Mathematical methods have been applied successfully to population
genet ics for a long time. Even the quite elementary ideas used
initially proved amazingly effective. For example, the famous
Hardy-Weinberg Law (1908) is basic to many calculations in
population genetics. The mathematics in the classical works of
Fisher, Haldane and Wright was also not very complicated but was of
great help for the theoretical understanding of evolutionary pro
cesses. More recently, the methods of mathematical genetics have
become more sophisticated. In use are probability theory,
stochastic processes, non linear differential and difference
equations and nonassociative algebras. First contacts with topology
have been established. Now in addition to the tra ditional movement
of mathematics for genetics, inspiration is flowing in the opposite
direction, yielding mathematics from genetics. The present mono
grapll reflects to some degree both patterns but especially the
latter one. A pioneer of this synthesis was S. N. Bernstein. He
raised-and partially solved- -the problem of characterizing all
stationary evolutionary operators, and this work was continued by
the author in a series of papers (1971-1979). This problem has not
been completely solved, but it appears that only cer tain operators
devoid of any biological significance remain to be addressed. The
results of these studies appear in chapters 4 and 5. The necessary
alge braic preliminaries are described in chapter 3 after some
elementary models in chapter 2."
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