By Jeremy Gray

This ebook is a examine of ways a specific imaginative and prescient of the team spirit of arithmetic, known as geometric functionality conception, was once created within the nineteenth century. The valuable concentration is at the convergence of 3 mathematical themes: the hypergeometric and similar linear differential equations, staff conception, and non-Euclidean geometry.

The textual content for this re-creation has been tremendously elevated and revised, and the prevailing appendices enriched with historic money owed of the Riemann--Hilbert challenge, the uniformization theorem, Picard-Vessiot thought, and the hypergeometric equation in better dimensions. The routines were retained, making it attainable to exploit the publication as a significant other to arithmetic classes on the graduate point.

This paintings remains to be the single up to date scholarly account of the heritage of a department of arithmetic that keeps to generate vital examine, for which the math has been the social gathering for the most profound paintings by way of various nineteenth century figures: Riemann, Fuchs, Dedekind, Klein, and Poincar.

Reviews of the 1st variation:

"The volume...is one amongst the main attention-grabbing books at the historical past of arithmetic released in contemporary times.... very stimulating examining for either historians of contemporary arithmetic and mathematicians as well." ---Mathematical stories

"The booklet includes an grand wealth of fabric in relation to the algebra, geometry, and research of the 19th century.... Written with exact historic point of view and transparent exposition, this booklet is actually not easy to place down." ---Zentralblatt fur Mathematik

"...must analyzing for each severe scholar of 19th century mathematics...represents a considerable contribution towards filling what's regularly said to be a tremendous hole within the old literature."---ISIS

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**Additional resources for Linear Differential Equations and Group Theory from Riemann to Poincare**

**Sample text**

1/ D30 D3C . 1. 2. Let O be a point group. 1). D2C /1 denotes the split crystallographic group generated by the point group D2C and the standard cubical lattice. 1. S4C . 1//. Chapter 5 A Splitting Formula for Lower Algebraic K -Theory Let be a three-dimensional crystallographic group with lattice L and point group H . ) In this chapter, we describe a simple construction of EVC . / and derive a splitting formula for the lower algebraic K-theory of any three-dimensional crystallographic group. 1 A Construction of EF IN .

V2 C v3 / ; v3 : 3 3 Indeed, we can assume L0 is the first lattice if H C D C6C or D6C , or that it is one of the first two lattices if H C D C3C . Proof. Suppose first that H C D C6C or D6C . L0 ; H /, where LP Ä L0 and each of the subgroups 0 hv2 ; v3 i, hv1 i is full in L . 3(4) shows that L0 D hv1 ; v2 ; v3 i. Next suppose H C D C3C . L0 ; H /, where LP Ä L0 and each of the subgroups hv2 ; v3 i, hv1 i 0 is full in L . One possibility is that L0 D hv1 ; v2 ; v3 i; we suppose otherwise. Let us consider a typical v D ˛v1 C ˇv2 C v3 2 L0 .

1. If is a three-dimensional crystallographic group, then there is an equivariant cell structure on R3 making it a model for EF IN . /. Proof. For every crystallographic group , there is a crystallographic group 0 of the same dimension, called the splitting group of ([Ra94, pp. 312–313]), and an embedding W ! 0 . The group 0 is a split crystallographic group in our sense, by Lemma 7 on page 313 of [Ra94]. It is therefore sufficient to prove the proposition for every split three-dimensional crystallographic group.