By Nicholas M. Katz

Riemann brought the idea that of a "local method" on P^{1}-{a finite set of issues} approximately one hundred forty years in the past. His inspiration used to be to check *n*th order linear differential equations by way of learning the rank *n* neighborhood platforms (of neighborhood holomorphic suggestions) to which they gave upward thrust. His first program was once to check the classical Gauss hypergeometric functionality, which he did via learning rank-two neighborhood structures on P^{1}- {0,1,infinity}. His research used to be profitable, principally simply because this type of (irreducible) neighborhood procedure is inflexible within the experience that it's globally decided once one is familiar with individually every one of its neighborhood monodromies. It turned transparent that success performed a task in Riemann's luck: so much neighborhood platforms usually are not inflexible. but many classical features are options of differential equations whose neighborhood structures are inflexible, together with either one of the normal *n*th order generalizations of the hypergeometric functionality, _{n} *F* _{n-1}'s, and the Pochhammer hypergeometric functions.

This ebook is dedicated to developing all (irreducible) inflexible neighborhood structures on P^{1}-{a finite set of issues} and spotting which collections of independently given neighborhood monodromies come up because the neighborhood monodromies of irreducible inflexible neighborhood systems.

Although the issues addressed the following return to Riemann, and appear to be difficulties in complicated research, their suggestions rely primarily on loads of very contemporary mathematics algebraic geometry, together with Grothendieck's etale cohomology conception, Deligne's evidence of his far-reaching generalization of the unique Weil Conjectures, the speculation of perverse sheaves, and Laumon's paintings at the *l*-adic Fourier Transform.