The theory of singularities lies at the crossroads between those branches of mathematics which are the most abstract and those which are the most applied. Algebraic and differential geometry and topology, commutative algebra and group theory are as intimately connected to singularity theory as are dynamical systems theory, control theory, differential equations, quantum mechanical and quasi-classical asymptotics, optics, and functional analysis. This collection of papers incorporates recent results of participants in the editor's ongoing seminar in singularity theory, held in the Mechanics and Mathematics Department of Moscow University for over twenty years. With its broad range of subject matter, this volume will appeal to a wide range of readers in various areas of the mathematical sciences.Among the topics covered are: construction of new knot invariants, stable cohomology of complementary spaces to diffusion diagrams, topological properties of spaces of Legendre maps, application of Weierstrass bifurcation points in projective curve flattenings, classification of singularities of projective surfaces with boundary, nonsmoothness of visible contours of smooth convex hypersurfaces, flag manifolds, hyperbolic partial differential systems, and control theory.

Almost every major mathematical theory, from 19th century classical analysis and geometry to the newest abstract constructions of category theory, have recently acquired a ""physical flavour"". In the case of representation theory, two new areas of mathematical physics - the theory of completely integrable systems and string theory - have had a great influence. In addition, the idea of supersymmetry has become a general mathematical principle that has had important ramifications in representation theory. Together with this wave of new connections and new trends in representation theory, more traditional activity, dealing mostly with the study of classical objects, has also flourished. The papers in this volume were written by members of the seminar on representation theory at Moscow University, which has been running continuously since 1961. The papers reflect some of the new influences seen in representation theory today. Among the topics included are representation theory of ""large"" groups, indecomposable representations of the affine unimodular group of the plane, dual objects for certain real reductive Lie groups, and geometrical interpretations of a certain infinite-dimensional Lie algebra.

Dynamical systems and statistical mechanics have been developing in close interaction during the past decade, and the papers in this book attest to the productiveness of this interaction. The first paper in the collection contains a new result in the theory of quantum chaos, a burgeoning line of inquiry that combines mathematics and physics and is likely in time to produce many new connections and applications. Another paper, related to the renormalization group method for the study of maps of the circle with singularities due to a jump in the derivative, demonstrates that the fixed point of the renormgroup can be sufficiently described in this case. In certain situations, the renormgroup methods work better than the traditional KAM method.Other topics covered include thermodynamic formalism for certain infinite-dimensional dynamical systems, numerical simulation of dynamical systems with hyperbolic behavior, periodic points of holomorphic maps, the theory of random media, statistical properties of the leading eigenvalue in matrix ensembles of large dimension, spectral properties of the one-dimensional Schrodinger operator. This volume will appeal to many readers, as it covers a broad range of topics and presents a view of the some of the frontier research in the Soviet Union today.

This volume contains previously unpublished papers on algebraic $K$-theory written by Leningrad mathematicians over the last few years. The main topic of the first part is the computation of $K$-theory and $K$-cohomology for special varieties, such as group varieties and their principal homogeneous spaces, flag fiber bundles and their twisted forms, $\lambda$-operations in higher $K$-theory, and Chow groups of nonsingular quadrics. The second part deals with Milnor $K$-theory: Gersten's conjecture for $K^M_3$ of a discrete valuation ring, the absence of $p$-torsion in $K^M_*$ for fields of characteristic $p$, Milnor $K$-theory and class field theory for multidimensional local fields, and the triviality of higher Chern classes for the $K$-theory of global fields.

This collection deals with several different topics related to the construction and spectral analysis of Hamiltonians of various systems arising in mathematical physics. Included are a study of the disposition and character of resonances for certain operators, with applications to solid body physics; a survey of work in the perturbation of Hamiltonians in fermion systems; an examination of the construction of the Hamiltonian for three different pointwise interacting quantum particles; and a study of the lower branches of the Hamiltonian of the lattice model for chromodynamics. The final paper presents an extensive survey of problems related to the spectrum of finite-particle lattice Hamiltonians, which arise in quantum field theory and in models in the theory of solid bodies. The book provides an introduction of sorts to a series of new methods and problems in mathematical physics.

In recent years, researchers have found new topological invariants of integrable Hamiltonian systems of differential equations and have constructed a theory for their topological classification. Each paper in this important collection describes one of the 'building blocks' of the theory, and several of the works are devoted to applications to specific physical equations. In particular, this collection covers the new topological invariants of integrable equations, the new topological obstructions to integrability, a new Morse-type theory of Bott integrals, and classification of bifurcations of the Liouville tori in integrable systems. The papers collected here grew out of the research seminar 'Contemporary Geometrical Methods' at Moscow University, under the guidance of A. T. Fomenko, V. V. Trofimov, and A. V. Bolsinov. Bringing together contributions by some of the experts in this area, this collection is the first publication to treat this theory in a comprehensive way.

