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János Bolyai International Mathematics Award
In honour of the 100th anniversary of the birth of world-famous Hungarian mathematician János Bolyai, the Hungarian Academy of Sciences established an international award of ten-thousand crowns for outstanding mathematical works in 1902.
In addition to preserving the memory of Bolyai, the original aim of the award was to fill the gap left by the Nobel Prize in mathematics.
The first laureate in 1905 was Henri Poincaré of France, one of the most versatile mathematicians of the 19th century, and in 1910 David Hilbert of Germany received the award. The awarding of the medal was interrupted after the outbreak of the First World War.
The Hungarian Academy of Sciences re-founded the award in 1994, calling it the János Bolyai International Mathematics Award. The award provides the winner USD 25,000 and a gold-plated bronze medallion made using the original designs.
The Bolyai Prize is awarded every fifth year by the Hungarian Academy of Sciences to the author of the most outstanding, ground-breaking mathematical monograph presenting his/her own new results and methods published anywhere and in any language in the preceding fifteen (previously ten) years, taking into account the author's previous scientific work.
One year before the prize is awarded, the Section of Mathematics elects a committee consisting of five regular members and five outstanding foreign mathematicians and appoints its chairman. The committee reports on its decision to the department chairman no later than three months before the award is presented. The committee itself chooses its presenter from among its members, who presents the awardee's work in detail and prepares a written report. The president also votes in the committee, with his/her vote being the deciding one in the event of a tie.
János Bolyai Award winners.
1905
Henri Poincaré.
The Bolyai Prize was awarded for the first time in 1905. The committee, whose members were Jean Gaston Darboux, Felix Klein, Gyula Kőnig and Gusztáv Rados, awarded it to Henri Poincaré. Poincaré did not go to Budapest for the presentation but the prize was delivered to him through official channels.
1910
David Hilbert.
In 1910 the committee composed by Gyula Kőnig, Gusztáv Rados, Henri Poincaré and Gösta Mittag-Leffler awarded the prize to the Göttingen university professor David Hilbert for his outstanding work. Hilbert did not go to Budapest for the presentation but the prize was delivered to him through official channels. He wrote a letter of thanks which included the line:-
Saharon Shelah.
The monograph for which Professor Saharon Shelah received the award is his work Cardinal Arithmetic, published by Oxford University Press in 1994, in which he presented his profound and sophisticated pcf (possible cofinalities) theory, a new approach to cardinal arithmetic. This work attests to his view that there is an essential core of set theory unaffected by the famous independence results of the last four decades, in which a great deal of work remains to be done, His applications of pcf theory, many of which are given in his monograph, constitute a breakthrough in many areas of pure, as well as applied set theory.
In his 700-plus papers and several books Professor Shelah has had an unparalleled influence on modern set theory and model theory. His solutions to deep and longstanding problems and his independence techniques forever changed the landscape of set theory; this is even more true in model theory where his concepts and methods have completely revolutionised the area. He also solved a number of famous problems arising in other branches of mathematics, such as algebra, combinatorics, and topology.
Laudation for Saharon Shelah.
The award winner Saharon Shelah is a phenomenal mathematician, preeminent both in model theory and set theory. His work, beginning in the early 1970's, has tremendously advanced both subjects, and even now, in his mid fifties, he is continuing to produce results at a furious pace. He has over 700 items in his bibliography, the majority of them long or substantial papers. The one other modern mathematician who sustained a comparable level of productivity on paper was Paul Erdős, and in an interview that appeared in 1985 he singled out Shelah among all mathematicians for praise.
The book "Cardinal Arithmetic" that is being awarded discusses only a specific, but very significant part of Shelah's work, the theory of pcf ("possible cofinalities") and its applications. This theory, created by Shelah, became a major branch of set-theoretic research since the late 1980's, illuminating many issues involving singular cardinals in combinatorial set theory and the theory of large cardinals.
One of the features of the theory, emphasised by Shelah himself as well, is that it has led to a plethora of direct theorems of set theory, as opposed to relative consistency results. The advent of forcing and large cardinals in the 1960's led to a continuing investigation through the next two decades of strong propositions independent of set theory and their consistency strength. What pcf theory did was to broaden set-theoretic research by infusing a complex of new direct theorems of set theory.
