Cambridge Mathematical Library Séries

This book is intended to serve as an introduction to the theory of semistable sheaves and at the same time to provide a survey of recent research results on the geometry of moduli spaces.The first part introduces the basic concepts in the Hilbert polynomial, slope, stability, Harder-Narasimhan filtration, Grothendieck's Quot-scheme. It presents detailed proofs of the Grauert-Mülich Theorem, the Bogomolov Inequality, the semistability of tensor products, and the boundedness of the family of semistable sheaves. It also gives a self-contained account of the construction of moduli spaces of semistable sheaves on a projective variety à la Gieseker, Maruyama, and Simpson.The second part presents some of the recent results of the geometry of moduli spaces of sheaves on an algebraic surface, following work of Mukai, O'Grady, Gieseker, Li and many others. In particular, moduli spaces of sheaves on K3 surfaces and determinant line bundles on the moduli spaces are treated in some detail. Other topics include the Serre correspondence, restriction of stable bundles to curves, symplectic structures, irreducibility and Kodaira-dimension of moduli spaces.

The book presents groundbreaking insights into the theory of functions of a complex variable, making it the first comprehensive resource in English at the undergraduate level. It explores special functions and their differential equations, reflecting E T Whittaker's pioneering contributions to mathematics. As a prominent mathematician and professor, Whittaker's work laid the foundation for future studies in this field, highlighting his significant role in advancing mathematical analysis during the early 20th century.

This classic calculus text remains a must-read for all students of introductory mathematical analysis. Clear, rigorous explanations of the mathematics of analytical number theory and calculus cover single-variable calculus, sequences, number series, more. 1921 edition.