Fields Medal

Periods and Harmonic Analysis on Spherical Varieties

Bulletin of the American Mathematical Society

Vol. 45, No. 1, Pages 1-184, January 2008

Perfectoid Spaces: Lectures from the 2017 Arizona Winter School

Introduced by Peter Scholze in 2011, perfectoid spaces are a bridge between geometry in characteristic 0 and characteristic p, and have been used to solve many important problems, including cases of the weight-monodromy conjecture and the association of Galois representations to torsion classes in cohomology. In recognition of the transformative impact perfectoid spaces have had on the field of arithmetic geometry, Scholze was awarded a Fields Medal in 2018.

Berkeley Lectures on p-adic Geometry

Berkeley Lectures on p-adic Geometry

The Monge-Ampere Equation and Its Applications

The Monge-Ampère equation is one of the most important partial differential equations, appearing in many problems in analysis and geometry. This monograph is a comprehensive introduction to the existence and regularity theory of the Monge-Ampère equation and some selected applications; the main goal is to provide the reader with a wealth of results and techniques he or she can draw from to understand current research related to this beautiful equation.

Optimal Transportation and Action-Minimizing Measures

In this book we describe recent developments in the theory of optimal transportation, and some of its applications to fluid dynamics. Moreover we explore new variants of the original problem, and we try to figure out some common (and sometimes unexpected) features in this emerging variety of problems.

Optimal Control With Applications in Space and Quantum Dynamics

Several complete textbooks of mathematics on geometric optimal control theory exist in the literature, but little has been done with relevant applications in control engineering. This monograph is intended to fill this gap.

Partial Differential Equations and Geometric Measure Theory

This book collects together lectures by some of the leaders in the field of partial differential equations and geometric measure theory. It features a wide variety of research topics in which a crucial role is played by the interaction of fine analytic techniques and deep geometric observations, combining the intuitive and geometric aspects of mathematics with analytical ideas and variational methods.

Nonlocal and Nonlinear Diffusions and Interactions

Presenting a selection of topics in the area of nonlocal and nonlinear diffusions, this book places a particular emphasis on new emerging subjects such as nonlocal operators in stationary and evolutionary problems and their applications, swarming models and applications to biology and mathematical physics, and nonlocal variational problems.

Higher Dimensional Algebraic Geometry

In Honour of Professor Yujiro Kawamata's Sixtieth Birthday

A Course on Rough Paths: With an Introduction to Regularity Structures

Rough path analysis provides the means for constructing a pathwise solution theory for stochastic differential equations which, in many respects, behaves like the theory of deterministic differential equations and permits a clean break between analytical and probabilistic arguments. Together with the theory of regularity structures, it forms a robust toolbox, allowing the recovery of many classical results without having to rely on specific probabilistic properties such as adaptedness or the martingale property.

Singular Random Dynamics

Written by leading experts in an emerging field, this book offers a unique view of the theory of stochastic partial differential equations, with lectures on the stationary KPZ equation, fully nonlinear SPDEs, and random data wave equations.

Quadratic and Higher Degree Forms

In the last decade, the areas of quadratic and higher degree forms have witnessed dramatic advances. This volume is an outgrowth of three seminal conferences on these topics held in 2009, two at the University of Florida and one at the Arizona Winter School.

Finite Simple Groups: Thirty Years of the Atlas and Beyond

Cocycles Over Partially Hyperbolic Maps

Perspectives in Analysis: Essays in Honor of Lennart Carleson's 75th Birthday

The Conference “Perspectives in Analysis” was held during May 26–28, 2003 at the Royal Institute of Technology in Stockholm, Sweden. The purpose of the conference was to consider the future of analysis along with its relations to other areas of mathematics and physics, and to celebrate the seventy-fifth birthday of Lennart Carleson.

On the Stabilization of the Trace Formula

The books are intended primarily for two groups of readers. The first group is interested in the structure of automorphic forms on reductive groups over number fields, and specifically in qualitative information about the multiplicities of automorphic representations. The second group is interested in the problem of classifying l-adic representations of Galois groups of number fields.

Homogeneous Flows, Moduli Spaces and Arithmetic

This book contains a wealth of material concerning two very active and interconnected directions of current research at the interface of dynamics, number theory and geometry.

Probability and Statistical Physics in Two and More Dimensions

In the past ten to fifteen years, various areas of probability theory related to statistical physics, disordered systems and combinatorics have undergone intensive development. A number of these developments deal with two-dimensional random structures at their critical points, and provide new tools and ways of coping with at least some of the limitations of Conformal Field Theory that had been so successfully developed in the theoretical physics community to understand phase transitions of two-dimensional systems.

Lectures on Probability Theory and Statistics

This is yet another indispensable volume for all probabilists and collectors of the Saint-Flour series, and is also of great interest for mathematical physicists.

Analysis I

This is part one of a two-volume book on real analysis and is intended for senior undergraduate students of mathematics who have already been exposed to calculus. The emphasis is on rigour and foundations of analysis. Beginning with the construction of the number systems and set theory, the book discusses the basics of analysis - limits, series, continuity, differentiation, Riemann integration.

Analysis II

This is part two of a two-volume book on real analysis and is intended for senior undergraduate students of mathematics who have already been exposed to calculus. The emphasis is on rigour and foundations of analysis. Beginning with the construction of the number systems and set theory, the book discusses the basics of analysis - limits, series, continuity, differentiation, Riemann integration.

Solving Mathematical Problems: A Personal Perspective

Authored by a leading name in mathematics, this engaging and clearly presented text leads the reader through the various tactics involved in solving mathematical problems at the Mathematical Olympiad level. Covering number theory, algebra, analysis, Euclidean geometry, and analytic geometry, Solving Mathematical Problems includes numerous exercises and model solutions throughout.

An Introduction to Measure Theory

This is a graduate text introducing the fundamentals of measure theory and integration theory, which is the foundation of modern real analysis. The text focuses first on the concrete setting of Lebesgue measure and the Lebesgue integral (which in turn is motivated by the more classical concepts of Jordan measure and the Riemann integral), before moving on to abstract measure and integration theory, including the standard convergence theorems, Fubini's theorem, and the Carathéodory extension theorem.

Nonlinear Dispersive Equations: Local and Global Analysis

Among nonlinear PDEs, dispersive and wave equations form an important class of equations. These include the nonlinear Schrödinger equation, the nonlinear wave equation, the Korteweg de Vries equation, and the wave maps equation. This book is an introduction to the methods and results used in the modern analysis (both locally and globally in time) of the Cauchy problem for such equations.

Higher Order Fourier Analysis

Traditional Fourier analysis, which has been remarkably effective in many contexts, uses linear phase functions to study functions. Some questions, such as problems involving arithmetic progressions, naturally lead to the use of quadratic or higher order phases. Higher order Fourier analysis is a subject that has become very active only recently. Gowers, in groundbreaking work, developed many of the basic concepts of this theory in order to give a new, quantitative proof of Szemerédi's theorem on arithmetic progressions.

Additive Combinatorics

Additive combinatorics is the theory of counting additive structures in sets. This theory has seen exciting developments and dramatic changes in direction in recent years thanks to its connections with areas such as number theory, ergodic theory and graph theory.

Structure and Randomness: Pages from Year One of a Mathematical Blog

There are many bits and pieces of folklore in mathematics that are passed down from advisor to student, or from collaborator to collaborator, but which are too fuzzy and non-rigorous to be discussed in the formal literature. Traditionally, it was a matter of luck and location as to who learned such folklore mathematics. But today, such bits and pieces can be communicated effectively and efficiently via the semiformal medium of research blogging.

Poincare's Legacies, Part I: Pages From Year Two of a Mathematical Blog

There are many bits and pieces of folklore in mathematics that are passed down from advisor to student, or from collaborator to collaborator, but which are too fuzzy and non-rigorous to be discussed in the formal literature. Traditionally, it was a matter of luck and location as to who learned such folklore mathematics. But today, such bits and pieces can be communicated effectively and efficiently via the semiformal medium of research blogging.

