Noga Alon

Wolf Prize Laureate in Mathematics 2024

Noga Alon

 

Affiliation at the time of the award:

Princeton University, USA

 

Award citation:

“for his fundamental contributions to Combinatorics and Theoretical Computer Science”.

 

Prize share:

Noga Alon

Adi Shamir 

 

“for their pioneering contributions to mathematical cryptography, combinatorics, and the theory of computer science”.

 

Noga Alon (born in Israel, 1956) is a Professor of Mathematics at Princeton University, a Baumritter Professor Emeritus of Mathematics and Computer Science at Tel Aviv University, and one of the most influential mathematicians worldwide. His research and developments changed the face of the field, created new concepts and original methods, and contributed greatly to the development of theoretical research and their applications in discrete mathematics, information theory, graph theory, and their uses in the theory of computer science. He is one of the most prolific mathematicians in the world, published hundreds of articles, and trained many research students in mathematics and computer science.

Alon showed a profound interest in mathematics from an early age, drawn to its objectivity and pursuit of absolute truth. Encouraged by his parents and his math teacher to follow his passions, Alon delved into mathematics and participated in math competitions. After graduating in mathematics at the Technion, he continued to earn his master’s and doctoral degrees at the Hebrew University of Jerusalem in 1983 and held visiting positions in various research institutes, including MIT, Harvard, the Institute for Advanced Study in Princeton, IBM Almaden Research Center, Bell Laboratories, Bellcore and Microsoft Research (Redmond and Israel). He joined Tel Aviv University in 1985, served as the head of the School of Mathematical Sciences in 1999-2000, retired from Tel Aviv, and moved to Princeton in 2018, where he works until today, and supervised many PhD students. He serves on the editorial boards of more than a dozen international professional journals and has given invited lectures at many conferences. He was the head of the scientific committee of the World Congress of Mathematics (Madrid, 2006) and a member of various prestigious prize committees worldwide. He published more than six hundred research papers and one book.

Alon’s contributions to mathematics are broad and have influenced many related areas in the theoretical and applied sciences. With his collaborators he established the tight connection between the expansion properties of a graph and its spectral properties and found numerous applications of expanders in Combinatorics and Theoretical Computer Science. His results stimulated a great amount of further work and are cited in essentially all subsequent extensive work in the area. In related work, he pioneered the application of spectral methods in the study of algorithmic problems. Alon proved the Combinatorial Nullstellensatz (1995), a powerful algebraic technique that yielded highly significant applications in Graph Theory, Combinatorics, and Additive Number Theory, including an extension of the Four-Color Theorem. With Nathanson and Ruzsa (1996) he obtained generalizations of the Cauchy-Davenport Theorem. In joint work with Kleitman (1992), he settled a problem of Hadwiger and Debrunner in Combinatorial Geometry raised in 1957, proving a far-reaching generalization of Helly’s Theorem. The method has proven to be highly influential and is described in most recent books and survey articles on the subject. Alon (1998) disproved a conjecture of Shannon, raised in 1956, proving the surprising fact that the Shannon capacity of a disjoint union of two channels can be much bigger than the sum of their capacities or even than any fixed power of this sum.

Alon played a major role in the development of probabilistic methods in Combinatorics, and his book with Spencer on the subject (first edition in 1992, fourth edition in 2016) is the undisputed leading text in this central area. His Color-Coding method, developed with Yuster and Zwick (1995), found applications in several other fields including the theory of Fixed Parameter Tractability and Bioinformatics. His joint work with Matias and Szegedy (1999) initiated the study of streaming algorithms investigating which statistical properties of a stream of data can be sampled and estimated on the fly. This has literally created a new active area of streaming and sketching algorithms and has numerous theoretical and applied applications.

Alon with his collaborators (1994) developed an algorithmic version of Szemerédi’s Regularity Lemma, discovered its connection to a classical inequality of Grothendieck, and used it to settle essentially all major open problems in the theory of Property Testing for dense graphs. This generated extensive research and played an important role in the subsequent development of Lovasz and his collaborators’ theory of convergent graph sequences.

