Modem geometric methods combine the intuitiveness of spatial
visualization with the rigor of analytical derivation. Classical
analysis is shown to provide a foundation for the study of geometry
while geometrical ideas lead to analytical concepts of intrinsic
beauty. Arching over many subdisciplines of mathematics and
branching out in applications to every quantitative science, these
methods are, notes the Russian mathematician A.T. Fomenko, in tune
with the Renais sance traditions. Economists and finance theorists
are already familiar with some aspects of this synthetic tradition.
Bifurcation and catastrophe theo ries have been used to analyze the
instability of economic models. Differential topology provided
useful techniques for deriving results in general equilibrium
analysis. But they are less aware of the central role that Felix
Klein and Sophus Lie gave to group theory in the study of
geometrical systems. Lie went on to show that the special methods
used in solving differential equations can be classified through
the study of the invariance of these equations under a continuous
group of transformations. Mathematicians and physicists later
recognized the relation between Lie's work on differential
equations and symme try and, combining the visions of Hamilton,
Lie, Klein and Noether, embarked on a research program whose
vitality is attested by the innumerable books and articles written
by them as well as by biolo gists, chemists and philosophers."
General
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