Poincare's Legacies, Part II: Pages From Year Two of a Mathematical Blog

There are many bits and pieces of folklore in mathematics that are passed down from advisor to student, or from collaborator to collaborator, but which are too fuzzy and non-rigorous to be discussed in the formal literature. Traditionally, it was a matter of luck and location as to who learned such folklore mathematics. But today, such bits and pieces can be communicated effectively and efficiently via the semiformal medium of research blogging.

An Epsilon of Room Real Analysis, Part I: Pages from Year Three of a Mathematical Blog

There are many bits and pieces of folklore in mathematics that are passed down from advisor to student, or from collaborator to collaborator, but which are too fuzzy and non-rigorous to be discussed in the formal literature. Traditionally, it was a matter of luck and location as to who learned such folklore mathematics. But today, such bits and pieces can be communicated effectively and efficiently via the semiformal medium of research blogging.

An Epsilon of Room, Part II: Pages From Year Three of a Mathematical Blog

There are many bits and pieces of folklore in mathematics that are passed down from advisor to student, or from collaborator to collaborator, but which are too fuzzy and non-rigorous to be discussed in the formal literature. Traditionally, it was a matter of luck and location as to who learned such folklore mathematics. But today, such bits and pieces can be communicated effectively and efficiently via the semiformal medium of research blogging.

Topics in Random Matrix Theory

The text is largely self-contained and starts with a review of relevant aspects of probability theory and linear algebra. With over 200 exercises, the book is suitable as an introductory text for beginning graduate students seeking to enter the field.

Hilbert's Fifth Problem and Related Topics

Compactness and Contradiction

Expansion in Finite Simple Groups of Lie Type

Expander graphs are an important tool in theoretical computer science, geometric group theory, probability, and number theory. Furthermore, the techniques used to rigorously establish the expansion property of a graph draw from such diverse areas of mathematics as representation theory, algebraic geometry, and arithmetic combinatorics.

Quantum Groups and Quantum Cohomology

Cycles, Transfers, and Motivic Homology Theories

The original goal that ultimately led to this volume was the construction of "motivic cohomology theory," whose existence was conjectured by A. Beilinson and S. Lichtenbaum. This is achieved in the book's fourth paper, using results of the other papers whose additional role is to contribute to our understanding of various properties of algebraic cycles. The material presented provides the foundations for the recent proof of the celebrated "Milnor Conjecture" by Vladimir Voevodsky.

Lecture Notes on Motivic Cohomology

The notion of a motive is an elusive one, like its namesake "the motif" of Cezanne's impressionist method of painting. Its existence was first suggested by Grothendieck in 1964 as the underlying structure behind the myriad cohomology theories in Algebraic Geometry. We now know that there is a triangulated theory of motives, discovered by Vladimir Voevodsky, which suffices for the development of a satisfactory Motivic Cohomology theory. However, the existence of motives themselves remains conjectural. The lecture notes format is designed for the book to be read by an advanced graduate student or an expert in a related field.

Motivic Homotopy Theory

This book is based on lectures given at a summer school on motivic homotopy theory at the Sophus Lie Centre in Nordfjordeid, Norway, in August 2002. Vladimir Voevodsky is one of the founders of the theory and received the Fields medal for his work.

Renormalization and 3-Manifolds Which Fiber Over the Circle

Many parallels between complex dynamics and hyperbolic geometry have emerged in the past decade. Building on work of Sullivan and Thurston, this book gives a unified treatment of the construction of fixed-points for renormalization and the construction of hyperbolic 3- manifolds fibering over the circle.

Complex Dynamics and Renormalization

Addressing researchers and graduate students in the active meeting ground of analysis, geometry, and dynamics, this book presents a study of renormalization of quadratic polynomials and a rapid introduction to techniques in complex dynamics. Its central concern is the structure of an infinitely renormalizable quadratic polynomial f(z) = z2 + c.

Homological Mirror Symmetry and Tropical Geometry

The relationship between Tropical Geometry and Mirror Symmetry goes back to the work of Kontsevich and Y. Soibelman (2000), who applied methods of non-archimedean geometry (in particular, tropical curves) to Homological Mirror Symmetry.

Pseudoperiodic Topology

This volume offers an account of the present state of the art in pseudoperiodic topology---a young branch of mathematics, born at the boundary between the ergodic theory of dynamical systems, topology, and number theory. Related topics include the theory of algorithms, convex integer polyhedra, Morse inequalities, real algebraic geometry, statistical physics, and algebraic number theory.

The Princeton Companion to Mathematics

This is a one-of-a-kind reference for anyone with a serious interest in mathematics. Edited by Timothy Gowers, a recipient of the Fields Medal, it presents nearly two hundred entries, written especially for this book by some of the world's leading mathematicians, that introduce basic mathematical tools and vocabulary; trace the development of modern mathematics; explain essential terms and concepts; examine core ideas in major areas of mathematics; describe the achievements of scores of famous mathematicians; explore the impact of mathematics on other disciplines such as biology, finance, and music--and much, much more.

Mathematics: A Very Short Introduction

The aim of this book is to explain, carefully but not technically, the differences between advanced, research-level mathematics, and the sort of mathematics we learn at school. The most fundamental differences are philosophical, and readers of this book will emerge with a clearer understanding of paradoxical-sounding concepts such as infinity, curved space, and imaginary numbers.

Mathematics: A Brief Insight

Selected Works of A. I. Shirshov

This book presents translations of selected works of the famous Russian mathematician A. I. Shirshov. He was a pioneer in several directions of associative, Lie, Jordan, and alternative algebras, as well as groups and projective planes.

From Number Theory to Physics

The present book contains fourteen expository contributions on various topics connected to Number Theory, or Arithmetics, and its relationships to Theoreti­cal Physics. The first part is mathematically oriented; it deals mostly with ellip­tic curves, modular forms, zeta functions, Galois theory, Riemann surfaces, and p-adic analysis. The second part reports on matters with more direct physical interest, such as periodic and quasiperiodic lattices, or classical and quantum dynamical systems.

Dynamical Systems and Small Divisors

Lectures given at the C.I.M.E. Summer School held in Cetraro Italy, June 13-20, 1998

Dynamical Systems

Michael Robert Herman had a profound impact on the theory of dynamical systems over the last 30 years. His seminar at the École Polytechnique had major worldwide influence and was the main vector in the development of the theory of dynamical systems in France. His interests covered most aspects of the subject though closest to his heart were the so-called small divisors problems, in particular those related to the stability of quasiperiodic motions.

The Master Equation and the Convergence Problem in Mean Field Games

This book describes the latest advances in the theory of mean field games, which are optimal control problems with a continuum of players, each of them interacting with the whole statistical distribution of a population. While it originated in economics, this theory now has applications in areas as diverse as mathematical finance, crowd phenomena, epidemiology, and cybersecurity.

Mathematical Topics in Fluid Mechanics, Volume 1: Incompressible Models

The main emphasis in Volume 1 is on the mathematical analysis of incompressible models. After recalling the fundamental description of Newtonian fluids, an original and self-contained study of both the classical Navier-Stokes equations (including the inhomogenous case) and the Euler equations is given. Known results and many new results about the existence and regularity of solutions are presented with complete proofs. The discussion contains many interesting insights and remarks. The text highlights in particular the use of modern analytical tools and methods and also indicates many open problems.

Mathematical Topics in Fluid Mechanics, Volume 2: Compressible Models

The second volume is an attempt to achieve a mathematical understanding of compressible Navier-Stokes equations. It is probably the first reference covering the issue of global solutions in the large. It includes unique material on compactness properties of solutions for the Cauchy problem, the existence and regularity of stationary solutions, and the existence of global weak solutions. Written by one of the world's leading researchers in nonlinear partial differential equations, Mathematical Topics in Fluid Mechanics will be an indispensable reference for every serious researcher in the field.

The Mathematical Theory of Thermodynamic Limits: Thomas-Fermi Type Models

The thermodynamic limit is a mathematical technique for modeling crystals or other macroscopic objects by considering them as infinite periodic arrays of molecules. The technique allows models in solid state physics to be derived directly from models in quantum chemistry. This book presents new results, many previously unpublished, for a large class of models and provides a survey of the mathematics of thermodynamic limit problems. The authors both work closely with Fields Medal-winner Pierre-Louis Lion, and the book will be a valuable tool for applied mathematicians and mathematical physicists studying nonlinear partial differential equations.