Some of Noga Alon’s most influential works deal with “expander graphs”. These are sparse networks with strong connectivity properties. They were originally conceived as a way to build economical, robust networks (phone or computer) and have found extensive applications in computer science, in designing algorithms, error-correcting codes, pseudorandom generators, and more. Alon, in part with Milman, established a tight connection between the expansion properties of a graph and its “spectral” properties, reminiscent of a relation between classical and quantum mechanics, and found numerous applications of expanders in combinatorics and in theoretical computer science. Alon’s results stimulated a great amount of further work and are cited in essentially all subsequent extensive work in the area.
Noga Alon is being awarded the 2024 Wolf Prize for his profound impact on Discrete Mathematics and related areas. His seminal contributions include the development of ingenious techniques in Combinatorics, Graph Theory, and Theoretical Computer Science, and the solution of long-standing problems in these fields as well as in Analytical Number Theory, Combinatorial Geometry, and Information Theory.

Adi Shamir

Wolf Prize Laureate in Mathematics 2024

Adi Shamir

 

Affiliation at the time of the award:

The Weizmann Institute of Science, Israel

 

Award citation:

“for his fundamental contributions to Mathematical Cryptography”.

 

Prize share:

Adi Shamir

Noga Alon

 

“for their pioneering contributions to mathematical cryptography, combinatorics, and the theory of computer science”.

 

Adi Shamir (born in Israel in 1952), a professor in the Department of Computer Science and Applied Mathematics at the Weizmann Institute of Science, is one of the most senior computer scientists globally. He is a top expert in the fields of information encryption and decryption. Shamir was among the developers of the RSA method that changed the face of computer communication in the world and was a fundamental pillar in electronic commerce and information security.
From a young age, Shamir showed an interest in science and participated in youth academic programs and science summer camps at the Weizmann Institute of Science. After graduating with high honors with his bachelor’s degree in mathematics at Tel Aviv University (1973), Shamir furthered his studies at the Weizmann Institute, focusing on computer science and earning an MSc in 1975, followed by a PhD in 1977. In his doctoral thesis, he examined the properties of certain mathematical functions that are relevant to the semantics of programming languages. Following the completion of his doctoral studies, he pursued a short-term postdoctoral position at the University of Warwick in Coventry, England, and continued his academic journey at the Massachusetts Institute of Technology (MIT), USA, where he began to study the theory of encryption and the theory of decoding.
In traditional encryption, a key is crucial for both message encryption and decryption, posing a security challenge in key distribution. Seeking a solution, in 1977, MIT researchers Ron Rivest, Adi Shamir, and Leonard Adelman devised a groundbreaking public-key encryption method known as RSA (initials of Rivest, Shamir, and Adelman – its three developers). This method utilized a one-way mathematical function based on the multiplication of prime numbers, ensuring that the original solution couldn’t be retrieved. RSA employs two distinct but mathematically linked keys: a public key for encryption and a private key for decryption, eliminating the need for key distribution. Recognized globally, RSA Cryptography is a cornerstone in securing online communication, e-commerce, and confidential data in transactions. Its significance extends beyond practical use, garnering attention from mathematicians, companies, governments, and intelligence agencies. The RSA method has become a fundamental and nearly exclusive element in safeguarding computer information and electronic commerce.
Among his numerous additional contributions to information security, Shamir introduced the groundbreaking secret-sharing method. This technique transforms secrets into sets of random numbers, requiring a specific combination to reconstruct the original secret, forming the basis for secure computations. Collaborating with peers, he advanced identification and signature methods through zero-knowledge proofs and devised the ring signature for group-based encryption. Shamir’s ingenuity extended to TV broadcast encryption, allowing encrypted transmissions exclusively for paying recipients. In recent years, his research delved into T-functions, intricate mathematical tools for information encryption. Shamir’s impact also extends to exposing vulnerabilities in encryption systems, developing general mathematical methods for attacks, and pioneering Side Channel Attacks on hardware and software implementations. Beyond information security, his contributions resonate in core computer science, notably shaping the theory of computational complexity.
Adi Shamir is awarded the Wolf Prize for being a truly exceptional scientist and has been the leading force in transforming cryptography into a scientific discipline that is heavily based on Mathematics. His foundational discoveries combine mathematical ingenuity with a range of analytical tools. They had a huge impact on several mathematical areas, advancing both mathematics and society in an unparalleled manner.

