Topology and Analysis The Atiyah-Singer Index Formula and Gauge-Theoretic Physics - Booss, B. (Springer US, 1985)
The Motivation. With intensified use of mathematical ideas, the methods and techniques of the various sciences and those for the solution of practical problems demand of the mathematician not only greater readiness for extra-mathematical applications but also more comprehensive orientations within mathematics. In applications, it is frequently less important to draw the most far-reaching conclusions from a single mathematical idea than to cover a subject or problem area tentatively by a proper "variety" of mathematical theories. To do this the mathematician must be familiar with the shared as well as specific features of different mathematical approaches, and must have experience with their interconnections. The Atiyah-Singer Index Formula, "one of the deepest and hardest results in mathematics", "probably has wider ramifications in topology and analysis than any other single result" (F. Hirzebruch) and offers perhaps a particularly fitting example for such an introduction to "Mathematics": In spite of its difficulty and immensely rich interrelations, the realm of the Index Formula can be delimited, and thus its ideas and methods can be made accessible to students in their middle * semesters. In fact, the Atiyah-Singer Index Formula has become progressively "easier" and "more transparent" over the years. The discovery of deeper and more comprehensive applications (see Chapter 111. 4) brought with it, not only a vigorous exploration of its methods particularly in the manyfacetted and always new presentations of the material by M. F.