Paris-Princeton Lectures on Mathematical Finance 2004

This is the third volume in the Paris-Princeton Lectures in Financial Mathematics, which publishes, on an annual basis, cutting-edge research in self-contained, expository articles from outstanding specialists, both established and upcoming.

Paris-Princeton Lectures on Mathematical Finance 2010

This is the fourth volume in the Paris-Princeton Lectures in Financial Mathematics, which publishes, on an annual basis, cutting-edge research in self-contained, expository articles from outstanding specialists, both established and upcoming.

Multimodal User Interfaces: From Signals to Interaction

The book presents a common theoretical framework for fusion and fission of multimodal information using the most advanced signal processing algorithms constrained by HCI rules, described in detail and integrated in the context of a common distributed software platform for easy and efficient development and usability assessment of multimodal tools.

On Euler Equations and Statistical Physics

The general goal is to describe mathematically some coherent structures observed in turbulent flows. The first problem we study concerns 2-dimensional flows: we begin with a system of N points vortices interacting with the natural Coulomb-like force and we consider the associated Gibbs measure. We then show how the measure goes, as N goes to infinity, to a stationary measure which is concentrated on very particular stationary solutions of the two-dimensional Euler equations.

Parabolic Equations with Irregular Data and Related Issues: Applications to Stochastic Differential Equations

This book studies the existence and uniqueness of solutions to parabolic-type equations with irregular coefficients and/or initial conditions. It elaborates on the DiPerna-Lions theory of renormalized solutions to linear transport equations and related equations, and also examines the connection between the results on the partial differential equation and the well-posedness of the underlying stochastic/ordinary differential equation.

Nonequilibrium Problems in Many-Particle Systems

This volume contains the text of four sets of lectures delivered at the third session of the Summer School organized by C.I.M.E. (Centro Internazionale Matematico Estivo). These texts are preceded by an introduction written by C. Cercignani and M. Pulvirenti which summarizes the present status in the area of Nonequilibrium Problems in Many-Particle Systems and tries to put the contents of the different sets of lectures in the right perspective, in order to orient the reader.

Stochastic Differential Systems, Stochastic Control Theory and Applications

Proceedings of a Workshop, Held at IMA, June 9-19, 1986

Global Solutions of Nonlinear Schrodinger Equations

This volume presents recent progress in the theory of nonlinear dispersive equations, primarily the nonlinear Schrodinger (NLS) equation. The Cauchy problem for defocusing NLS with critical nonlinearity is discussed. New techniques and results are described on global existence and properties of solutions with large Cauchy data.

Green's Function Estimates for Lattice Schrödinger Operators and Applications

Intended primarily for graduate students and researchers in the general area of dynamical systems and mathematical physics, the book provides a coherent account of a large body of work that is presently scattered in the literature. It does so in a refreshingly contained manner that seeks to convey the present technological "state of the art."

New Classes of Lp-Spaces

Mathematical Aspects of Nonlinear Dispersive Equations

This collection of new and original papers on mathematical aspects of nonlinear dispersive equations includes both expository and technical papers that reflect a number of recent advances in the field. The expository papers describe the state of the art and research directions.

Visions in Mathematics, Part 1

The goals of the conference were to discuss the importance, the methods, the past and the future of mathematics as we enter the 21st century and to consider the connection between mathematics and related areas.

Visions in Mathematics, Part 2

The goals of the conference were to discuss the importance, the methods, the past and the future of mathematics as we enter the 21st century and to consider the connection between mathematics and related areas.

Lectures on Geometry

This volume contains a collection of papers based on lectures delivered by distinguished mathematicians at Clay Mathematics Institute events over the past few years. It is intended to be the first in an occasional series of volumes of CMI lectures. Although not explicitly linked, the topics in this inaugural volume have a common flavour and a common appeal to all who are interested in recent developments in geometry.

Superstring Theory, Volume 1: Introduction

This two-volume book attempts to meet the need for a systematic exposition of superstring theory and its applications accessible to as wide an audience as possible.

Superstring Theory, Volume 2: Loop Amplitudes, Anomalies and Phenomenology

This two-volume book attempts to meet the need for a systematic exposition of superstring theory and its applications accessible to as wide an audience as possible.

Quantum Fields and Strings: A Course for Mathematicians, Vol. 1

In 1996-97 the Institute for Advanced Study organized a special year-long program designed to teach mathematicians the basic physical ideas which underlie the mathematical applications. These volumes are a written record of the program.

Quantum Fields and Strings: A Course for Mathematicians, Vol. 2

In 1996-97 the Institute for Advanced Study organized a special year-long program designed to teach mathematicians the basic physical ideas which underlie the mathematical applications. These volumes are a written record of the program.

Birational Geometry of Algebraic Varieties

One of the major discoveries of the past two decades in algebraic geometry is the realization that the theory of minimal models of surfaces can be generalized to higher dimensional varieties. This generalization, called the minimal model program, or Mori's program, has developed into a powerful tool with applications to diverse questions in algebraic geometry and beyond.

The Resolution of Singular Algebraic Varieties

Resolution of Singularities has long been considered as being a difficult to access area of mathematics. The more systematic and simpler proofs that have appeared in the last few years in zero characteristic now give us a much better understanding of singularities. They reveal the aesthetics of both the logical structure of the proof and the various methods used in it. The present volume is intended for readers who are not yet experts but always wondered about the intricacies of resolution.

Moduli Spaces and Arithmetic Geometry

Since its birth algebraic geometry has been closely related to and deeply motivated by number theory. Particularly the modern study of moduli spaces and arithmetic geometry have many important techniques and ideas in common. With this close relation in mind, the RIMS conference Moduli Spaces and Arithmetic Geometry was held at Kyoto University during September 8-15, 2004 as the 13th International Research Institute of the Mathematical Society of Japan. This volume is the outcome of this conference and consists of thirteen papers by invited speakers, including C Soulé, A Beauville and C Faber, and participants.

Higher Dimensional Birational Geometry

Advanced Studies in Pure Mathematics

Knots, Low-Dimensional Topology and Applications

This proceedings volume presents a diverse collection of high-quality, state-of-the-art research and survey articles written by top experts in low-dimensional topology and its applications. The focal topics include the wide range of historical and contemporary invariants of knots and links and related topics such as three- and four-dimensional manifolds, braids, virtual knot theory, quantum invariants, braids, skein modules and knot algebras, link homology, quandles and their homology; hyperbolic knots and geometric structures of three-dimensional manifolds; the mechanism of topological surgery in physical processes, knots in Nature in the sense of physical knots with applications to polymers, DNA enzyme mechanisms, and protein structure and function.

Coxeter Graphs and Towers of Algebras

Our subject is certain algebraic and von Neumann algebraic topics closely related to the original paper. However, in order to promote, in a modest way, the contact between diverse fields of mathematics, we have tried to make this work accessible to the broadest audience. Consequently, this book contains much elementary expository material.

Subfactors and Knots

Provides an extensive introduction to the theory of von Neumann algebras and to knot theory and braid groups. The presentation follows the historical development of the theory of subfactors and the ensuing applications to knot theory, including full proofs of some of the major results.

Knots at Hellas 98

Proceedings of the International Conference on Knot Theory and Its Ramifications

Introduction to Subfactors

These notes give an introduction to the subject suitable for a student who has only a little familiarity with the theory of Hilbert space. Subfactors have been a subject of considerable research activity for about fifteen years and are known to have significant relations with other fields such as low dimensional topology and algebraic quantum field theory.

Chiral Algebras

This long-awaited publication contains the results of the research of two distinguished professors from the University of Chicago, Alexander Beilinson and Fields Medalist Vladimir Drinfeld. Years in the making, this is a one-of-a-kind book featuring previously unpublished material. Chiral algebras form the primary algebraic structure of modern conformal field theory.

