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 s(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.
IMPORTANT: before the start of the course, we will give an "intensive reminder" (several full days of lectures) in which we will review some of the basics of differential geometry needed for the rest of the course. More details about the intensive reminder will be announced later on the website http://www.staff.science.uu.nl/~crain101/AS-2013/