 

 

Ingrid Daubechies

Wolf Prize Laureate in Mathematics 2023

Ingrid Daubechies

 

Affiliation at the time of the award:

Duke University, USA

 

Award citation:

“for work in wavelet theory and applied harmonic analysis”.

 

Prize share:

None

 

Ingrid Daubechies is a Belgian mathematician and physicist at Duke University in Durham, North Carolina. She earned her bachelor’s degree in physics from the Free University of Brussels in 1975. She then continued her research at the same university, earning her doctorate in physics with a thesis on the Representation of quantum mechanical operators by kernels on Hilbert spaces of analytic functions.

Ingrid Daubechies’ love for math and science was nurtured from a young age. Her father fostered her curiosity and interest in these subjects while she was in school. As a child, she was fascinated by how things worked and how to construct them, as well as the mechanisms behind machinery and the truth behind mathematical concepts. She would even calculate large numbers in her head when she couldn’t sleep, finding it captivating to see the numbers quickly grow.

Professor Ingrid Daubechies has made significant contributions to the field of wavelet theory. Her research has revolutionized the way images and signals are processed numerically, providing standard and flexible algorithms for data compression. This has led to a wide range of innovations in various technologies, including medical imaging, wireless communication, and even digital cinema.
The Wavelet theory, as presented by the work of Professor Daubechies, has become a crucial tool in many areas of signal and image processing. For example, it has been used to enhance and reconstruct images from the early days of the Hubble Telescope, to detect forged documents and fingerprints. In addition, wavelets are a vital component of wireless communication and are used to compress sound sequences into MP3 files.

Beyond her scientific contributions, Professor Daubechies also advocates for equal opportunities in science and math education, particularly in developing countries. As President of the International Mathematical Union, she worked to promote this cause. She is aware of the barriers women face in these fields and works to mentor young women scientists and increase representation and opportunities for them.

Daubechies’s most important contribution is her introduction in 1988 of smooth compactly supported orthonormal wavelet bases. These bases revolutionized signal processing, leading to highly efficient methods for digitizing, storing, compressing, and analyzing data, such as audio and video signals, computed tomography, and magnetic resonance imaging. The compact support of these wavelets made it possible to digitize a signal in time linearly dependent on the length of the signal. This was a critical ingredient for researchers and engineers in signal processing to be able to rapidly decompose a signal as a superposition of contributions at various scales.
In subsequent joint work with A. Cohen and J.C. Feauveau, Daubechies introduced symmetrical biorthogonal wavelet bases. These wavelet bases give up orthonormality in favor of symmetry. Such bases are much more suitable for treating the discontinuities arising at the boundaries of finite-length signals and improving image quality. Her biorthogonal wavelets became the basis for the JPEG 2000 image compression and coding system.

Ingrid Daubechies is awarded the Wolf Prize for her work in the creation and development of wavelet theory and modern time-frequency analysis. Her discovery of smooth, compactly supported wavelets, and the development of biorthogonal wavelets transformed image and signal processing and filtering.
Her work is of tremendous importance in image compression, medical imaging, remote sensing, and digital photography. Daubechies has also made unparalleled contributions to developing real-world applications of harmonic analysis, introducing sophisticated image-processing techniques to fields ranging from art to evolutionary biology and beyond.

