Books in the Translations of Mathematical Monographs series

This book, together with the companion volume, Fermat's Last Theorem: The Proof, presents in full detail the proof of Fermat's Last Theorem given by Wiles and Taylor. With these two books, the reader will be able to see the whole picture of the proof to appreciate one of the deepest achievements in the history of mathematics.

Focuses on the fundamentals of a theory, which is an analog of affine algebraic geometry for partial differential equations. This work describes applications of Secondary Calculus ranging from algebraic geometry to field theory, classical and quantum, including areas such as characteristic classes, differential invariants and variational calculus.

Intended for students wishing to deepen their knowledge of mathematical analysis and for those teaching courses in this area, this title contains problems, some of which are well-known theorems in analysis. It also includes brief but detailed solutions to most of the problems.

Aims to give an exposition of generalized (co)homology theories that can be read by mathematicians who are not experts in algebraic topology. This work starts with basic notions of homotopy theory and then introduces the axioms of generalized (co)homology theory. It focuses on spectral sequences and their use in generalized cohomology theories.

Algebraic geometry plays an important role in several branches of science and technology. This book discusses dimension theory, flat and proper morphisms, regular schemes, smooth morphisms, completion, and Zariski's main theorem. It also presents the theory of algebraic curves and their Jacobians.

Presents an understanding of scattering theory, starting with basic principles and extending to research. This work presents a comprehensive exposition of the theory, covering different methods (of trace class and smooth perturbations) and approaches (time dependent and stationary) and discussing the relationships among them.

Covers topics including fundamental notions such as homeomorphisms, homotopy equivalence, fundamental groups and higher homotopy groups, homology and cohomology, fiber bundles, spectral sequences and characteristic classes. This work considers objects and examples including the torus, the Mobius strip, the Klein bottle and closed surfaces.

Treats inverse problems in the theory of small oscillations of systems with finitely many degrees of freedom, which requires finding the potential energy of a system from the observations of its oscillations.

In a very broad sense, 'spaces' are objects of study in geometry, and 'functions' are objects of study in analysis. There are, however, deep relations between functions defined on a space and the shape of the space, and the study of these relations is the main theme of Morse theory. This book describes Morse theory for finite dimensions.

The main goal of this book is to present the so-called birational Arakelov geometry, which can be viewed as an arithmetic analog of the classical birational geometry. After explaining classical results about the geometry of numbers, the author starts with Arakelov geometry for arithmetic curves, and continues with Arakelov geometry of arithmetic surfaces and higher-dimensional varieties.

Based on a course given by the author at the Mechanics-Mathematics Faculty of Moscow University, this book covers mathematical analysis. It includes bibliography and indexes.

Determining the invariant subspaces of any given transformation and writing the transformation as an integral in terms of invariant subspaces is a fundamental problem. This book presents the foundations of the theory of triangular and Jordan representations of bounded linear operators in Hilbert space.

Based on courses taught by the author at Moscow State University, this book features such topics as the theory of Banach and Hilbert tensor products, the theory of distributions and weak topologies, and Borel operator calculus. It contains many examples illustrating the general theory presented, as well as multiple exercises to help the reader.

The space of all Riemann surfaces (the so-called moduli space) plays an important role in algebraic geometry and its applications to quantum field theory. This book focuses on the study of topological properties of this space and of similar moduli spaces, such as the space of real algebraic curves, and the space of mappings.