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Lectures on Kähler Geometry
Kähler geometry is a beautiful and intriguing area of mathematics, of substantial research interest to both mathematicians and physicists. This self-contained graduate text provides a concise and accessible introduction to the topic. The book begins with a review of basic differential geometry, before moving on to a description of complex manifolds and holomorphic vector bundles. Kähler manifolds are discussed from the point of view of Riemannian geometry, and Hodge and Dolbeault theories are outlined, together with a simple proof of the famous Kähler identities. The final part of the text studies several aspects of compact Kähler manifolds: the Calabi conjecture, Weitzenböck techniques, Calabi–Yau manifolds, and divisors. All sections of the book end with a series of exercises and students and researchers working in the fields of algebraic and differential geometry and theoretical physics will find that the book provides them with a sound understanding of this theory.
Riemann Surfaces and Algebraic Curves
Classroom-tested and featuring over 100 exercises, this text introduces the key algebraic geometry field of Hurwitz theory.
Quaternionic Structures In Mathematics And Physics - Proceedings Of The Second Meeting
During the last five years, after the first meeting on “Quaternionic Structures in Mathematics and Physics”, 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 Kähler, hyper-Kähler, 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-Kähler manifolds with singularities) appeared naturally and were studied. Some of those results are published in this book.
Inverse Problems and Data Assimilation
A clear and concise mathematical introduction to the subjects of inverse problems and data assimilation, and their inter-relations.
Dispersive Partial Differential Equations
Introduces nonlinear dispersive partial differential equations in a detailed yet elementary way without compromising the depth and richness of the subject.
The Riemann Hypothesis for Function Fields
This book provides a lucid exposition of the connections between non-commutative geometry and the famous Riemann Hypothesis, focusing on the theory of one-dimensional varieties over a finite field. The reader will encounter many important aspects of the theory, such as Bombieri's proof of the Riemann Hypothesis for function fields, along with an explanation of the connections with Nevanlinna theory and non-commutative geometry. The connection with non-commutative geometry is given special attention, with a complete determination of the Weil terms in the explicit formula for the point counting function as a trace of a shift operator on the additive space, and a discussion of how to obtain the explicit formula from the action of the idele class group on the space of adele classes. The exposition is accessible at the graduate level and above, and provides a wealth of motivation for further research in this area.
Fast Track to Forcing
For those who wonder if the forcing theory is beyond their means: no. Directions to research in forcing are given.
Lectures on Lagrangian Torus Fibrations
Comprehensive and visual introduction to the geometry of 4-dimensional symplectic manifolds via 2-dimensional almost-toric diagrams.
Representation Theorems in Hardy Spaces
This self-contained text provides an introduction to a wide range of representation theorems and provides a complete description of the representation theorems with direct proofs for both classes of Hardy spaces: Hardy spaces of the open unit disc and Hardy spaces of the upper half plane.