2005
Misha Gromov.
Misha Gromov was awarded the 2005 János Bolyai International Mathematical Prize of the Hungarian Academy of Sciences for Metric structures for Riemannian and non-Riemannian spaces, Birkhäuser, 1999.
Laudation for Misha Gromov.
It is a great pleasure to announce that the International Bolyai Prize was awarded to Misha Gromov for his book "Metric Structures for Riemannian and Non-Riemannian Spaces". This decision was reached by a committee appointed by the Academy, composed of Sándor Csörgő, Zoltán Daróczy, Ciprian Foias, András Hajnal, János Kollár, Miklós Laczkovich, Yakov Pesin, Claus Michael Ringel, Andrzej Schinzel, András Recski as Secretary, and myself (László Lovász) as Chair.
Misha Gromov is one of the very top leading mathematicians of our time; András Szucs will give a summary of his research. However, this prize is not given primarily for his pathbreaking theories and beautiful theorems; rather, it is given for this book, as a recognition of the additional work invested in summarising these results in a unified monograph, and of the great additional value this presentation gives to all of us.
Gromov's book is based on his 1981 French lecture notes (edited by Lafontaine and Pansu), but in the time that elapsed since then the subject developed and expanded so dramatically, and accordingly the book is enriched with so much new material that it is almost four times longer than the original, and can be considered as an entirely new book. This remarkable development over two decades is largely due to the powerful ideas that Gromov introduced, and the compelling questions that he asked. It is a wide-spread opinion that Gromov's book is the most influential mathematical book published in the past decade. This book embodies Gromov's philosophy of distilling simple and elegant geometric and analytic conditions, which are the core of various deep phenomena in Riemannian geometry.
One remarkable aspect of Gromov's achievements is that they make this topic widely accessible: researchers whose background and main interest is not Riemannian geometry can understand, through Gromov's book, this important subject, and even contribute to it. Thus, this book exemplifies a fundamental aspect of Gromov's work, which goes beyond its mathematical depth: mathematical leadership.
The basic approach of Gromov is that many phenomena in Riemannian geometry are based on properties of certain manifolds which have simple characterisations in terms of their metric structure. Thus, for example, when studying negatively curved manifolds it is beneficial to understand specific hyperbolic and metric spaces. Moreover, the interaction between metric and measure is presented as a fundamental tool in high dimensional geometry. This is exemplified in the thorough discussion of the concentration of measure phenomenon, the observable diameter, and volume comparison theorems under Ricci curvature bounds. The material connects up seemingly distant topics such as packing inequalities, entropy, simplicial norms, and Betti numbers.
Gromov's philosophy is to study phenomena rather than particular structures. This is why this book, and through it his ideas, are influential not only in differential geometry, but also in seemingly very distant areas like graph theory.
For example, Gromov's book discusses in depth the phenomenon of concentration of measure in high dimension, going back to the work of Paul Lévy and Vitali Milman. This leads to a "Law of Large Numbers" for metric spaces with measures and dimension going to infinity. Many fundamental results and constructions in functional analysis, information theory, probability, graph theory, and other areas can be viewed as manifestations of this phenomenon. The concentration phenomenon is connected to phase transitions in statistical physics, another exciting general phenomenon that bridges traditional areas. Other such general phenomena discussed in the book include expansion and isoperimetry.
May I add a further example that influenced my own current research. Gromov's book introduces the paradigm that one can define a metric on the space of metric spaces (now called the Gromov-Hausdorff metric), and study their convergence in various contexts, this way defining limits of sequences of metric spaces that are not part of a single structure at all. It is surprising that such an abstractly defined space is useful at all, but in fact the study of parameters that are continuous and properties that are preserved when going to the limit leads to important compactness results and finite classifications.
These abstract formulations of geometric phenomena are not made just for the sake of generalisation. The author keeps the reader "well grounded" by illustrating the general treatment by the analysis of numerous concrete examples, which often go beyond Riemannian spaces. The reader meets a dazzling wealth of ideas, questions, and possibilities for further research.
