The topic of the 2010 Abel Symposium, hosted at the Norwegian Academy of Science and Letters, Oslo, was Nonlinear Partial Differential Equations, the study of which is of fundamental importance in mathematics and in almost all of natural sciences, economics, and engineering. This area of mathematics is currently in the midst of an unprecedented development worldwide. Differential equations are used to model phenomena of increasing complexity, and in areas that have traditionally been outside the realm of mathematics. New analytical tools and numerical methods are dramatically improving our understanding of nonlinear models. Nonlinearity gives rise to novel effects reflected in the appearance of shock waves, turbulence, material defects, etc., and offers challenging mathematical problems. On the other hand, new mathematical developments provide new insight in many applications. These proceedings present a selection of the latest exciting results by world leading researchers.

This volume contains the collected works of the eminent chemist and physicist Lars Onsager, one of the most influential scientists of the 20th Century. The volume includes Onsager's previously unpublished PhD thesis, a biography by H C Longuet-Higgins and M E Fisher, an autobiographical commentary, selected photographs, and a list of Onsager discussion remarks in print. Onsager's scientific achievements were characterized by deep insights into the natural sciences. His two best-known accomplishments are his reciprocal relations for irreversible processes, for which he received the 1968 Nobel Prize in Chemistry, and his explicit solution of the two-dimensional Ising model, a mathematical tour de force that created a sensation when it appeared. In addition, he made significant theoretical contributions to other fields, including electrolytes, colloids, superconductivity, turbulence, ice, electrons in metals, and dielectrics. In this volume, Onsager's contributions are divided into the following fields: irreversible processes; the Ising model; electrolytes; colloids; helium II and vortex quantization; off-diagonal long-range order and flux quantization; electrons in metal; turbulence; ion recombination; fluctuation theory; dielectrics; ice and water; biology; Mathieu functions. The different fields are evaluated by leading experts. The commentators are P W Anderson, R Askey, A Chorin, C Domb, R J Donnelly, W Ebeling, J-C Justice, H N W Lekkerkerker, P Mazur, H P McKean, J F Nagle, T Odijk, A B Pippard, G Stell, G H Weiss, and C N Yang.

This book is based on research that, to a large extent, started around 1990, when a research project on fluid flow in stochastic reservoirs was initiated by a group including some of us with the support of VISTA, a research coopera tion between the Norwegian Academy of Science and Letters and Den norske stats oljeselskap A.S. (Statoil). The purpose of the project was to use stochastic partial differential equations (SPDEs) to describe the flow of fluid in a medium where some of the parameters, e.g., the permeability, were stochastic or "noisy." We soon realized that the theory of SPDEs at the time was insufficient to handle such equations. Therefore it became our aim to develop a new mathematically rigorous theory that satisfied the following conditions. 1) The theory should be physically meaningful and realistic, and the corre sponding solutions should make sense physically and should be useful in applications. 2) The theory should be general enough to handle many of the interesting SPDEs that occur in reservoir theory and related areas. 3) The theory should be strong and efficient enough to allow us to solve th, se SPDEs explicitly, or at least provide algorithms or approximations for the solutions."

The Workshop on Hyperbolic Conservation Laws and Related Analysis with Applications at the International Centre for Mathematical Sciences (Edinburgh, UK) held in Edinburgh, September 2011, produced this fine collection of original research and survey articles. Many leading mathematicians attended the event and submitted their contributions for this volume.This book presents thirteen papers, representing the most significant advances and current trends in nonlinear hyperbolic conservation laws, related analysisand applications. Topics covered include a survey on multidimensional systems of conservation laws as well as novel results on liquid crystals, conservation laws with discontinuous flux functions, and applications to sedimentation. Also included are articles on recent advances in the Euler equations and the Navier Stokes Fourier Poisson system, in addition to new results on collective phenomena described by the Cucker Smale model.

The present volume is addressed toresearchers and graduate students interested in partial differential equations and related analysis with applications.

Covering the years 2008-2012, this bookprofilesthe life and work of recent winners of the Abel Prize:
. John G. Thompson and Jacques Tits, 2008
. MikhailGromov, 2009
. John T. Tate Jr., 2010
. John W. Milnor, 2011
. Endre Szemeredi, 2012.
The profiles feature autobiographical information as well as a description of each mathematician's work. In addition, each profile contains a complete bibliography, a curriculum vitae, as well as photos old and new. As an added feature, interviews with the Laureates are presented on an accompanying web site (http: //extras.springer.com/).

The book also presents a history of the Abel Prize written by the historian Kim Helsvig, and includes a facsimile of aletter from Niels Henrik Abel, which is transcribed, translated into English, and placed into historical perspectiveby Christian Skau.

This book follows onThe Abel Prize: 2003-2007, The First Five Years(Springer, 2010), which profiles the work of the first Abel Prize winners.