George Lusztig

Wolf Prize Laureate in Mathematics 2022

George Lusztig

 

Affiliation at the time of the award:

Massachusetts Institute of Technology, USA

 

Award citation:

“for Groundbreaking contributions to representation theory and related areas”.

 

Prize share:

None

 

Lusztig is a Romanian-American mathematician, who works on geometric finite reductive groups, representation theory and algebraic groups. Lusztig’s work is characterized by a very high degree of originality, an enormous breadth of subject matter, remarkable technical virtuosity, and great profundity in getting to the heart of the problems involved. Lusztig’s groundbreaking contributions mark him as one of the great mathematicians of our time.

His passion for mathematics began at a young age. In fact, it was in math competitions at school which made him realize that he was talented in mathematics. After finishing 10th grade, Lusztig represented Romania in the International Mathematical Olympiad in 1962 and then again, in 1963: being awarded a Silver Medal on both occasions. Lusztig graduated from the University of Bucharest in 1968 and received both the M.A. and Ph.D. from Princeton University in 1971 under the direction of Michael Atiyah and William Browder. He joined the MIT mathematics faculty in 1978 following a professorship appointment at the University of Warwick, 1974-77. He was appointed Norbert Wiener Professor at MIT 1999-2009.

Lusztig is known for his work on representation theory, in particular for the objects closely related to algebraic groups, such as finite reductive groups, Hecke algebras, P-adic groups, quantum groups, and Weyl groups. He essentially paved the way for modern representation theory. This has included fundamental new concepts, including the character sheaves, the “Deligne–Lusztig” varieties, and the “Kazhdan–Lusztig” polynomials.

Lusztig’s first breakthrough came with Deligne around 1975, with the construction of Deligne-Lusztig representations. He then obtained a complete description of the irreducible representations of reductive groups over finite fields. Lusztig’s description of the character table of a finite reductive group rates as one of the most  extraordinary achievements of a single mathematician in the 20th century. To achieve his goal, he developed a panoply of techniques which are in use today by hundreds of mathematicians. The highlights include the use of étale cohomology; the role played by the dual group; the use of intersection cohomology, and the ensuing theory of character sheaves, almost characters, and the noncommutative Fourier transform.

In 1979 Kazhdan and Lusztig defined the “Kazhdan-Lusztig” basis of the Hecke algebra of a Coxeter group and stated the “Kazhdan-Lusztig” conjecture. The “Kazhdan-Lusztig” conjecture led directly to the “Beilinson-Bernstein” localization theorem, which four decades later, remains our most powerful tool for understanding representations
of reductive Lie algebras. Lusztig’s work with Vogan then introduced a variant of the “Kazhdan-Lusztig” algorithm to produce “Lusztig-Vogan” polynomials. These polynomials are fundamental to our understanding of real reductive groups and their unitary representations.

In the 1990s, Lusztig made seminal contributions to the theory of quantum groups. His contributions include the introduction of the canonical basis; the introduction of the Lusztig form (which allows specialization to a root of unity, and connections to modular representations); the quantum Frobenius and a small quantum group; and connections to the representation theory of affine Lie algebras. Lusztig’s theory of the canonical basis (and Kashiwara’s parallel theory of crystal bases) has led to deep results in combinatorics and representation theory. Recently there has been significant progress in representation theory and low-dimensional topology via “categorification”; the roots of this work go back to Lusztig’s geometric categorification of quantum groups via perverse sheaves on quiver moduli.

Yakov Eliashberg

Wolf Prize Laureate in Mathematics 2020

Yakov Eliashberg

 

Affiliation at the time of the award:

Stanford University, USA

 

Award citation:

“for their contributions to differential geometry and topology.”

 

Prize Share:

Yakov Eliashberg

Simon Donaldson 

 

Yakov Eliashberg is one of the founders of symplectic and contact topology, a discipline originated as mathematical language for qualitative problems of classical mechanics, and having deep connections with modern physics. The emergence of symplectic and contact topology has been one of the most striking long-term advances in mathematical research over the past four decades. Eliashberg is among the main exponents of this development.