2010
Yuri Ivanovich Manin.
In December 2010 the János Bolyai International Mathematical Prize of the Hungarian Academy of Sciences was awarded to Professor Yuri Ivanovich Manin for Frobenius Manifolds, Quantum Cohomology and Moduli Spaces, American Mathematical Society, 1999.
This is the first monograph dedicated to the systematic exposition of the whole variety of topics related to quantum cohomology. The subject first originated in theoretical physics (quantum string theory) and has continued to develop extensively over the last decade.
The author's approach to quantum cohomology is based on the notion of the Frobenius manifold. The first part of the book is devoted to this notion and its extensive interconnections with algebraic formalism of operads, differential equations, perturbations, and geometry. In the second part of the book, the author describes the construction of quantum cohomology and reviews the algebraic geometry mechanisms involved in this construction (intersection and deformation theory of Deligne-Artin and Mumford stacks).
2015
Barry Simon.
In 2015 the János Bolyai International Mathematical Prize of the was awarded to Barry Simon for Orthogonal Polynomials on the Unit Circle,American Mathematical Society , 2005.
This two-part volume gives a comprehensive overview of the theory of probability measures on the unit circle, viewed especially in terms of the orthogonal polynomials defined by those measures. A major theme involves the connections between the Verblunsky coefficients (the coefficients of the recurrence equation for the orthogonal polynomials) and the measures, an analogue of the spectral theory of one-dimensional Schrödinger operators.
Among the topics discussed along the way are the asymptotics of Toeplitz determinants (Szegő's theorems), limit theorems for the density of the zeros of orthogonal polynomials, matrix representations for multiplication by z (CMV matrices), periodic Verblunsky coefficients from the point of view of meromorphic functions on hyperelliptic surfaces, and connections between the theories of orthogonal polynomials on the unit circle and on the real line.
2020
Terence Tao.
In 2020 the János Bolyai International Mathematical Prize of the Hungarian Academy of Sciences was awarded to Terence Tao for Nonlinear Dispersive Equations, American Mathematical Society, 2006.
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. The book Nonlinear Dispersive Equations 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.
Starting only with a basic knowledge of graduate real analysis and Fourier analysis, the text first presents basic nonlinear tools such as the bootstrap method and perturbation theory in the simpler context of nonlinear ODE, then introduces the harmonic analysis and geometric tools used to control linear dispersive PDE. These methods are then combined to study four model nonlinear dispersive equations. Through extensive exercises, diagrams, and informal discussion, the book gives a rigorous theoretical treatment of the material, the real-world intuition and heuristics that underlie the subject, as well as mentioning connections with other areas of PDE, harmonic analysis, and dynamical systems.
As the subject is vast, the book does not attempt to give a comprehensive survey of the field, but instead concentrates on a representative sample of results for a selected set of equations, ranging from the fundamental local and global existence theorems to very recent results, particularly focusing on the recent progress in understanding the evolution of energy-critical dispersive equations from large data. The book is suitable for a graduate course on nonlinear PDE.
2025
János Kollár.
In 2020 the János Bolyai International Mathematical Prize of the Hungarian Academy of Sciences was awarded to János Kollár for Families of varieties of general type (2023).
The book Families of varieties of general type establishes the moduli theory of stable varieties, giving the optimal approach to understanding families of varieties of general type. Starting from the Deligne-Mumford theory of the moduli of curves and using Mori's program as a main tool, the book develops the techniques necessary for a theory in all dimensions. The main results give all the expected general properties, including a projective coarse moduli space. A wealth of previously unpublished material is also featured, including Chapter 5 on numerical flatness criteria, Chapter 7 on K-flatness, and Chapter 9 on hulls and husks.
One important aspect of Kollár's theory is its wide-ranging applicability. What Kollár achieves through this book is fundamental, across all moduli theories of varieties. As is true of all his books, Families of varieties of general type, although presenting the reader with some very serious technical challenges and being full of profound insights, has a clear structure, clear explanations and provides detailed exposition of the motivation of the concepts, making it an enjoyable read and an attractive way to study the moduli of varieties of general type; it is required reading for anyone working in the moduli theory of varieties. This book is written by the master of the field - it is a crowning achievement in algebraic geometry.
