Gray, John J. (1981) Differential equations and group theory from Riemann to Poincare. PhD thesis, University of Warwick.

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Abstract

The origins of the theory of modular and
automorphic functions are found in the work of Legendre, Gauss, Jacobi, and Kummer on elliptic functions and the hypergeometric equation.
Riemann's work on this differential equation gave a decisive impulse to
the global theory of the solutions to such equations, and was extended
by Fuchs who raised the problem: when are all solutions to a linear
differential equation algebraic? This problem was tackled in various
ways by Schwarz, Fuchs himself, Gordan, Jordan, and Klein, with results that
displayed the new methods of group theory to advantage. At the same time,
or a little earlier, the theory of modular transformations proclaimed by
Galois was explored by several mathematicians, notably Hermite, and Klein
was able to unite that work with his geometrical methods and the crucial
observations of Dedekind. This work marks the origin of the Galois theory
of function fields and the systematic study of modular functions. The
theory of linear differential equations was then further extended by
Poincare, who brought to it geometric
and group-theoretic insights strikingly
similar to, but at first independent of, those of Klein, and who opened-up
the theory of automorphic functions.

Item Type:	Thesis (PhD)
Subjects:	Q Science > QA Mathematics
Library of Congress Subject Headings (LCSH):	Differential equations, Linear -- History, Group theory -- History
Official Date:	June 1981
Dates:	Date Event June 1981 Submitted
Institution:	University of Warwick
Theses Department:	Mathematics Institute
Thesis Type:	PhD
Publication Status:	Unpublished
Supervisor(s)/Advisor:	Fowler, D. H. ; Stewart, Ian, 1945-
Extent:	128 p.
Language:	eng

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