"

These volumes contain a collection of essays by many of the closest co-workers of Raphael Hoegh-Krohn, 1938-88, one of the outstanding mathematical physicists of our age. The contributions vary in style, purpose and content - some are surveys where leading experts sum up and clarify a subject area, others are new and adventurous expeditions into unknown territory. The topics cover most aspects of modern mathematical physics with special emphasis on methods from operator theory and stochastic analysis. Many of the papers are based on talks given at a symposium in honour of Hoegh-Krohn at the University of Oslo; however the final volumes are far more than proceedings and great care has been taken to attract contributions from the leading researchers.

These volumes contain a collection of essays by many of the closest co-workers of Raphael Hoegh-Krohn, 1938-88, one of the outstanding mathematical physicists of our age. The contributions vary in style, purpose, and content - some are surveys where leading experts sum up and clarify a subject area, others are new and adventurous expeditions into unknown territory. The topics cover most aspects of modern mathematical physics with special emphasis on methods from operator theory and stochastic analysis. Many of the papers are based on talks given at a symposium in honor of Hoegh-Krohn at the University of Oslo: however the final volumes are far more than proceedings and great care has been taken to attract contributions from the leading researchers.

As a partner to Volume 1: Dimensional Continuous Models, this monograph provides a self-contained introduction to algebro-geometric solutions of completely integrable, nonlinear, partial differential-difference equations, also known as soliton equations. The systems studied in this volume include the Toda lattice hierarchy, the Kac-van Moerbeke hierarchy, and the Ablowitz-Ladik hierarchy. An extensive treatment of the class of algebro-geometric solutions in the stationary as well as time-dependent contexts is provided. The theory presented includes trace formulas, algebro-geometric initial value problems, Baker-Akhiezer functions, and theta function representations of all relevant quantities involved. The book uses basic techniques from the theory of difference equations and spectral analysis, some elements of algebraic geometry and especially, the theory of compact Riemann surfaces. The presentation is constructive and rigorous, with ample background material provided in various appendices. Detailed notes for each chapter, together with an exhaustive bibliography, enhance understanding of the main results.

The book presents the winners of the Abel Prize in mathematics for the period 2013-17: Pierre Deligne (2013); Yakov G. Sinai (2014); John Nash Jr. and Louis Nirenberg (2015); Sir Andrew Wiles (2016); and Yves Meyer (2017). The profiles feature autobiographical information as well as a scholarly description of each mathematician's work. In addition, each profile contains a Curriculum Vitae, a complete bibliography, and the full citation from the prize committee. The book also includes photos for the period 2003-2017 showing many of the additional activities connected with the Abel Prize. As an added feature, video interviews with the Laureates as well as videos from the prize ceremony are provided at an accompanying website (http://extras.springer.com/). This book follows on The Abel Prize: 2003-2007. The First Five Years (Springer, 2010) and The Abel Prize 2008-2012 (Springer 2014), which profile the work of the previous Abel Prize winners.

This book presents thirteen papers, representing the most significant advances and current trends in nonlinear hyperbolic conservation laws and related analysis with applications. Topics covered include a survey on multidimensional systems of conservation laws as well as novel results  on liquid crystals, conservation laws with discontinuous flux functions, and applications to sedimentation.  Also included are articles on recent advances in the Euler equations and the Navier-Stokes-Fourier-Poisson system, in addition to new results on collective phenomena described by the Cucker-Smale model.    The Workshop on Hyperbolic Conservation Laws and Related Analysis with Applications at the International Centre for Mathematical Sciences (Edinburgh, UK) held in Edinburgh, September 2011, produced this fine collection of original research and survey articles. Many leading mathematicians attended the event and submitted their contributions for this volume. It is addressed to researchers and graduate students interested in partial differential equations and related analysis with applications.  

This book is about algebro-geometric solutions of completely integrable nonlinear partial differential equations in (1+1)-dimensions; also known as soliton equations. Explicitly treated integrable models include the KdV, AKNS, sine-Gordon, and Camassa-Holm hierarchies as well as the classical massive Thirring system. An extensive treatment of the class of algebro-geometric solutions in the stationary and time-dependent contexts is provided. The formalism presented includes trace formulas, Dubrovin-type initial value problems, Baker-Akhiezer functions, and theta function representations of all relevant quantities involved. The book uses techniques from the theory of differential equations, spectral analysis, and elements of algebraic geometry (most notably, the theory of compact Riemann surfaces).

This book presents the theory of hyperbolic conservation laws from basic theory to the forefront of research. The text treats the theory of scalar conservation laws in one dimension in detail, showing the stability of the Cauchy problem using front tracking. The extension to multidimensional scalar conservation laws is obtained using dimensional splitting. The book includes detailed discussion of the recent proof of well-posedness of the Cauchy problem for one-dimensional hyperbolic conservation laws, and a chapter on traditional finite difference methods for hyperbolic conservation laws with error estimates and a section on measure valued solutions.

Understanding quantum mechanics inevitably leads to an in-depth study of the Schroedinger operator. This set of review lectures informs researchers and advanced students of the most recent developments in the analysis of the Schroedinger operator occurring in solid-state physics, nuclear physics, etc. The topics covered are nonlinear and random potentials, magnetic fields, and many-body problems. Inverse spectral theory is also treated. The results are mathematically rigorous and many physical implications are discussed. The book is suitable for advanced courses in mathematical physics.