Topology of 4-Manifolds

One of the great achievements of contemporary mathematics is the new understanding of four dimensions. Michael Freedman and Frank Quinn have been the principals in the geometric and topological development of this subject, proving the Poincar and Annulus conjectures respectively.

Selected Applications of Geometry to Low-Dimensional Topology

By the late 1950s, algebra and topology had produced a successful and beautiful fusion. Geometric methods and insight, now vitally important in topology, encompass analytic objects such as instantons and minimal surfaces, as well as nondifferentiable constructions. Keeping technical details to a minimum, the authors lead the reader on a fascinating exploration of several developments in geometric topology.

Lectures on the Arithmetic Riemann-Roch Theorem

The arithmetic Riemann-Roch Theorem has been shown recently by Bismut-Gillet-Soul. The proof mixes algebra, arithmetic, and analysis. The purpose of this book is to give a concise introduction to the necessary techniques, and to present a simplified and extended version of the proof. It should enable mathematicians with a background in arithmetic algebraic geometry to understand some basic techniques in the rapidly evolving field of Arakelov-theory.

Degeneration of Abelian Varieties

A new and complete treatment of semi-abelian degenerations of abelian varieties, and their application to the construction of arithmetic compactifications of Siegel moduli space, with most of the results being published for the first time. Highlights of the book include a classification of semi-abelian schemes, construction of the toroidal and the minimal compactification over the integers, heights for abelian varieties over number fields, and Eichler integrals in several variables, together with a new approach to Siegel modular forms.

Rational Points

This book consists of the notes from the seminar Bonn/ Wuppertal 1983/ 84 on Arithmetic Geometry. It contains a proof for the Mordell conjecture and may be useful as an introduction to Arakelov's point of view in diophantine geometry. The third edition includes an appendix in which a detailed survey on the spectacular recent developments in arithmetic algebraic geometry is given.

Arithmetic and Geometry

The 'Arithmetic and Geometry' trimester, held at the Hausdorff Research Institute for Mathematics in Bonn, focussed on recent work on Serre's conjecture and on rational points on algebraic varieties. The resulting proceedings volume provides a modern overview of the subject for graduate students in arithmetic geometry and Diophantine geometry.

Riemann Surfaces

The theory of Riemann surfaces occupies a very special place in mathematics. It is a culmination of much of traditional calculus, making surprising connections with geometry and arithmetic. It is an extremely useful part of mathematics, knowledge of which is needed by specialists in many other fields.

Different Faces of Geometry

Edited by the world renowned geometers S. Donaldson, Ya. Eliashberg, and M. Gromov - presents the current state, new results, original ideas and open questions from the following important topics in modern geometry:

The Shape of a Life: One Mathematician's Search for the Universe's Hidden Geometry

Harvard geometer and Fields medalist Shing-Tung Yau has provided a mathematical foundation for string theory, offered new insights into black holes, and mathematically demonstrated the stability of our universe.

The Shape of Inner Space

String theory says we live in a ten-dimensional universe, but that only four are accessible to our everyday senses. According to theorists, the missing six are curled up in bizarre structures known as Calabi-Yau manifolds. In The Shape of Inner Space, Shing-Tung Yau, the man who mathematically proved that these manifolds exist, argues that not only is geometry fundamental to string theory, it is also fundamental to the very nature of our universe.

Selected Works of Shing-Tung Yau: 5-Volume Set (1971-1991)

These five volumes reproduce a comprehensive selection of Yau's published mathematical papers of the years 1971 to 1991 -- a period of groundbreaking accomplishments in numerous disciplines including geometric analysis, Kähler geometry, and general relativity.

A History in Sum: 150 Years of Mathematics at Harvard

In the twentieth century, American mathematicians began to make critical advances in a field previously dominated by Europeans. Harvard's mathematics department was at the center of these developments. A History in Sum is an inviting account of the pioneers who trailblazed a distinctly American tradition of mathematics--in algebraic geometry and topology, complex analysis, number theory, and a host of esoteric subdisciplines that have rarely been written about outside of journal articles or advanced textbooks.

Lectures on Differential Geometry

This volume presents lectures given by Richard Schoen and Shing-Tung Yau at the Institute for Advanced Studies at Princeton University in 1984 and 1985. The lectures describe the major advances in differential geometry, which progressed rapidly in the twentieth century.

Current Developments in Mathematics, 2018

This volume presents five papers based on selected lectures given at the Current Developments in Mathematics conference, held in November 2018 at Harvard University.

Computational Conformal Geometry

This new volume presents thorough introductions to the theoretical foundations -- as well as to the practical algorithms -- of computational conformal geometry. These have direct applications to engineering and digital geometric processing, including surface parameterization, surface matching, brain mapping, 3-D face recognition and identification, facial expression and animation, dynamic face tracking, mesh-spline conversion, and more.

The Founders of Index Theory

Index Theory is one of the most exciting and consequential accomplishments of twentieth-century mathematics. The Founders of Index Theory contemplates the four great mathematicians who developed index theory—Sir Michael Atiyah, Raoul Bott, Friedrich Hirzebruch, and I.M. Singer—through the eyes of their students, collaborators and colleagues, their friends and family members, and themselves.

From the Great Wall to the Great Collider: China and the Quest to Uncover the Inner Workings of the Universe

The 2012 discovery of the Higgs boson was a sensational triumph -- the culmination of a 48-year-long search that put the finishing touches on the so-called "Standard Model" of particle physics. While the celebrations were still underway, researchers in China were making plans to continue the centuries-old quest to identify the fundamental building blocks of nature.

Elliptic Curves, Modular Forms and Fermat's Last Theorem

Uniformization, Riemann-Hilbert Correspondence, Calabi-Yau Manifolds & Picard-Fuchs Equations

The uniformization theorem of Riemann surfaces is one of the most beautiful and important theorems in mathematics. Besides giving a clean classification of Riemann surfaces, its proof has motivated many new methods, such as the Riemann-Hilbert correspondence, Picard-Fuchs equations, and higher-dimensional generalizations of the uniformization theorem, which include Calabi-Yau manifolds.

Three-Dimensional Geometry and Topology: Volume 1

This book develops some of the extraordinary richness, beauty, and power of geometry in two and three dimensions, and the strong connection of geometry with topology.

Confoliations

This book presents the first steps of a theory of confoliations designed to link geometry and topology of three-dimensional contact structures with the geometry and topology of codimension-one foliations on three-dimensional manifolds. Developing almost independently, these theories at first glance belonged to two different worlds: The theory of foliations is part of topology and dynamical systems, while contact geometry is the odd-dimensional "brother" of symplectic geometry.

Noncommutative Geometry

This English version of the path-breaking French book on this subject gives the definitive treatment of the revolutionary approach to measure theory, geometry, and mathematical physics developed by Alain Connes.

Noncommutative Geometry, Quantum Fields and Motives

The unifying theme of this book is the interplay among noncommutative geometry, physics, and number theory.

Triangle of Thought

Our view of the world today is fundamentally influenced by twentieth century results in physics and mathematics. Here, three members of the French Academy of Sciences: Alain Connes, André Lichnerowicz, and Marcel Paul Schützenberger, discuss the relations among mathematics, physics and philosophy, and other sciences.

Conversations on Mind, Matter, and Mathematics

Do numbers and the other objects of mathematics enjoy a timeless existence independent of human minds, or are they the products of cerebral invention? Do we discover them, as Plato supposed and many others have believed since, or do we construct them?

Daniel Quillen's Notebooks

Notebooks of Daniel QUillen provided freely by the Clay Mathematics Institute.

Discrete Subgroups of Semisimple Lie Groups

On Some Aspects of the Theory of Anosov Systems

Uses ergodic theoretic techniques to study the distribution of periodic orbits of Anosov flows. The thesis introduces the "Margulis measure" and uses it to obtain a precise asymptotic formula for counting periodic orbits.

The Ambient Metric

This book develops and applies a theory of the ambient metric in conformal geometry. This is a Lorentz metric in n+2 dimensions that encodes a conformal class of metrics in n dimensions.

