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This book surveys progress in the domains described in the hitherto unpublished manuscript 'Esquisse d'un Programme' (Sketch of a Program) by Alexander Grothendieck. It will be of wide interest amongst workers in algebraic geometry, number theory, algebra and topology.

Geometry, Algebraic. --- Moduli theory. --- Theory of moduli --- Analytic spaces --- Functions of several complex variables --- Geometry, Algebraic --- Algebraic geometry --- Geometry

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Vector bundles and their associated moduli spaces are of fundamental importance in algebraic geometry. In recent decades this subject has been greatly enhanced by its relationships with other areas of mathematics, including differential geometry, topology and even theoretical physics, specifically gauge theory, quantum field theory and string theory. Peter E. Newstead has been a leading figure in this field almost from its inception and has made many seminal contributions to our understanding of moduli spaces of stable bundles. This volume has been assembled in tribute to Professor Newstead and his contribution to algebraic geometry. Some of the subject's leading experts cover foundational material, while the survey and research papers focus on topics at the forefront of the field. This volume is suitable for both graduate students and more experienced researchers.

Vector bundles. --- Moduli theory. --- Theory of moduli --- Analytic spaces --- Functions of several complex variables --- Geometry, Algebraic --- Fiber spaces (Mathematics)

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This volume presents the construction of canonical modular compactifications of moduli spaces for polarized Abelian varieties (possibly with level structure), building on the earlier work of Alexeev, Nakamura, and Namikawa. This provides a different approach to compactifying these spaces than the more classical approach using toroical embeddings, which are not canonical. There are two main new contributions in this monograph: (1) The introduction of logarithmic geometry as understood by Fontaine, Illusie, and Kato to the study of degenerating Abelian varieties; and (2) the construction of canonical compactifications for moduli spaces with higher degree polarizations based on stack-theoretic techniques and a study of the theta group.

Abelian varieties. --- Moduli theory. --- Theory of moduli --- Analytic spaces --- Functions of several complex variables --- Geometry, Algebraic --- Varieties, Abelian --- Geometry, algebraic. --- Algebraic Geometry. --- Algebraic geometry --- Geometry --- Algebraic geometry.

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The relations between Frobenius manifolds and singularity theory are treated here in a rigorous yet accessible manner. For those working in singularity theory or other areas of complex geometry, this book will open the door to the study of Frobenius manifolds. This class of manifolds are now known to be relevant for the study of singularity theory, quantum cohomology, mirror symmetry, symplectic geometry and integrable systems. The first part of the book explains the theory of manifolds with a multiplication on the tangent bundle. The second presents a simplified explanation of the role of Frobenius manifolds in singularity theory along with all the necessary tools and several applications. Readers will find here a careful and sound study of the fundamental structures and results in this exciting branch of maths. This book will serve as an excellent resource for researchers and graduate students who wish to work in this area.

Singularities (Mathematics) --- Frobenius algebras. --- Moduli theory. --- Theory of moduli --- Analytic spaces --- Functions of several complex variables --- Geometry, Algebraic --- Algebras, Frobenius --- Associative algebras

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Algebraic geometry --- Geometry, Algebraic. --- Moduli theory. --- Singularities (Mathematics) --- 512.55 --- Geometry, Algebraic --- Moduli theory --- Theory of moduli --- Analytic spaces --- Functions of several complex variables --- Geometry --- Rings and modules --- Singularities (Mathematics). --- 512.55 Rings and modules