For the past thirty years, E. B. Vinberg and L. A. Onishchik have conducted a seminar on Lie groups at Moscow University; about five years ago V. L. Popov became the third co-director, and the range of topics expanded to include invariant theory. Today, the seminar encompasses such areas as algebraic groups, geometry and topology of homogeneous spaces, and Kac-Moody groups and algebras. This collection of papers presents a snapshot of the research activities of this well-established seminar, including new results in Lie groups, crystallographic groups, and algebraic transformation groups. These papers will not be published elsewhere. Readers will find this volume useful for the new results it contains as well as for the open problems it poses.

This volume presents research conducted between 1989 and 1991 by the participants in the Leningrad Seminar on representation theory, dynamical systems, and their applications, headed by A. M. Vershik. The primary areas covered here are mathematical physics, Lie groups and their representations, infinite-dimensional groups, topology, and dynamical systems. The book contains a number of useful introductory surveys; for example, one paper by Vaksman and Soibelman provides a systematic description of the theory of quantum groups in the spirit of representation theory - a new and popular area for which there are few introductory surveys. A portion of the book is devoted to adic transformations and substitutions, a new area of ergodic theory. With a balance of survey papers and frontier research results, this book will appeal to graduate students and researchers alike.

The four papers in this volume examine attractors of partial differential equations, with a focus on investigation of elements of attractors. Unlike the finite-dimensional case of ordinary differential equations, an element of the attractor of a partial differential equation is itself a function of spatial variables. This dependence on spatial variables is investigated by asymptotic methods. For example, the asymptotics show that the turbulence generated in a tube by a large localized external force does not propagate to infinity along the tube if the flux of the flow is not too large. Another topic considered here is the dependence of attractors on singular perturbations of the equations. The theory of unbounded attractors of equations without bounded attracting sets is also covered. All of the articles are systematic and detailed, furnishing an excellent review of new approaches and techniques developed by the Moscow school.

The papers in this collection, written by participants of the Research Seminar on the Theory of Functions at Kharkov University, primarily address the theory of entire and subharmonic functions. Founded in 1953 by B. Ya. Levin and still functioning today, this seminar ranges over different problems in the theory of functions, functional analysis, and related problems in calculus and mathematical physics. ""Entire and Subharmonic Functions"" contains works presented recently in the seminar.

This book contains papers presented at the Seminar on Mathematical Statistics held at the Institute for Problems of Information Transmission of the Academy of Sciences in the former Soviet Union. Founded in the mid-1960s, this seminar is still active today and attracts most of the researchers in Moscow who are interested in mathematical statistics. The topics covered include density, regression, and image estimation, adaptive estimation, stochastic approximation, median estimation, sequential experimental design, and large deviations for empirical measures. This collection is distinguished by the high scientific level of the papers and their modern approach. This book will be of interest to scientists and engineers who use probability and statistics, to mathematicians and applied statisticians who work in approximation theory, and to computer scientists who work in image analysis.

The nonlinear Stokes phenomenon occurs in the local theory of differential equations (or, more concisely, local dynamics) and finds application in singularity theory. This book contains a number of papers on this subject, including a survey that begins with Stokes' pioneering works on linear theory and discusses the work of Voronin.

This book contains recent results from a group focusing on minimal surfaces in the Moscow State University seminar on modern geometrical methods, headed by A. V. Bolsinov, A. T. Fomenko, and V. V. Trofimov. The papers collected here fall into three areas: one-dimensional minimal graphs on Riemannian surfaces and the Steiner problem, two-dimensional minimal surfaces and surfaces of constant mean curvature in three-dimensional Euclidean space, and multidimensional globally minimal and harmonic surfaces in Riemannian manifolds. The volume opens with an exposition of several important problems in the modern theory of minimal surfaces that will be of interest to newcomers to the field. Prepared with attention to clarity and accessibility, these papers will appeal to mathematicians, physicists, and other researchers interested in the application of geometrical methods to specific problems.

This volume contains papers by participants in the celebrated seminar of I. M. Gelfand, which ran for more than forty years at Moscow State University. Among the authors are some of the world's most renowned mathematicians. The high scientific level of the articles makes this an important contribution to the literature.