In addition to preserving the memory of Bolyai, the original aim of the award was to fill the gap left by the Nobel Prize in mathematics.
The first laureate in 1905 was Henri Poincaré of France, one of the most versatile mathematicians of the 19th century, and in 1910 David Hilbert of Germany received the award. The awarding of the medal was interrupted after the outbreak of the First World War.
The Hungarian Academy of Sciences re-founded the award in 1994, calling it the János Bolyai International Mathematics Award. The award provides the winner USD 25,000 and a gold-plated bronze medallion made using the original designs.
The Bolyai Prize is awarded every fifth year by the Hungarian Academy of Sciences to the author of the most outstanding, ground-breaking mathematical monograph presenting his/her own new results and methods published anywhere and in any language in the preceding fifteen (previously ten) years, taking into account the author's previous scientific work.
One year before the prize is awarded, the Section of Mathematics elects a committee consisting of five regular members and five outstanding foreign mathematicians and appoints its chairman. The committee reports on its decision to the department chairman no later than three months before the award is presented. The committee itself chooses its presenter from among its members, who presents the awardee's work in detail and prepares a written report. The president also votes in the committee, with his/her vote being the deciding one in the event of a tie.
János Bolyai Award winners.
1905
Henri Poincaré.
The Bolyai Prize was awarded for the first time in 1905. The committee, whose members were Jean Gaston Darboux, Felix Klein, Gyula Kőnig and Gusztáv Rados, awarded it to Henri Poincaré. Poincaré did not go to Budapest for the presentation but the prize was delivered to him through official channels.
1910
David Hilbert.
In 1910 the committee composed by Gyula Kőnig, Gusztáv Rados, Henri Poincaré and Gösta Mittag-Leffler awarded the prize to the Göttingen university professor David Hilbert for his outstanding work. Hilbert did not go to Budapest for the presentation but the prize was delivered to him through official channels. He wrote a letter of thanks which included the line:-
I'm all the more proud of it, as up until now I have never been given such a distinctive honour.2000
Saharon Shelah.
The monograph for which Professor Saharon Shelah received the award is his work Cardinal Arithmetic, published by Oxford University Press in 1994, in which he presented his profound and sophisticated pcf (possible cofinalities) theory, a new approach to cardinal arithmetic. This work attests to his view that there is an essential core of set theory unaffected by the famous independence results of the last four decades, in which a great deal of work remains to be done, His applications of pcf theory, many of which are given in his monograph, constitute a breakthrough in many areas of pure, as well as applied set theory.
In his 700-plus papers and several books Professor Shelah has had an unparalleled influence on modern set theory and model theory. His solutions to deep and longstanding problems and his independence techniques forever changed the landscape of set theory; this is even more true in model theory where his concepts and methods have completely revolutionised the area. He also solved a number of famous problems arising in other branches of mathematics, such as algebra, combinatorics, and topology.
Laudation for Saharon Shelah.
The award winner Saharon Shelah is a phenomenal mathematician, preeminent both in model theory and set theory. His work, beginning in the early 1970's, has tremendously advanced both subjects, and even now, in his mid fifties, he is continuing to produce results at a furious pace. He has over 700 items in his bibliography, the majority of them long or substantial papers. The one other modern mathematician who sustained a comparable level of productivity on paper was Paul Erdős, and in an interview that appeared in 1985 he singled out Shelah among all mathematicians for praise.
The book "Cardinal Arithmetic" that is being awarded discusses only a specific, but very significant part of Shelah's work, the theory of pcf ("possible cofinalities") and its applications. This theory, created by Shelah, became a major branch of set-theoretic research since the late 1980's, illuminating many issues involving singular cardinals in combinatorial set theory and the theory of large cardinals.
One of the features of the theory, emphasised by Shelah himself as well, is that it has led to a plethora of direct theorems of set theory, as opposed to relative consistency results. The advent of forcing and large cardinals in the 1960's led to a continuing investigation through the next two decades of strong propositions independent of set theory and their consistency strength. What pcf theory did was to broaden set-theoretic research by infusing a complex of new direct theorems of set theory.