Partial Differential Equations in Fluid Mechanics

The Euler and Navier–Stokes equations are the fundamental mathematical models of fluid mechanics, and their study remains central in the modern theory of partial differential equations.

Essays on Fourier Analysis in Honor of Elias M. Stein

This book contains the lectures presented at a conference held at Princeton University in May 1991 in honor of Elias M. Stein's sixtieth birthday. The lectures deal with Fourier analysis and its applications.

Advances in Analysis: The Legacy of Elias M. Stein

Princeton University's Elias Stein was the first mathematician to see the profound interconnections that tie classical Fourier analysis to several complex variables and representation theory.

Quantum Fields and Strings: A Course for Mathematicians

In 1996-97 the Institute for Advanced Study organized a special year-long program designed to teach mathematicians the basic physical ideas which underlie the mathematical applications. These volumes are a written record of the program.

Quantum Fields and Strings: A Course for Mathematicians, Vol. 2

In 1996-97 the Institute for Advanced Study organized a special year-long program designed to teach mathematicians the basic physical ideas which underlie the mathematical applications. These volumes are a written record of the program.

Commensurabilities among Lattices in PU (1,n)

The first part of this monograph is devoted to a characterization of hypergeometric-like functions, that is, twists of hypergeometric functions in n-variables.

Hodge Cycles, Motives and Shimura Varieties (Lecture Notes in Mathematics)

Modular Functions of One Variable II

Pattern Theory: The Stochastic Analysis of Real-World Signals

Pattern theory is a distinctive approach to the analysis of all forms of real-world signals. At its core is the design of a large variety of probabilistic models whose samples reproduce the look and feel of the real signals, their patterns, and their variability. Bayesian statistical inference then allows you to apply these models in the analysis of new signals.

Indra's Pearls: The Vision of Felix Klein

Felix Klein, one of the great nineteenth-century geometers, discovered in mathematics an idea prefigured in Buddhist mythology: the heaven of Indra contained a net of pearls, each of which was reflected in its neighbour, so that the whole Universe was mirrored in each pearl.

Tata Lectures on Theta I

This volume is the first of three in a series surveying the theory of theta functions. Based on lectures given by the author at the Tata Institute of Fundamental Research in Bombay, these volumes constitute a systematic exposition of theta functions, beginning with their historical roots as analytic functions in one variable (Volume I), touching on some of the beautiful ways they can be used to describe moduli spaces (Volume II), and culminating in a methodical comparison of theta functions in analysis, algebraic geometry, and representation theory (Volume III).

Tata Lectures on Theta II: Jacobian Theta Functions and Differential Equations

The second in a series of three volumes that survey the theory of theta functions, this volume emphasizes the special properties of the theta functions associated with compact Riemann surfaces and how they lead to solutions of the Korteweg-de-Vries equations as well as other non-linear differential equations of mathematical physics.

The Red Book of Varieties and Schemes

Mumford's famous "Red Book" gives a simple, readable account of the basic objects of algebraic geometry, preserving as much as possible their geometric flavor and integrating this with the tools of commutative algebra.

Algebraic Geometry I: Complex Projective Varieties

Although several textbooks on modern algebraic geometry have been published in the meantime, Mumford's "Volume I" is, together with its predecessor the red book of varieties and schemes, now as before one of the most excellent and profound primers of modern algebraic geometry. Both books are just true classics!

Calculus: Single and Multivariable

Striking a balance between concepts, modeling, and skills, this highly acclaimed book arms readers with an accessible introduction to calculus. It builds on the strengths from previous editions, presenting key concepts graphically, numerically, symbolically, and verbally.

Geometric Invariant Theory

This standard reference on applications of invariant theory to the construction of moduli spaces is a systematic exposition of the geometric aspects of classical theory of polynomial invariants.

Selected Papers: On the Classification of Varieties and Moduli Spaces

Curves and Their Jacobians

Heights in Diophantine Geometry

Diophantine geometry has been studied by number theorists for thousands of years, since the time of Pythagoras, and has continued to be a rich area of ideas such as Fermat's Last Theorem, and most recently the ABC conjecture.

Number Theory, Trace Formulas and Discrete Groups

Symposium in Honor of Atle Selberg, Oslo, Norway, July 14–21, 1987

Seminar on Minimal Submanifolds

Volume 103 (Annals of Mathematics Studies)

Modern Geometry ― Methods and Applications, Part I: The Geometry of Surfaces, Transformation Groups, and Fields

This is the first volume of a three-volume introduction to modern geometry which emphasizes applications to other areas of mathematics and theoretical physics. Topics covered include tensors and their differential calculus, the calculus of variations in one and several dimensions, and geometric field theory.

Modern Geometry― Methods and Applications, Part II: The Geometry and Topology of Manifolds

This is the second volume of a three-volume introduction to modern geometry which emphasizes applications to other areas of mathematics and theoretical physics.

Modern Geometry―Methods and Applications, Part III: Introduction to Homology Theory

This is the third volume of a three-volume introduction to modern geometry which emphasizes applications to other areas of mathematics and theoretical physics.

Topological Library, Part 1: Cobordisms and Their Applications

Series on Knots and Everything

Topological Library, Part 2: Characteristic Classes and Smooth Structures on Manifolds

Series on Knots and Everything

Modern Geometric Structures and Fields

The book presents the basics of Riemannian geometry in its modern form as geometry of differentiable manifolds and the most important structures on them. The authors' approach is that the source of all constructions in Riemannian geometry is a manifold that allows one to compute scalar products of tangent vectors.

Basic Elements of Differential Geometry and Topology

Mathematics and its Applications Book 60

Solitons and Geometry

In this book, Professor Novikov describes recent developments in soliton theory and their relations to so-called Poisson geometry. This formalism, which is related to symplectic geometry, is extremely useful for the study of integrable systems that are described in terms of differential equations (ordinary or partial) and quantum field theories.

Topology I: General Survey

The book gives an overview of various subfields, beginning with the elements and proceeding right up to the present (1996) frontiers of research.

Topology II: Homotopy and Homology, Classical Manifolds

Two top experts in topology, O.Ya. Viro and D.B. Fuchs, give an up-to-date account of research in central areas of topology and the theory of Lie groups. They cover homotopy, homology and cohomology as well as the theory of manifolds, Lie groups, Grassmanians and low-dimensional manifolds.

Elements of Mathematical Logic

Dynamical Systems IV: Symplectic Geometry and its Applications

This book takes a snapshot of the mathematical foundations of classical and quantum mechanics from a contemporary mathematical viewpoint. It covers a number of important recent developments in dynamical systems and mathematical physics and places them in the framework of the more classical approaches.

Dynamical Systems VII: Integrable Systems Nonholonomic Dynamical Systems

A collection of five surveys on dynamical systems, indispensable for graduate students and researchers in mathematics and theoretical physics. Written in the modern language of differential geometry, the book covers all the new differential geometric and Lie-algebraic methods currently used in the theory of integrable systems.

Mathematics for Elementary School, Grades 1-6

One reason for the high quality of Japanese mathematics teaching and learning is focused and rigorous textbooks that develop problem solving abilities, mathematical thinking, and calculation skills in a systematic and coherent way.

Complex Analytic Desingularization

The main ideas of the proof of resolution of singularities of complex-analytic spaces presented here were developed by Heisuke Hironaka in the late 1960s and early 1970s. Since then, a number of proofs, all inspired by Hironaka's general approach, have appeared, the validity of some of them extending beyond the complex analytic case.

The Works of Hamanaka Gesson

Selected Papers Of Masatake Kuranishi

This book is a selection of Masatake Kuranishi's papers. Born in 1924, Kuranishi produced deep and far-reaching results in geometry over his career. Of his voluminous contributions, this book focuses on his later works: (i) his work on locally complete families of deformation of compact complex manifolds. Kuranishi was first to prove the fundamental results of the existence of complete families whose parameter spaces are today called Kuranishi spaces.

A Comprehensive Course in Number Theory

Developed from the author's popular text, A Concise Introduction to the Theory of Numbers, this book provides a comprehensive initiation to all the major branches of number theory.

