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The Geometry of Four-manifolds
This text provides an accessible account to the modern study of the geometry of four-manifolds. Prerequisites are a firm grounding in differential topology and geometry, as may be gained from the first year of a graduate course.
The Seiberg-Witten Equations and Applications to the Topology of Smooth Four-Manifolds. (MN-44), Volume 44
The recent introduction of the Seiberg-Witten invariants of smooth four-manifolds has revolutionized the study of those manifolds. The invariants are gauge-theoretic in nature and are close cousins of the much-studied SU(2)-invariants defined over fifteen years ago by Donaldson. On a practical level, the new invariants have proved to be more powerful and have led to a vast generalization of earlier results. This book is an introduction to the Seiberg-Witten invariants. The work begins with a review of the classical material on Spin c structures and their associated Dirac operators. Next comes a discussion of the Seiberg-Witten equations, which is set in the context of nonlinear elliptic oper...
Nonlinear Methods in Riemannian and Kählerian Geometry
In this book, I present an expanded version of the contents of my lectures at a Seminar of the DMV (Deutsche Mathematiker Vereinigung) in Diisseldorf, June, 1986. The title "Nonlinear methods in complex geometry" already indicates a combination of techniques from nonlinear partial differential equations and geometric concepts. In older geometric investigations, usually the local aspects attracted more attention than the global ones as differential geometry in its foundations provides approximations of local phenomena through infinitesimal or differential constructions. Here, all equations are linear. If one wants to consider global aspects, however, usually the presence of curvature leads to...
A Transverse Dreamer
The final text of the Book of Micah provokes a series of questions: - Can the Book be read as a coherent composition or is it the result of a complex redaction history? - Was Micah a prophet of doom whose literary heritage was later softened by the inclusion of oracles of salvation? The essays in this book center around these questions. Some of them are of a more general character, while others analyze specific passages. Some articles discuss the Book of Micah by looking at specific themes (prophecy; religious polemics; metaphors). The others are concerned with the proclamation of a peaceful future (Micah 4:1-5); the famous moral incentive in Micah 6:8 and the question of prophetic and divine gender in Micah 7:8-13. They have two features in common: - A thorough reading of the Hebrew text informed by grammar and syntax. - A comparative approach: the Book of Micah is seen as part of the ancient Near Eastern culture. All in all, the author defends the view that the Book of Micah contains three independent literary elements: Micah 1: a prophecy of doom; Micah 2-5 a two-sided futurology, and 6-8 a later appropriation of Micah’s message.
Quaternionic Structures in Mathematics and Physics
During the last five years, after the first meeting on OC Quaternionic Structures in Mathematics and PhysicsOCO, interest in quaternionic geometry and its applications has continued to increase. Progress has been made in constructing new classes of manifolds with quaternionic structures (quaternionic Knhler, hyper-Knhler, hyper-complex, etc.), studying the differential geometry of special classes of such manifolds and their submanifolds, understanding relations between the quaternionic structure and other differential-geometric structures, and also in physical applications of quaternionic geometry. Some generalizations of classical quaternion-like structures (like HKT structures and hyper-Kn...
Recent Advances in Riemannian and Lorentzian Geometries
This volume covers material presented by invited speakers at the AMS special session on Riemannian and Lorentzian geometries held at the annual Joint Mathematics Meetings in Baltimore. Topics covered include classification of curvature-related operators, curvature-homogeneous Einstein 4-manifolds, linear stability/instability singularity and hyperbolic operators of spacetimes, spectral geometry of holomorphic manifolds, cut loci of nilpotent Lie groups, conformal geometry of almost Hermitian manifolds, and also submanifolds of complex and contact spaces. This volume can serve as a good reference source and provide indications for further research. It is suitable for graduate students and research mathematicians interested in differential geometry.
Emmy Noether in Bryn Mawr
Sponsored by the Association for Women in Mathematics
Mathematische Werke / Mathematical Works
For most mathematicians and many mathematical physicists the name Erich Kähler is strongly tied to important geometric notions such as Kähler metrics, Kähler manifolds and Kähler groups. They all go back to a paper of 14 pages written in 1932. This, however, is just a small part of Kähler's many outstanding achievements which cover an unusually wide area: From celestial mechanics he got into complex function theory, differential equations, analytic and complex geometry with differential forms, and then into his main topic, i.e. arithmetic geometry where he constructed a system of notions which is a precursor and, in large parts, equivalent to the now used system of Grothendieck and Dieu...