2005
Misha Gromov.
Misha Gromov was awarded the 2005 János Bolyai International Mathematical Prize of the Hungarian Academy of Sciences for Metric structures for Riemannian and non-Riemannian spaces, Birkhäuser, 1999.
Laudation for Misha Gromov.
It is a great pleasure to announce that the International Bolyai Prize was awarded to Misha Gromov for his book "Metric Structures for Riemannian and Non-Riemannian Spaces". This decision was reached by a committee appointed by the Academy, composed of Sándor Csörgő, Zoltán Daróczy, Ciprian Foias, András Hajnal, János Kollár, Miklós Laczkovich, Yakov Pesin, Claus Michael Ringel, Andrzej Schinzel, András Recski as Secretary, and myself (László Lovász) as Chair.
Misha Gromov is one of the very top leading mathematicians of our time; András Szucs will give a summary of his research. However, this prize is not given primarily for his pathbreaking theories and beautiful theorems; rather, it is given for this book, as a recognition of the additional work invested in summarising these results in a unified monograph, and of the great additional value this presentation gives to all of us.
Gromov's book is based on his 1981 French lecture notes (edited by Lafontaine and Pansu), but in the time that elapsed since then the subject developed and expanded so dramatically, and accordingly the book is enriched with so much new material that it is almost four times longer than the original, and can be considered as an entirely new book. This remarkable development over two decades is largely due to the powerful ideas that Gromov introduced, and the compelling questions that he asked. It is a wide-spread opinion that Gromov's book is the most influential mathematical book published in the past decade. This book embodies Gromov's philosophy of distilling simple and elegant geometric and analytic conditions, which are the core of various deep phenomena in Riemannian geometry.
One remarkable aspect of Gromov's achievements is that they make this topic widely accessible: researchers whose background and main interest is not Riemannian geometry can understand, through Gromov's book, this important subject, and even contribute to it. Thus, this book exemplifies a fundamental aspect of Gromov's work, which goes beyond its mathematical depth: mathematical leadership.
The basic approach of Gromov is that many phenomena in Riemannian geometry are based on properties of certain manifolds which have simple characterisations in terms of their metric structure. Thus, for example, when studying negatively curved manifolds it is beneficial to understand specific hyperbolic and metric spaces. Moreover, the interaction between metric and measure is presented as a fundamental tool in high dimensional geometry. This is exemplified in the thorough discussion of the concentration of measure phenomenon, the observable diameter, and volume comparison theorems under Ricci curvature bounds. The material connects up seemingly distant topics such as packing inequalities, entropy, simplicial norms, and Betti numbers.
Gromov's philosophy is to study phenomena rather than particular structures. This is why this book, and through it his ideas, are influential not only in differential geometry, but also in seemingly very distant areas like graph theory.
For example, Gromov's book discusses in depth the phenomenon of concentration of measure in high dimension, going back to the work of Paul Lévy and Vitali Milman. This leads to a "Law of Large Numbers" for metric spaces with measures and dimension going to infinity. Many fundamental results and constructions in functional analysis, information theory, probability, graph theory, and other areas can be viewed as manifestations of this phenomenon. The concentration phenomenon is connected to phase transitions in statistical physics, another exciting general phenomenon that bridges traditional areas. Other such general phenomena discussed in the book include expansion and isoperimetry.
May I add a further example that influenced my own current research. Gromov's book introduces the paradigm that one can define a metric on the space of metric spaces (now called the Gromov-Hausdorff metric), and study their convergence in various contexts, this way defining limits of sequences of metric spaces that are not part of a single structure at all. It is surprising that such an abstractly defined space is useful at all, but in fact the study of parameters that are continuous and properties that are preserved when going to the limit leads to important compactness results and finite classifications.
These abstract formulations of geometric phenomena are not made just for the sake of generalisation. The author keeps the reader "well grounded" by illustrating the general treatment by the analysis of numerous concrete examples, which often go beyond Riemannian spaces. The reader meets a dazzling wealth of ideas, questions, and possibilities for further research.
2010
Yuri Ivanovich Manin.