A Concise Introduction to the Theory of Numbers

The book is based on Professor Baker's lectures given at the University of Cambridge and is intended for undergraduate students of mathematics. Professor Baker describes the rudiments of number theory in a concise, simple and direct manner.

Logarithmic Forms and Diophantine Geometry

This book gives an account of the theory of linear forms in the logarithms of algebraic numbers with special emphasis on the important developments of the past twenty-five years. The first part covers basic material in transcendental number theory but with a modern perspective.

Transcendental Number Theory

First published in 1975, this classic book gives a systematic account of transcendental number theory, that is those numbers which cannot be expressed as the roots of algebraic equations having rational coefficients.

New Advances in Transcendence Theory

This is an account of the proceedings of a very successful symposium of Transcendental Number Theory held in Durham in 1986. Most of the leading international specialists were present and the lectures reflected the great advances that have taken place in this area.

Differential Equations, Dynamical Systems, and an Introduction to Chaos

Hirsch, Devaney, and Smale’s classic Differential Equations, Dynamical Systems, and an Introduction to Chaos has been used by professors as the primary text for undergraduate and graduate level courses covering differential equations.

Differential Equations, Dynamical Systems, and Linear Algebra

This book is about dynamical aspects of ordinary differential equations and the relations between dynamical systems and certain fields outside pure mathematics.

The Smale Collection: Beauty in Natural Crystals

The acclaimed photography of Jeff Scovil (70 plates) and Steve Smale (30 plates) capture the presence of masterpieces from one of the world's finest private collections--that of celebrated mathematician Steve Smale and his wife Clara.

The Collected Papers of Stephen Smale: Volume I

The Collected Papers of Stephen Smale: Volume II

The Collected Papers of Stephen Smale: Volume III

Grothendieck-Serre Correspondence (English and French Edition)

This extraordinary volume contains a large part of the mathematical correspondence between A. Grothendieck and J-P. Serre. It forms a vivid introduction to the development of algebraic geometry during the years 1955-1965. During this period, algebraic geometry went through a remarkable transformation

Topological Vector Spaces

Local Cohomology: A Seminar Given by A. Groethendieck, Harvard University. Fall, 1961

The Tame Fundamental Group of a Formal Neighbourhood of a Divisor with Normal Crossings on a Scheme

Set Theory and the Continuum Hypothesis

This exploration of a notorious mathematical problem is the work of the man who discovered the solution. The independence of the continuum hypothesis is the focus of this study by Paul J. Cohen. It presents not only an accessible technical explanation of the author's landmark proof but also a fine introduction to mathematical logic.

Michael Atiyah - Collected Works: 7 Volume Set

Atiyah's huge number of published papers, focusing on the areas of algebraic geometry and topology, have here been collected into seven volumes, with the first five volumes divided thematically and the sixth and seventh arranged by date.

Michael Atiyah: Collected Works: Volume 1: Early Papers; General Papers

Michael Atiyah: Collected Works: Volume 2: Early Papers on K-Theory

Michael Atiyah: Collected Works: Volume 3: Index Theory: 1

Michael Atiyah: Collected Works: Volume 4: Index Theory: 2

Michael Atiyah: Collected Works: Volume 5: Gauge Theories

Michael Atiyah Collected Works: Volume 6: 1987-2002

Michael Atiyah Collected Works: Volume 7: 2002-2013

Introduction To Commutative Algebra

This book grew out of a course of lectures given to third year undergraduates at Oxford University and it has the modest aim of producing a rapid introduction to the subject. It is designed to be read by students who have had a first elementary course in general algebra.

K-theory

These notes are based on the course of lectures Atiyah gave at Harvard in the fall of 1964. They constitute a self-contained account of vector bundles and K-theory assuming only the rudiments of point-set topology and linear algebra. One of the features of the treatment is that no use is made of ordinary homology or cohomology theory.

The Geometry and Physics of Knots

Professor Atiyah presents an introduction to Witten's ideas from the mathematical point of view. The book will be essential reading for all geometers and gauge theorists as an exposition of new and interesting ideas in a rapidly developing area.

A Community of Scholars: Impressions of the Institute for Advanced Study

This beautifully illustrated anthology celebrates eighty years of history and intellectual inquiry at the Institute for Advanced Study, one of the world's leading centers for theoretical research.

Paul Dirac: The Man and his Work

Together the lectures in this volume, originally presented on the occasion of the dedication ceremony for a plaque honoring Dirac in Westminster Abbey, give a unique insight into the relationship between Dirac's character and his scientific achievements.

The Geometry and Dynamics of Magnetic Monopoles

In this book a particular system, describing the interaction of magnetic monopoles, is investigated in detail. The use of new geometrical methods produces a reasonably clear picture of the dynamics for slowly moving monopoles.

Collected Papers of V.K. Patodi

Vijay Kumar Patodi was a brilliant Indian mathematicians who made, during his short life, fundamental contributions to the analytic proof of the index theorem and to the study of differential geometric invariants of manifolds.

Elliptic Operators and Compact Groups

Geometry of Yang-Mills Fields

These Lecture Notes are an expanded version of the Fermi Lectures Atiyah gave at Scuola Normale Superiore in Pisa, the Loeb Lectures at Harvard and the Whittemore Lectures at Yale, in 1978.

Vector Bundles on Algebraic Varieties: Papers presented at the Bombay Colloquium 1984

The purpose of this authoritative volume is to highlight recent developments in the general area of vector bundles as well as principal bundles on both affine and projective varieties.

Notes on the Lefschetz Fixed Point Theorem for Elliptic Complexes

Physics and Mathematics of Strings

Fields Medallist's Lectures

Vector Fields on Manifolds

This paper is a contribution to the topological study of vector fields on manifolds. In particular we shall be concerned with the problems of exist­ ence of r linearly independent vector fields.

Morse Theory

One of the most cited books in mathematics, John Milnor's exposition of Morse theory has been the most important book on the subject for more than forty years. Morse theory was developed in the 1920s by mathematician Marston Morse. (Morse was on the faculty of the Institute for Advanced Study, and Princeton published his Topological Methods in the Theory of Functions of a Complex Variable in the Annals of Mathematics Studies series in 1947.)

John Milnor Collected Papers, Volume 1: Geometry

This volume contains papers on geometry of one of the best modern geometers and topologists, John Milnor. This book covers a wide variety of topics and includes several previously unpublished works. It is delightful reading for any mathematician with an interest in geometry and topology and for any person with an interest in mathematics.

John Milnor Collected Papers, Volume II: The Fundamental Group

The volume contains sixteen papers and is partitioned into four parts: Knot theory, Free action on spheres, Torsion, and Three-dimensional manifolds.

Collected Papers of John Milnor, Volume III: Differential Topology

Collected Papers of John Milnor, Volume IV: Homotopy, Homology and Manifolds

The book is divided into four parts: Homotopy Theory, Homology and Cohomology, Manifolds, and Expository Papers. Introductions to each part provide some historical context and subsequent development.

Collected Papers of John Milnor, Volume V: Algebra

These papers, together with other (some of them previously unpublished) works in algebra are assembled here in this fifth volume of Milnor's Collected Papers. They constitute not only an important historical archive, but also, thanks to the clarity and elegance of Milnor's mathematical exposition, a valuable resource for work in the fields treated.

Collected Papers of John Milnor, Volume VI: Dynamical Systems

Contains all of Milnor's work on Real and Complex Dynamics from 1953 to 1999, plus one paper from 2000. These papers provide important and fundamental material in real and complex dynamical systems.

Collected Papers of John Milnor, Volume VII: Dynamical Systems

Together with the preceding Volume VI, it contains all of Milnor's papers in dynamics, through the year 2012. Most of the papers are in holomorphic dynamics; however, there are two in real dynamics and one on cellular automata.

Lectures on the H-Cobordism Theorem

These are notes for Lectures of John Milnor that were given as a seminar on differential topology in October and November, 1963 Princeton University.

Prospects in Mathematics

Five papers by distinguished American and European mathematicians describe some current trends in mathematics in the perspective of the recent past and in terms of expectations for the future.