The topic of the 2010 Abel Symposium, hosted at the Norwegian Academy of Science and Letters, Oslo, was Nonlinear Partial Differential Equations, the study of which is of fundamental importance in mathematics and in almost all of natural sciences, economics, and engineering. This area of mathematics is currently in the midst of an unprecedented development worldwide. Differential equations are used to model phenomena of increasing complexity, and in areas that have traditionally been outside the realm of mathematics. New analytical tools and numerical methods are dramatically improving our understanding of nonlinear models. Nonlinearity gives rise to novel effects reflected in the appearance of shock waves, turbulence, material defects, etc., and offers challenging mathematical problems. On the other hand, new mathematical developments provide new insight in many applications. These proceedings present a selection of the latest exciting results by world leading researchers.

Next to the harmonic oscillator and the Coulomb potential the class of two-body models with point interactions is the only one where complete solutions are available. All mathematical and physical quantities can be calculated explicitly which makes this field of research important also for more complicated and realistic models in quantum mechanics. The detailed results allow their implementation in numerical codes to analyse properties of alloys, impurities, crystals and other features in solid state quantum physics. This monograph presents in a systematic way the mathematical approach and unifies results obtained in recent years. The student with a sound background in mathematics will get a deeper understanding of Schrodinger Operators and will see many examples which may eventually be used with profit in courses on quantum mechanics and solid state physics. The book has textbook potential in mathematical physics and is suitable for additional reading in various fields of theoretical quantum physics."

This book is based on research that, to a large extent, started around 1990, when a research project on fluid flow in stochastic reservoirs was initiated by a group including some of us with the support of VISTA, a research coopera tion between the Norwegian Academy of Science and Letters and Den norske stats oljeselskap A.S. (Statoil). The purpose of the project was to use stochastic partial differential equations (SPDEs) to describe the flow of fluid in a medium where some of the parameters, e.g., the permeability, were stochastic or "noisy." We soon realized that the theory of SPDEs at the time was insufficient to handle such equations. Therefore it became our aim to develop a new mathematically rigorous theory that satisfied the following conditions. 1) The theory should be physically meaningful and realistic, and the corre sponding solutions should make sense physically and should be useful in applications. 2) The theory should be general enough to handle many of the interesting SPDEs that occur in reservoir theory and related areas. 3) The theory should be strong and efficient enough to allow us to solve th, se SPDEs explicitly, or at least provide algorithms or approximations for the solutions."

The first edition of Stochastic Partial Differential Equations: A Modeling, White Noise Functional Approach, gave a comprehensive introduction to SPDEs. In this, the second edition, the authors build on the theory of SPDEs driven by space-time Brownian motion, or more generally, space-time L vy process noise. Applications of the theory are emphasized throughout. The stochastic pressure equation for fluid flow in porous media is treated, as are applications to finance.

Graduate students in pure and applied mathematics as well as researchers in SPDEs, physics, and engineering will find this introduction indispensible. Useful exercises are collected at the end of each chapter.

This is the second edition of a well-received book providing the fundamentals of the theory hyperbolic conservation laws. Several chapters have been rewritten, new material has been added, in particular, a chapter on space dependent flux functions and the detailed solution of the Riemann problem for the Euler equations. Hyperbolic conservation laws are central in the theory of nonlinear partial differential equations and in science and technology. The reader is given a self-contained presentation using front tracking, which is also a numerical method. The multidimensional scalar case and the case of systems on the line are treated in detail. A chapter on finite differences is included. From the reviews of the first edition: "It is already one of the few best digests on this topic. The present book is an excellent compromise between theory and practice. Students will appreciate the lively and accurate style." D. Serre, MathSciNet "I have read the book with great pleasure, and I can recommend it to experts as well as students. It can also be used for reliable and very exciting basis for a one-semester graduate course." S. Noelle, Book review, German Math. Soc. "Making it an ideal first book for the theory of nonlinear partial differential equations...an excellent reference for a graduate course on nonlinear conservation laws." M. Laforest, Comp. Phys. Comm.

The book presents the winners of the Abel Prize in mathematics for the period 2018–2022:  - Robert P. Langlands (2018) - Karen K. Uhlenbeck (2019) - Hillel Furstenberg and Gregory Margulis (2020) - Lászlo Lóvász and Avi Wigderson (2021) - Dennis P. Sullivan (2022) The profiles feature autobiographical information as well as a scholarly description of each mathematician’s work. In addition, each profile contains a Curriculum Vitae, a complete bibliography, and the full citation from the prize committee.   The book also includes photos from the period 2018–2022 showing many of the additional activities connected with the Abel Prize. This book follows on The Abel Prize: 2003–2007. The First Five Years (Springer, 2010) and The Abel Prize 2008–2012 (Springer, 2014) as well as on The Abel Prize 2013–2017 (Springer, 2019), which profile the previous Abel Prize laureates.