The theory of singularities lies at the crossroads between those branches of mathematics which are the most abstract and those which are the most applied. Algebraic and differential geometry and topology, commutative algebra and group theory are as intimately connected to singularity theory as are dynamical systems theory, control theory, differential equations, quantum mechanical and quasi-classical asymptotics, optics, and functional analysis. This collection of papers incorporates recent results of participants in the editor's ongoing seminar in singularity theory, held in the Mechanics and Mathematics Department of Moscow University for over twenty years. With its broad range of subject matter, this volume will appeal to a wide range of readers in various areas of the mathematical sciences.Among the topics covered are: construction of new knot invariants, stable cohomology of complementary spaces to diffusion diagrams, topological properties of spaces of Legendre maps, application of Weierstrass bifurcation points in projective curve flattenings, classification of singularities of projective surfaces with boundary, nonsmoothness of visible contours of smooth convex hypersurfaces, flag manifolds, hyperbolic partial differential systems, and control theory.

Almost every major mathematical theory, from 19th century classical analysis and geometry to the newest abstract constructions of category theory, have recently acquired a ""physical flavour"". In the case of representation theory, two new areas of mathematical physics - the theory of completely integrable systems and string theory - have had a great influence. In addition, the idea of supersymmetry has become a general mathematical principle that has had important ramifications in representation theory. Together with this wave of new connections and new trends in representation theory, more traditional activity, dealing mostly with the study of classical objects, has also flourished. The papers in this volume were written by members of the seminar on representation theory at Moscow University, which has been running continuously since 1961. The papers reflect some of the new influences seen in representation theory today. Among the topics included are representation theory of ""large"" groups, indecomposable representations of the affine unimodular group of the plane, dual objects for certain real reductive Lie groups, and geometrical interpretations of a certain infinite-dimensional Lie algebra.

Dynamical systems and statistical mechanics have been developing in close interaction during the past decade, and the papers in this book attest to the productiveness of this interaction. The first paper in the collection contains a new result in the theory of quantum chaos, a burgeoning line of inquiry that combines mathematics and physics and is likely in time to produce many new connections and applications. Another paper, related to the renormalization group method for the study of maps of the circle with singularities due to a jump in the derivative, demonstrates that the fixed point of the renormgroup can be sufficiently described in this case. In certain situations, the renormgroup methods work better than the traditional KAM method.Other topics covered include thermodynamic formalism for certain infinite-dimensional dynamical systems, numerical simulation of dynamical systems with hyperbolic behavior, periodic points of holomorphic maps, the theory of random media, statistical properties of the leading eigenvalue in matrix ensembles of large dimension, spectral properties of the one-dimensional Schrodinger operator. This volume will appeal to many readers, as it covers a broad range of topics and presents a view of the some of the frontier research in the Soviet Union today.

This volume contains previously unpublished papers on algebraic $K$-theory written by Leningrad mathematicians over the last few years. The main topic of the first part is the computation of $K$-theory and $K$-cohomology for special varieties, such as group varieties and their principal homogeneous spaces, flag fiber bundles and their twisted forms, $\lambda$-operations in higher $K$-theory, and Chow groups of nonsingular quadrics. The second part deals with Milnor $K$-theory: Gersten's conjecture for $K^M_3$ of a discrete valuation ring, the absence of $p$-torsion in $K^M_*$ for fields of characteristic $p$, Milnor $K$-theory and class field theory for multidimensional local fields, and the triviality of higher Chern classes for the $K$-theory of global fields.

This collection deals with several different topics related to the construction and spectral analysis of Hamiltonians of various systems arising in mathematical physics. Included are a study of the disposition and character of resonances for certain operators, with applications to solid body physics; a survey of work in the perturbation of Hamiltonians in fermion systems; an examination of the construction of the Hamiltonian for three different pointwise interacting quantum particles; and a study of the lower branches of the Hamiltonian of the lattice model for chromodynamics. The final paper presents an extensive survey of problems related to the spectrum of finite-particle lattice Hamiltonians, which arise in quantum field theory and in models in the theory of solid bodies. The book provides an introduction of sorts to a series of new methods and problems in mathematical physics.

In recent years, researchers have found new topological invariants of integrable Hamiltonian systems of differential equations and have constructed a theory for their topological classification. Each paper in this important collection describes one of the 'building blocks' of the theory, and several of the works are devoted to applications to specific physical equations. In particular, this collection covers the new topological invariants of integrable equations, the new topological obstructions to integrability, a new Morse-type theory of Bott integrals, and classification of bifurcations of the Liouville tori in integrable systems. The papers collected here grew out of the research seminar 'Contemporary Geometrical Methods' at Moscow University, under the guidance of A. T. Fomenko, V. V. Trofimov, and A. V. Bolsinov. Bringing together contributions by some of the experts in this area, this collection is the first publication to treat this theory in a comprehensive way.