Professor Eliashberg was born in 1946 in Leningrad (now St. Petersburg), Russia. He received his doctoral degree in Leningrad University in 1972 under the direction of V.A. Rokhlin, and in the same year he joined Syktyvkar University in northern Soviet Union. Eliashberg’s route passed through the refusenik years in Leningrad (1980-1987) where he had to do software engineering in order to feed his family, and where he was virtually cut off from normal mathematical life. In 1988 he emigrated to the United States and in 1989 became a Professor at Stanford University. He is a Member of U.S. National Academy of Sciences. For his contributions, Eliashberg has received a number of prestigious awards, including the Guggenheim Fellowship in 1995, the Oswald Veblen Prize in 2001, the Heinz Hopf Prize in 2013 and the Crafoord Prize in 2016. Eliashberg is currently the Herald L. and Caroline L. Ritch Professor at Stanford University.

Eliashberg developed a highly ingenious and very visual combinatorial technique that led him to the first manifestation of symplectic rigidity: the group of symplectomorphisms is closed in the group of all diffeomorphisms in the uniform topology. This fundamental result, proved in a different way also by Gromov and called nowadays the Eliashberg-Gromov theorem, is considered as one of the wonders and cornerstones of symplectic topology. In a series of papers (1989-1992), Eliashberg introduced and explored a fundamental dichotomy “tight vs overtwisted” contact structure that shaped the face of modern contact topology. Using this dichotomy, he gave the complete classification of contact structures on

the 3-sphere (1992). In these papers Eliashberg laid foundations of modern contact topology and introduced mathematical language which is widely used by researchers in this rapidly developing field.

In a seminal 2000 paper Eliashberg (with Givental and Hofer) pioneered foundations of symplectic field theory, a powerful, rich and notoriously sophisticated algebraic structure behind Gromov’s pseudo-holomorphic curves. It had a huge impact and became one of the most central and exciting directions in symplectic and contact topology. It has led to a significant progress on numerous areas including topology of Lagrangian submanifolds and geometry and dynamics of contact transformations, and it exhibited surprising links with classical and quantum integrable systems.

In recent years (2013-2015), Eliashberg found a number of astonishing appearances of homotopy principle in symplectic and contact topology leading him to a solution of a number of outstanding open problems and leading to a “mentality shift” in the field. Before these developments the consensus among experts was that the symplectic world is governed by rigidity coming from Gromov’s theory of pseudo-holomorphic curves or, equivalently, by Morse theory on the loop spaces of symplectic manifolds. The current impression based on Eliashberg’s discoveries is that rigidity is just a drop in the ocean of flexible phenomena.

Professor Yakov Eliashberg is awarded the Wolf Prize for his foundational work on symplectic and contact topology changing the face of these fields, and for his ground-breaking contribution to homotopy principles for partial differential relations and to topological foundations of multi-dimensional complex analysis.

 

Simon K. Donaldson

Wolf Prize Laureate in Mathematics 2020

Sir Simon Kirwan Donaldson

 

Affiliation at the time of the award:

Imperial College London, UK

Simons Center, Stony Brook, UK

 

Award citation:

“for their contributions to differential geometry and topology”

 

Prize Share:

Simon Donaldson 

Yakov Eliashberg

 

Sir Simon Kirwan Donaldson (born 1957, Cambridge, U.K.) is an English mathematician known for his work on the topology of smooth (differentiable) four-dimensional manifolds and Donaldson–Thomas theory.

Donaldson’s passion of youth was sailing. Through this, he became interested in the design of boats, and in turn in mathematics. Donaldson gained a BA degree in mathematics from Pembroke College, Cambridge in 1979, and in 1980 began postgraduate work at Worcester College, Oxford.

