Intensive reminder: Before the start of the course, we will have an "intensive reminder" consisting of one day in which we will review some of the basics of differential geometry and one day in which we will review some basics of functional analysis.
Lecture notes: Lecture notes will be made available during the course.
Intensive reminder 1 and 2 (Wednesday, January 30): Differential Geometry day.
Intensive reminder 3 and 4 (Thursday, January 31): Functional Analysis day.
WEEK 6/Lecture 1 (February 7): A short overview of the course and of the main key-words starting from the (preliminary) version of the Atiyah-Singer index theorem (ellipticity, Fredholmness, pseudo-differential operators, characteristic classes). Then we started with chapter 1: differential operator and symbols (first trivial coefficients, then general vector bundles).
Hand-in exercise (to be handed on February 21, at the start of the lecture) : 1.1.9 from the 2012/2013 lecture notes.
WEEK 7/Lecture 2 (February 14): Densities and integration, formal adjoints of differential operators, ellipticity and Fredholmness of operators, differential complexes, elliptic complexes, how to deduce results about elliptic complexes from similar results about elliptic operators (e.g.: any elliptic complex is Fredholm). Then we started chapter 2. However, as it seemed that everyone knew the basics of locally convex vector spaces and distributions on opens in R^n, we aggreed that I will assume all of these known (kindly asking you to refresh your memory by looking at the notes), and I will continue next time assuming them (so, next time, we will go directly to manifolds).
Exercises for the werkcollege: (1.2.9 and 1.3.7), (1.3.3, 1.3.13 and 1.3.15), (1.3.5 and 1.3.16).
Hand-in exercise (to be handed on February 28, at the start of the lecture) : 1.3.14 from the 2012/2013 lecture notes.
WEEK 8/Lecture 3 (February 21): The rest of Chapter 2 (distributions on manifolds, kernels), then we started the chapter on functional spaces, discussing the main axioms and results, but without proofs.
Hand-in exercise: 2.4.6.
WEEK 9/Lecture 4 (February 28): We went back to the notion of functional spaces, the main axioms and the main results, giving the proofs. Then we showed how from a functional space on R^n (which is local and invarian) one obtains one on any open in R^n, and then one can pass to general n-dimensional manifolds and vector bundles; then we looked at the particular case of Sobolev spaces (hence we finished Chapter 3).
Exercises for the werkcollege: 3.6.3.
Hand-in exercise: 3.8.7.
WEEK 10/Lecture 5 (March 7):
WEEK 11/Lecture 6 (March 14):
WEEK 12/Lecture 7 (March 21):
WEEK 13/Lecture 8 (March 28):
WEEK 14/Lecture 9 (April 4):
WEEK 15/Lecture 10 (April 11):
WEEK 16/Lecture 11 (April 18):
WEEK 17/Lecture 12 (April 25):
WEEK 18/Lecture 13 (May 2):
WEEK 19/Lecture 14 (May 9):
WEEK 20/Lecture 15 (May 16):
WEEK 21/Lecture 16 (May 23):
Aim/content of the course:
The aim of this course is to develop the mathematical language needed to understand the Atiyah-Singer index formula.
In the 1960's M. Atiyah and I. Singer proved their index formula, which expresses the analytic index of an elliptic differential operator on a compact manifold in topological terms constructed out of the operator. This formula is one of the main bridges between analysis and topology- a bridge which stimulated a lot of further research and interplay between geometry, analysis and mathematical physics. In 2004 both mathematicians were awarded the Abel prize for their mathematical work. The goal of this course is to develop the mathematical
language needed to understand the Atiyah-Singer index formula.
In the first part of the course we will discuss the language of vector bundles on a manifold, and of differential operators between the spaces of smooth sections of these bundles. Such operators have a principal symbol. An operator with invertible principal symbol is called elliptic. An elliptic operator $D$ between vector bundles on a compact manifold is a Fredholm operator on the level of Sobolev spaces. We will discuss the proof of this result, which makes use of the construction of parametrices (inverses modulo smoothing operators) via pseudo-differential operators. The theory of pseudo-differential operators will be developed from the start, a quick review of distributions
and Sobolev spaces will be given.
The Fredholm property implies that the kernel of $D$ has finite dimension, and its image finite codimension. The difference of these natural numbers is called the analytic index of the operator.
The elliptic operator $D$ also has a topological index. The second half of the course will be devoted to the description of this index. The description makes use of the Chern classes of a complex vector bundle. These are cohomology classes on the base manifold, which can be described in terms of the curvature of a connection on the given bundle. The principal symbol $\gs(D)$ of the operator $D$ gives
rise to a particular vector bundle. The topological index of $D$ can be defined in terms of the Chern classes of this bundle.
The Atiyah-Singer index formula states that analytic and topological index of $D$ are equal. During the course we will also discuss special examples of the formula, such as the Hirzebruch-Riemann-Roch formula.
We will use lecture notes (see above).
Here is some extra-literature which may be useful to consult throughout the semester (some of which will also be used for the course):
Material on the Atiyah-Singer theorem: