The first systematic theory of generalized
functions (also known as distributions) was created in the early 1950s,
although some aspects were developed much earlier, most notably in the
definition of the Green's function in mathematics and in the work of
Paul Dirac on quantum electrodynamics in physics. The six-volume
collection, Generalized Functions, written by I. M. Gel′fand
and co-authors and published in Russian between 1958 and 1966, gives an
introduction to generalized functions and presents various
applications to analysis, PDE, stochastic processes, and
representation theory.
The unifying theme of Volume 6 is the study of representations of
the general linear group of order two over various fields and rings of
number-theoretic nature, most importantly over local fields
($p$-adic fields and fields of power series over finite
fields) and over the ring of adeles. Representation theory of the
latter group naturally leads to the study of automorphic functions and
related number-theoretic problems. The book contains a wealth of
information about discrete subgroups and automorphic representations,
and can be used both as a very good introduction to the subject and as
a valuable reference.
Readership
Graduate students and research mathematicians interested in
representation theory and automorphic forms.