As a graduate student, Donaldson made a spectacular discovery on the nature or 4-dimensional geometry and topology which is considered one of the great events of 20th century mathematics. He showed there are phenomena in 4-dlmenslons which have no counterpart in any other dimension. This was totally unexpected, running against the perceived wisdom of the time.

Not only did Donaldson make this discovery but he also produced new tools with which to study it, involving deep new ideas in global nonlinear analysis, topology, and algebraic geometry.

After gaining his DPhil degree from Oxford University in 1983, Donaldson was appointed a Junior Research Fellow at All Souls College, Oxford, he spent the academic year 1983–84 at the Institute for Advanced Study in Princeton, and returned to Oxford as Wallis Professor of Mathematics in 1985. After spending one year visiting Stanford University, he moved to Imperial College London in 1998. Donaldson is currently a permanent member of the Simons Center for Geometry and Physics at Stony Brook University and a Professor in Pure Mathematics at Imperial College London.

Donaldson’s work is remarkable in its reversal of the usual direction of ideas from mathematics being applied to solve problems in physics.

A trademark of Donaldson’s work is to use geometric ideas in infinite dimensions, and deep non-linear analysis, to give new ways to solve partial differential equations (PDE). In this way he used the Yang-Mills equations, which has its origin in quantum field theory, to solve problems in pure mathematics (Kähler manifolds) and changed our understanding of symplectic manifolds. These are the phase spaces of classical mechanics, and he has shown that large parts of the powerful theory of algebraic geometry can be extended to them.

Applying physics to problems or pure mathematics was a stunning reversal of the usual interaction between the subjects and has helped develop a new unification of the subjects over the last 20 years, resulting in great progress in both. His use of moduli (or parameter) spaces of solutions of physical equations – and the interpretation of this technique as a form of quantum field theory – is now pervasive throughout many branches of modem mathematics and physics as a way to produce “Donaldson-type Invariants” of geometries of all types. In the last 5 years he has been making great progress with special geometries crucial to string theory in dimensions six (“Donaldson-Thomas theory”), seven and eight.

Professor Simon Donaldson is awarded the Wolf Prize for his leadership in geometry in the last 35 years. His work has been a unique combination of novel ideas in global non-linear analysis, topology, algebraic geometry, and theoretical physics, following his fundamental work on 4-manifolds and gauge theory. Especially remarkable is his recent work on symplectic and Kähler geometry.

 

 

 

Jean Francois le Gall

Wolf Prize Laureate in Mathematics 2019

Jean Francois le Gall

 

Affiliation at the time of the award:

Université Paris-Sud ,France

 

Award citation:

“for his deep and elegant work on stochastic processes”.

 

Prize Share:

Jean Francois le Gall

Gregory Lawler

 

Jean-François Le Gall (born in 1959, France) is a French mathematician working in areas of probability theory such as Brownian motion, Lévy processes, superprocesses and their connections with partial differential equations, the Brownian snake, random trees, branching processes, stochastic coalescence and random planar maps. He received his Ph.D. in 1982 under the supervision of Marc Yor. He is currently professor at the University of Paris-Sud in Orsay and is a senior member of the Institut universitaire de France.

Le Gall’s generation has a great impact on Probability or demonstrated more power and beauty in their arguments. He has been at the forefront of Probability since 1983, when he established what still are the best results on pathwise uniqueness for one-dimensional stochastic differential equations. His current groundbreaking discoveries on the Brownian map ensure he remains at the cutting edge of the field today. He has received the 1986 Rollo Davidson Prize the 2005 Grand Prix Sophie Germain, French Academy of Science, the 2005 Fermat prize in mathematics and was elected to French academy of sciences, December 2013.

