Monodromy Representation

Ramified Integrals, Singularities and Lacunas

Introduces the Picard-Lefschetz theory, which controls the ramifications and qualitative behavior of many important functions of partial differential equations and integral geometry, and reviews the fundamentals of singularity (mathematical catastrophe) theory, on which it is based. Then uses Picard-Lefschetz as a topological tool to investigate the analytic properties of three famous classes of functions: the volume functions, which appear in the Archimedes-Newton problem on integrable bodies; the Newton-Coulomb potentials; and the Green functions of hyperbolic equations as studied in the Hadamard-Pretrovskii-Atiyah-Bott-Garding lacuna theory. Assumes a familiarity with the analysis of differential forms and homology theory. Annotation copyright by Book News, Inc., Portland, OR

Differential Topology of Complex Surfaces

This book is about the smooth classification of a certain class of algebraicsurfaces, namely regular elliptic surfaces of geometric genus one, i.e. elliptic surfaces with b1 = 0 and b2+ = 3. The authors give a complete classification of these surfaces up to diffeomorphism. They achieve this result by partially computing one of Donalson's polynomial invariants. The computation is carried out using techniques from algebraic geometry. In these computations both thebasic facts about the Donaldson invariants and the relationship of the moduli space of ASD connections with the moduli space of stable bundles are assumed known. Some familiarity with the basic facts of the theory of moduliof sheaves and bundles on a surface is also assumed. This work gives a good and fairly comprehensive indication of how the methods of algebraic geometry can be used to compute Donaldson invariants.

Bulletin (new Series) of the American Mathematical Society