will be up to 4-5 homework assignments throughout the semester.

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, 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 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 part will be to devoted to 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.

• Chapter 2: Operators on a Hilbert space.

• Chapter 3: Characteristic values and Operator Ideals.

• Chapter 4: Quantized calculus.

• Chapter 5: Pseudodifferential operators.

• Chapter 6: The Noncommutative residue.

• Chapter 7: Quantized calculus and lower dimensional volumes.

• Chapter 8: The Atiyah-Singer index theorem.

• Chapter 9: K-theory.

• Chapter 10: Cyclic cohomology.

• Chapter 11: The local index formula in noncommutative geometry.

• Chapter 12: The transverse index theorem in diffeomorphism-invariant geometry.

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. Ponge, R.: Introduction to Noncommutative Geometry. Lecture notes, graduate course, University of Tokyo, Oct. 2010-Jan. 2011 (Available online here.)

8. 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.