Teachers:
Location and time: TBA
Organization:
Exam:
There will be take home exercises. They will count for 50 percent of your grade. At the end of the course there will be a final examination. Each week we will list several exercises. We advise you to do them in the order suggested. You should be able to do at least three of them each week. You are not allowed to hand these exercises in for correction. However, three assistants will be present in class to help you with their expertise. In addition, each week there will be a take home exercise to be handed in two weeks later, at the start of the lectures. It will be graded by one of the teaching assistents with a grade from 1 to 10. At the end of the course, the average grade will contribute to your final grade with a weight of 50 percent.
Prerequisites:
Analysis of several variables, basic theory of manifolds, in particular differential forms. Basics of functional analysis. If you lack some of these, we strongly advise you to follow the intensive reminder (see below).
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.
The schedule week by week (here we will try to add, after each lecture, a description of what was discussed in the lectures + the exercises):
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.
Description:
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.
Literature:
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:
Material on vector bundles and characteristic classes (besides the material already mentioned):
Material on Functional Analysis, Analysis on Manifolds, Pseudodifferential operators (besides the papers of Atiyah-Singer and the book of Gilkey mentioned above):
Last update: 20/11-2012