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**Location and time:**

In these sessions we will review some of the basics of differential geometry needed for the rest of the course. For schedule and other details, see the programme.

**Aim:**

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.

** Lecture Notes**.

In the first half of the course, on elliptic operators, we will use lecture notes.

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):

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 during next week's exercise session. 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. count for 50 percent

Analysis of several variables, basic theory of manifolds, in particular differential forms. Basics of functional analysis.

Last update: 4/9-2009