Topology from the Differentiable Viewpoint

Beginning with basic concepts such as diffeomorphisms and smooth manifolds, Milnor goes on to examine tangent spaces, oriented manifolds, and vector fields. Key concepts such as homotopy, the index number of a map, and the Pontryagin construction are discussed.

Characteristic Classes

The theory of characteristic classes provides a meeting ground for the various disciplines of differential topology, differential and algebraic geometry, cohomology, and fiber bundle theory. As such, it is a fundamental and an essential tool in the study of differentiable manifolds.

Dynamics in One Complex Variable

This volume studies the dynamics of iterated holomorphic mappings from a Riemann surface to itself, concentrating on the classical case of rational maps of the Riemann sphere. This subject is large and rapidly growing. These lectures are intended to introduce some key ideas in the field, and to form a basis for further study.

Symmetric Bilinear Forms

The theory cf quadratic forms and the intimately related theory of sym- metrie bilinear forms have a lang and rich his tory, highlighted by the work of Legendre, Gauss, Minkowski, and Hasse.

Singular Points of Complex Hypersurfaces

Introduction to Algebraic K-Theory

Algebraic K-theory describes a branch of algebra that centers about two functors. K0 and K1, which assign to each associative ring ∧ an abelian group K0∧ or K1∧ respectively. Professor Milnor sets out, in the present work, to define and study an analogous functor K2, also from associative rings to abelian groups.

The Analysis of Linear Partial Differential Operators I: Distribution Theory And Fourier Analysis

The Analysis of Linear Partial Differential Operators II: Differential Operators with Constant Coefficients

The Analysis of Linear Partial Differential Operators III: Pseudo-Differential Operators

The Analysis of Linear Partial Differential Operators IV: Fourier Integral Operators

Unpublished Manuscripts: from 1951 to 2007

Hörmander himself organised the manuscripts and also wrote the notes explaining their origins, presenting the material in the form he fully intended it to be published in. As his daughter, Sofia Broström, mentions in the Foreword, towards the end of his life, Hörmander "carefully went through his unpublished manuscripts, checking and revising each of them with his very critical eye, deciding what should be kept for posterity and what should be thrown out".

Lectures on Nonlinear Hyperbolic Differential Equations

In this introductory textbook, a revised and extended version of well-known lectures by L. Hörmander from 1986, four chapters are devoted to weak solutions of systems of conservation laws. Apart from that the book only studies classical solutions.

Introduction to Complex Analysis in Several Variables

Collected Papers of Marcel Riesz

Riesz worked on summability theory, analytic functions, the moment problem, harmonic and functional analysis, potential theory and the wave equation.

Partial Differential Equations and Mathematical Physics

The most frequently occurring theme is the use of microlocal analysis which is now important also in the study of non-linear differential equations although it originated entirely within the linear theory. Perhaps it is less surprising that microlocal analysis has proved to be useful in the study of mathematical problems of classical quantum mechanics, for it re­ceived a substantial input of ideas from that field.

Notions of Convexity

The first two chapters of this book are devoted to convexity in the classical sense, for functions of one and several real variables respectively.

Seminar on Singularities of Solutions of Linear Partial Differential Equations

Singularities of solutions of differential equations forms the common theme of these papers taken from a seminar held at the Institute for Advanced Study in Princeton in 1977-1978.

Structural Stability And Morphogenesis

A topological investigation and introduction to Catastrophe Theory

To Predict IS NOT To Explain: Conversations on Mathematics, Science, Catastrophe Theory, Semiophysics, Morphogenesis and Natural Philosophy

These interviews with Rene Thom range from Mathematics to Semiophysics, from embryology and morphogenesis to linguistics and from Natural Philosophy to autobiographical remarks.

Semio Physics: A Sketch

Parables, Parabolas and Catastrophes: Conversations on Mathematics, Science and Philosophy

The author, after providing us with a broad outline of Catastrophe Theory, applies it to a passionate and critical overview of the major scientific themes of our age. This book succeeds remarkably to popularize these topics.

In Praise of Mathematics: A Collection of Articles On Mathematics, and Related Topics, 1972-1994

This collection of articles on mathematics and related topics from 1972-1994 have been put in chronological order so that the reader can "see", in Part 1, the evolution of some of Thom's professional ideas in mathematics and in Part 2, to "glimpse" his philosophical ideas in the mathematical sciences as they relate to the ideas in Part 1.

Mathematical Models of Morphogenesis

Mathematics and its Applications

Grothendieck-Serre Correspondence (English and French Edition)

This extraordinary volume contains a large part of the mathematical correspondence between A. Grothendieck and J-P. Serre. It forms a vivid introduction to the development of algebraic geometry during the years 1955-1965. During this period, algebraic geometry went through a remarkable transformation

Oeuvres - Collected Papers I: 1949 - 1959

As listed in the preface, the three volumes cover almost all articles published in mathematical journals between 1949 and 1984, the summaries of the author's courses at the Collège de France since 1956, some of his Séminaire notes, and some items not previously published.

Oeuvres - Collected Papers II: 1960-1971

As listed in the preface, the three volumes cover almost all articles published in mathematical journals between 1949 and 1984, the summaries of the author's courses at the Collège de France since 1956, some of his Séminaire notes, and some items not previously published.

Oeuvres - Collected Papers III: 1972 - 1984

Characteristic of Serre's publications are the many open questions he formulated suggesting further research directions. Four volumes specify how he has provided comments on and corrections to most articles, and described the present status of the open questions with reference to later results.

Oeuvres - Collected Papers IV: 1985 - 1998

This is the fourth volume of J-P. Serre's Collected Papers covering the period 1985-1998. Items, numbered 133-173, contain "the essence" of his work from that period and are devoted to number theory, algebraic geometry, and group theory. Half of them are articles and another half are summaries of his courses in those years and letters. Most courses have never been previously published, nor proofs of the announced results.

Algebraic Groups and Class Fields

Local Fields

The goal of this book is to present local class field theory from the cohomo­ logical point of view, following the method inaugurated by Hochschild and developed by Artin-Tate. This theory is about extensions-primarily abelian-of "local" (i.e., complete for a discrete valuation) fields with finite residue field.

Local Algebra

This is an English translation of the now classic "Algbre Locale - Multiplicits" originally published by Springer as LNM 11. It gives a short account of the main theorems of commutative algebra, with emphasis on modules, homological methods and intersection multiplicities.

Trees

The seminal ideas of this book played a key role in the development of group theory since the 70s. Several generations of mathematicians learned geometric ideas in group theory from this book. In it, the author proves the fundamental theorem for the special cases of free groups and tree products before dealing with the proof of the general case.

Finite Groups: An Introduction

This book is a short introduction to the subject, written both for beginners and for mathematicians at large. There are ten chapters: Preliminaries, Sylow theory, Solvable groups and nilpotent groups, Group extensions, Hall subgroups, Frobenius groups, Transfer, Characters, Finite subgroups of GLn, and Small groups.

Linear Representations of Finite Groups

This book consists of three parts, rather different in level and purpose. The first part was originally written for quantum chemists. It describes the correspondence, due to Frobenius, between linear representations and characters. The second part is a course given in 1966 to second-year students of l’Ecole Normale. It completes in a certain sense the first part. The third part is an introduction to Brauer Theory.

Lie Algebras and Lie Groups

The main general theorems on Lie Algebras are covered, roughly the content of Bourbaki's Chapter I. I have added some results on free Lie algebras, which are useful, both for Lie's theory itself (Campbell-Hausdorff formula) and for applications to pro-Jrgroups.

Abelian l-Adic Representations and Elliptic Curves

This classic book contains an introduction to systems of l-adic representations, a topic of great importance in number theory and algebraic geometry, as reflected by the spectacular recent developments on the Taniyama-Weil conjecture and Fermat's Last Theorem.

Complex Semisimple Lie Algebras

These short notes, already well-known in their original French edition, present the basic theory of semisimple Lie algebras over the complex numbers. The author begins with a summary of the general properties of nilpotent, solvable, and semisimple Lie algebras.