For the past thirty years, E. B. Vinberg and L. A. Onishchik have conducted a seminar on Lie groups at Moscow University; about five years ago V. L. Popov became the third co-director, and the range of topics expanded to include invariant theory. Today, the seminar encompasses such areas as algebraic groups, geometry and topology of homogeneous spaces, and Kac-Moody groups and algebras. This collection of papers presents a snapshot of the research activities of this well-established seminar, including new results in Lie groups, crystallographic groups, and algebraic transformation groups. These papers will not be published elsewhere. Readers will find this volume useful for the new results it contains as well as for the open problems it poses.

This volume presents research conducted between 1989 and 1991 by the participants in the Leningrad Seminar on representation theory, dynamical systems, and their applications, headed by A. M. Vershik. The primary areas covered here are mathematical physics, Lie groups and their representations, infinite-dimensional groups, topology, and dynamical systems. The book contains a number of useful introductory surveys; for example, one paper by Vaksman and Soibelman provides a systematic description of the theory of quantum groups in the spirit of representation theory - a new and popular area for which there are few introductory surveys. A portion of the book is devoted to adic transformations and substitutions, a new area of ergodic theory. With a balance of survey papers and frontier research results, this book will appeal to graduate students and researchers alike.

The four papers in this volume examine attractors of partial differential equations, with a focus on investigation of elements of attractors. Unlike the finite-dimensional case of ordinary differential equations, an element of the attractor of a partial differential equation is itself a function of spatial variables. This dependence on spatial variables is investigated by asymptotic methods. For example, the asymptotics show that the turbulence generated in a tube by a large localized external force does not propagate to infinity along the tube if the flux of the flow is not too large. Another topic considered here is the dependence of attractors on singular perturbations of the equations. The theory of unbounded attractors of equations without bounded attracting sets is also covered. All of the articles are systematic and detailed, furnishing an excellent review of new approaches and techniques developed by the Moscow school.

The papers in this collection, written by participants of the Research Seminar on the Theory of Functions at Kharkov University, primarily address the theory of entire and subharmonic functions. Founded in 1953 by B. Ya. Levin and still functioning today, this seminar ranges over different problems in the theory of functions, functional analysis, and related problems in calculus and mathematical physics. ""Entire and Subharmonic Functions"" contains works presented recently in the seminar.

This book contains papers presented at the Seminar on Mathematical Statistics held at the Institute for Problems of Information Transmission of the Academy of Sciences in the former Soviet Union. Founded in the mid-1960s, this seminar is still active today and attracts most of the researchers in Moscow who are interested in mathematical statistics. The topics covered include density, regression, and image estimation, adaptive estimation, stochastic approximation, median estimation, sequential experimental design, and large deviations for empirical measures. This collection is distinguished by the high scientific level of the papers and their modern approach. This book will be of interest to scientists and engineers who use probability and statistics, to mathematicians and applied statisticians who work in approximation theory, and to computer scientists who work in image analysis.

The nonlinear Stokes phenomenon occurs in the local theory of differential equations (or, more concisely, local dynamics) and finds application in singularity theory. This book contains a number of papers on this subject, including a survey that begins with Stokes' pioneering works on linear theory and discusses the work of Voronin.

This book contains recent results from a group focusing on minimal surfaces in the Moscow State University seminar on modern geometrical methods, headed by A. V. Bolsinov, A. T. Fomenko, and V. V. Trofimov. The papers collected here fall into three areas: one-dimensional minimal graphs on Riemannian surfaces and the Steiner problem, two-dimensional minimal surfaces and surfaces of constant mean curvature in three-dimensional Euclidean space, and multidimensional globally minimal and harmonic surfaces in Riemannian manifolds. The volume opens with an exposition of several important problems in the modern theory of minimal surfaces that will be of interest to newcomers to the field. Prepared with attention to clarity and accessibility, these papers will appeal to mathematicians, physicists, and other researchers interested in the application of geometrical methods to specific problems.

This volume contains papers by participants in the celebrated seminar of I. M. Gelfand, which ran for more than forty years at Moscow State University. Among the authors are some of the world's most renowned mathematicians. The high scientific level of the articles makes this an important contribution to the literature.