Global analysis has as its primary focus the interplay between the
local analysis and the global geometry and topology of a manifold.
This is seen classicallv in the Gauss-Bonnet theorem and its
generalizations. which culminate in the Ativah-Singer Index Theorem
[ASI] which places constraints on the solutions of elliptic systems
of partial differential equations in terms of the Fredholm index of
the associated elliptic operator and characteristic differential
forms which are related to global topologie al properties of the
manifold. The Ativah-Singer Index Theorem has been generalized in
several directions. notably by Atiyah-Singer to an index theorem
for families [AS4]. The typical setting here is given by a family
of elliptic operators (Pb) on the total space of a fibre bundle P =
F_M_B. where is defined the Hilbert space on Pb 2 L 1p
-llbl.dvollFll. In this case there is an abstract index class
indlPI E ROIBI. Once the problem is properly formulated it turns
out that no further deep analvtic information is needed in order to
identify the class. These theorems and their equivariant
counterparts have been enormously useful in topology. geometry.
physics. and in representation theory.
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