Cohomological Invariants in Galois Cohomology

This volume addresses algebraic invariants that occur in the confluence of several important areas of mathematics, including number theory, algebra, and arithmetic algebraic geometry. The invariants are analogues for Galois cohomology of the characteristic classes of topology, which have been extremely useful tools in both topology and geometry.

Topics in Galois Theory: Research Notes in Mathematics

This book is based on a course given by the author at Harvard University in the fall semester of 1988. The course focused on the inverse problem of Galois Theory: the construction of field extensions having a given finite group as Galois group.

Galois Cohomology

This is an updated English translation of Cohomologie Galoisienne, published more than thirty years ago as one of the very first versions of Lecture Notes in Mathematics. It includes a reproduction of an influential paper by R. Steinberg, together with some new material and an expanded bibliography.

Lectures on the Mordell-Weil Theorem

The book is based on a course given by J.-P. Serre at the Collège de France in 1980 and 1981. Basic techniques in Diophantine geometry are covered, such as heights, the Mordell-Weil theorem, Siegel's and Baker's theorems, Hilbert's irreducibility theorem, and the large sieve.

Prospects in Mathematics

Five papers by distinguished American and European mathematicians describe some current trends in mathematics in the perspective of the recent past and in terms of expectations for the future.

Motives

Complex Manifolds

This volume serves as an introduction to the Kodaira-Spencer theory of deformations of complex structures. Based on notes taken by James Morrow from lectures given by Kunihiko Kodaira at Stanford University in 1965-1966, the book gives the original proof of the Kodaira embedding theorem, showing that the restricted class of Kähler manifolds called Hodge manifolds is algebraic.

Introduction to Complex Analysis

This textbook is an introduction to the classical theory of functions of a complex variable. The author's aim is to explain the basic theory in an easy-to-understand and careful way. He emphasizes geometrical considerations and, to avoid topological difficulties associated with complex analysis, begins by deriving Cauchy's integral formula in a topologically simple case.

Complex Manifolds and Deformation of Complex Structures

Kunihiko Kodaira, Collected Works: Volume I

Kunihiko Kodaira's influence in mathematics has been fundamental and international, and his efforts have helped lay the foundations of modern complex analysis. These three volumes contain Kodaira's written contributions, published in a large number of journals and books between 1937 and 1971.

Kunihiko Kodaira, Collected Works: Volume II

Kunihiko Kodaira's influence in mathematics has been fundamental and international, and his efforts have helped lay the foundations of modern complex analysis. These three volumes contain Kodaira's written contributions, published in a large number of journals and books between 1937 and 1971.

Kunihiko Kodaira, Collected Works: Volume III

Kunihiko Kodaira's influence in mathematics has been fundamental and international, and his efforts have helped lay the foundations of modern complex analysis. These three volumes contain Kodaira's written contributions, published in a large number of journals and books between 1937 and 1971.

The Collected Papers of Teiji Takagi

Teiji Takagi one of the leading number theorists of this century, is most renowned as the founder of class field theory. This volume reflects the stages of his development of this theory. Inspired by a genial idea related to analytic number theory, he developed a beautiful general theory of abelian extensions of algebraic number fields which he addressed at the ICM 1920 at Strasbourg.

Harmonic Integrals

Lectures Delivered In A Seminar Conducted By Professors Hermann Weyl And Karl Ludwig Siegel At The Institute For Advanced Study, 1950.

Nevanlinna Theory

This book deals with the classical theory of Nevanlinna on the value distribution of meromorphic functions of one complex variable, based on minimum prerequisites for complex manifolds.

Japanese Grade 7 Mathematics

The University of Chicago School Mathematics Project

Japanese Grade 8 Mathematics

The University of Chicago School Mathematics Project

Japanese Grade 9 Mathematics

The University of Chicago School Mathematics Project

Mathematics 1: Japanese Grade 10

This is the translation from the Japanese textbook for the grade 10 course, "Basic Mathematics". The book covers the material which is a compulsory for Japanese high school students. The course comprises algebra (including quadratic functions, equations, and inequalities), trigonometric functions, and plane coordinate geometry.

Mathematics 2: Japanese Grade 11

This is the translation from the Japanese textbook for the grade 11 course, "General Mathematics". It is part of the easier of the three elective courses in mathematics offered at this level and is taken by about 40% of students. The book covers basic notions of probability and statistics, vectors, exponential, logarithmic, and trigonometric functions, and an introduction to differentiation and integration.

Basic Analysis: Japanese Grade 11

This is the translation of the Japanese textbook for the grade 11 course, "Basic Analysis", which is one of three elective courses offered at this level in Japanese high schools. The book includes a thorough treatment of exponential, logarithmic, and trigonometric functions, progressions, and induction method, as well as an extensive introduction to differential and integral calculus.

Algebra and Geometry: Japanese Grade 11

A textbook used by upper level secondary school students in Japan, covering plane and solid coordinate geometry, vectors, and matrices.

Collected Papers I

Springer Collected Works in Mathematics

Collected Papers II

Springer Collected Works in Mathematics

Mathematics for the physical sciences

Geometry and Probability in Banach Spaces

A Mathematician Grappling with His Century

Laurent Schwartz is one of the most remarkable intellects of the 20th century. His discovery of distributions, one of the most beautiful theories in mathematics, earned him a 1950 Fields Medal. Beyond this formidable achievement, his love for science and for teaching led him to think deeply and lecture broadly to the general public on the significance of science and mathematics to the well-being of the world.

Lectures on Mixed Problems in Partial Differential Equations and Representation of Semi-Groups

Application of Distributions to the Theory of Elementary Particles in Quantum Mechanics

Notes on Pure Mathematics

Galois Lectures

Addresses delivered by Jesse Douglas, Philip Franklin, Cassius Jackson Keyser, Leopold Infeld at the Galois Institute of Mathematics, Long Island University, Brooklyn, N.Y.

Complex Analysis

A standard source of information of functions of one complex variable, this text has retained its wide popularity in this field by being consistently rigorous without becoming needlessly concerned with advanced or overspecialized material.

Conformal Invariants: Topics in Geometric Function Theory

The book emphasizes the geometric approach as well as classical and semi-classical results which Lars Ahlfors felt every student of complex analysis should know before embarking on independent research. At the time of the book's original appearance, much of this material had never appeared in book form, particularly the discussion of the theory of extremal length.

Riemann Surfaces

The theory of Riemann surfaces has a geometric and an analytic part. The former deals with the axiomatic definition of a Riemann surface, methods of construction, topological equivalence, and conformal mappings of one Riemann surface on another. The analytic part is concerned with the existence and properties of functions that have a special character connected with the conformal structure, for instance: subharmonic, harmonic, and analytic functions.

Advances in the Theory of Riemann Surfaces

The present volume contains all but two of the papers read at the conference, as well as a few papers and short notes submitted afterwards. We hope that it reflects faithfully the present state of research in the fields covered, and that it may provide an access to these fields for future investigations.

Contributions to the Theory of Riemann Surfaces

Volume 30 (Annals of Mathematics Studies)

Contributions to Analysis: A Collection of Papers Dedicated to Lipman Bers

Contributions to Analysis: A Collection of Papers Dedicated to Lipman Bers is a compendium of papers provided by Bers, friends, students, colleagues, and professors. These papers deal with Teichmuller spaces, Kleinian groups, theta functions, algebraic geometry. Other papers discuss quasiconformal mappings, function theory, differential equations, and differential topology.

Analytic Functions

A survey of recent (1960) developments both in the classical and modern fields of the theory. Contents include: The complex analytic structure of the space of closed Riemann surfaces; Complex analysis on noncompact Riemann domains; Proof of the Teichmuller-Ahlfors theorem; The conformal mapping of Riemann surfaces; On certain coefficients of univalent functions

Lectures on Quasiconformal Mappings

Lars Ahlfors's Lectures on Quasiconformal Mappings, based on a course he gave at Harvard University in the spring term of 1964, was first published in 1966 and was soon recognized as the classic it was shortly destined to become. These lectures develop the theory of quasiconformal mappings from scratch, give a self-contained treatment of the Beltrami equation, and cover the basic properties of Teichmuller spaces

Collected Papers, Volumes 1 & 2