In December 2010 the János Bolyai International Mathematical Prize of the Hungarian Academy of Sciences was awarded to Professor Yuri Ivanovich Manin for Frobenius Manifolds, Quantum Cohomology and Moduli Spaces, American Mathematical Society, 1999.
This is the first monograph dedicated to the systematic exposition of the whole variety of topics related to quantum cohomology. The subject first originated in theoretical physics (quantum string theory) and has continued to develop extensively over the last decade.
The author's approach to quantum cohomology is based on the notion of the Frobenius manifold. The first part of the book is devoted to this notion and its extensive interconnections with algebraic formalism of operads, differential equations, perturbations, and geometry. In the second part of the book, the author describes the construction of quantum cohomology and reviews the algebraic geometry mechanisms involved in this construction (intersection and deformation theory of Deligne-Artin and Mumford stacks).
2015
Barry Simon.
In 2015 the János Bolyai International Mathematical Prize of the was awarded to Barry Simon for Orthogonal Polynomials on the Unit Circle,
This two-part volume gives a comprehensive overview of the theory of probability measures on the unit circle, viewed especially in terms of the orthogonal polynomials defined by those measures. A major theme involves the connections between the Verblunsky coefficients (the coefficients of the recurrence equation for the orthogonal polynomials) and the measures, an analogue of the spectral theory of one-dimensional Schrödinger operators.
Among the topics discussed along the way are the asymptotics of Toeplitz determinants (Szegő's theorems), limit theorems for the density of the zeros of orthogonal polynomials, matrix representations for multiplication by z (CMV matrices), periodic Verblunsky coefficients from the point of view of meromorphic functions on hyperelliptic surfaces, and connections between the theories of orthogonal polynomials on the unit circle and on the real line.
2020
Terence Tao.
In 2020 the János Bolyai International Mathematical Prize of the Hungarian Academy of Sciences was awarded to Terence Tao for Nonlinear Dispersive Equations, American Mathematical Society, 2006.
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. The book Nonlinear Dispersive Equations 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.
Starting only with a basic knowledge of graduate real analysis and Fourier analysis, the text first presents basic nonlinear tools such as the bootstrap method and perturbation theory in the simpler context of nonlinear ODE, then introduces the harmonic analysis and geometric tools used to control linear dispersive PDE. These methods are then combined to study four model nonlinear dispersive equations. Through extensive exercises, diagrams, and informal discussion, the book gives a rigorous theoretical treatment of the material, the real-world intuition and heuristics that underlie the subject, as well as mentioning connections with other areas of PDE, harmonic analysis, and dynamical systems.
As the subject is vast, the book does not attempt to give a comprehensive survey of the field, but instead concentrates on a representative sample of results for a selected set of equations, ranging from the fundamental local and global existence theorems to very recent results, particularly focusing on the recent progress in understanding the evolution of energy-critical dispersive equations from large data. The book is suitable for a graduate course on nonlinear PDE.
2025
János Kollár.
In 2020 the János Bolyai International Mathematical Prize of the Hungarian Academy of Sciences was awarded to János Kollár for Families of varieties of general type (2023).
The book Families of varieties of general type establishes the moduli theory of stable varieties, giving the optimal approach to understanding families of varieties of general type. Starting from the Deligne-Mumford theory of the moduli of curves and using Mori's program as a main tool, the book develops the techniques necessary for a theory in all dimensions. The main results give all the expected general properties, including a projective coarse moduli space. A wealth of previously unpublished material is also featured, including Chapter 5 on numerical flatness criteria, Chapter 7 on K-flatness, and Chapter 9 on hulls and husks.
One important aspect of Kollár's theory is its wide-ranging applicability. What Kollár achieves through this book is fundamental, across all moduli theories of varieties. As is true of all his books, Families of varieties of general type, although presenting the reader with some very serious technical challenges and being full of profound insights, has a clear structure, clear explanations and provides detailed exposition of the motivation of the concepts, making it an enjoyable read and an attractive way to study the moduli of varieties of general type; it is required reading for anyone working in the moduli theory of varieties. This book is written by the master of the field - it is a crowning achievement in algebraic geometry.