Jean-François Le Gall has made several deep and elegant contributions to the theory of stochastic processes. His work on the fine properties of Brownian motions solved many difficult problems, such as the characterization of sets visited multiple times and the behavior of the volume of its neighborhood – the Brownian sausage. Le Gall made groundbreaking advances in the theory of branching processes, which arise in many applications. In particular, his introduction of the Brownian snake and his studies of its properties revolutionized the theory of super-processes – generalizations of Markov processes to an evolving cloud of dying and splitting particles. He then used some of these tools for achieving a spectacular breakthrough in the mathematical understanding of 2D quantum gravity. Le Gall established the convergence of uniform planar maps to a canonical random metric object, the Brownian map, and showed that it almost surely has Hausdorff dimension 4 and is homeomorphic to the 2-spher.

Gregory Lawler

Wolf Prize Laureate in Mathematics 2019

Gregory Lawler

 

Affiliation at the time of the award:

The University of Chicago, USA

 

Award citation:

“for his extensive and groundbreaking research on random paths and loops”.

 

Prize Share:

Gregory Lawler

Jean Francois le Gall

 

Gregory Francis Lawler (born in 1955) is an American mathematician working in probability theory and best known for his work since 2000 on the Schramm–Loewner evolution. Gregory Lawler received his Ph.D. from Princeton University in 1979 under the supervision of Edward Nelson. He was on the faculty of Duke University from 1979 to 2001, of Cornell University from 2001 to 2006, and since 2006 is at the University of Chicago. Lawler received the 2006 SIAM George Pólya Prize, in 2012 he became a fellow of the American Mathematical Society and Member, National Academy of Sciences in 2013.

Lawler has made trailblazing contributions to the development of probability theory. He obtained outstanding results regarding a number of properties of Brownian motion, such as cover times, intersection exponents and dimensions of various subsets. Studying random curves, Lawler introduced a now classical model, the Loop-Erased Random Walk (LERW), and established many of its properties. While simple to define, it turned out to be of a fundamental nature, and was shown to be related to uniform spanning trees and dimer tilings. This work formed much of the foundation for a great number of spectacular breakthroughs, which followed Oded Schramm’s introduction of the SLE curves. Lawler, Schramm and Werner calculated Brownian intersection exponents, proved Mandelbrot’s conjecture that the Brownian frontier has Hausdorff dimension 4/3 and established that the LERW has a conformally invariant scaling limit. These results, in turn, paved the way for further exciting progress by Lawler and others.

Vladimir Drinfeld

Wolf Prize Laureate in Mathematics 2018

Vladimir Drinfeld

 

Affiliation at the time of the award:

University of Chicago, USA

 

Award citation:

“for their ground-breaking work in algebraic geometry (a field that integrates abstract algebra with geometry), in mathematical physics and in presentation theory, a field which helps to understand complex algebraic structures”.

 

Prize Share:

Vladimir Drinfeld

Alexander Beilinson 

 

An “algebraic structure” is a set of objects, including the actions that can be performed on those objectas, that obey certain axioms. One of the roles of modern algebra is to research, in the most general and abstract way possible, the properties of various algebraic structures (including their objects), many of which are amazingly complicated.

Vladimir Drinfeld, born in Kharkov, Ukraine (1954), represented the USSR at the International Mathematical Olympiad at the age of 15 and won a gold medal. In the same year he also began his studies at the University of Moscow. Since the eighties he has been considered one of the world’s leading mathematicians. In 1990 he won the prestigious Fields Medal and in 2008 was elected to the National Academy of Sciences (USA). Drinfeld has contributed greatly to various branches of pure mathematics, mainly algebraic geometry, arithmetic geometry and the theory of representation – as well as mathematical physics. The mathematical objects named after him – the “Drinfeld Modules”, the “Drinfeld Chtoucas”, the “Drinfeld Upper Half Plane”, the “Drinfeld Associator”, and so many others that one of his endorsers jokingly said, “one could think that “Drinfeld” was an adjective, not the name of a person”.

In the seventies, Drinfeld began his work on the aforementioned “Langlands Program”, the ambitious program that aimed at unifying the fields of mathematics. This program was proposed by the American-Canadian mathematician Robert Langlands (winner of the Wolf Prize in 1996) and discovered for the first-time tight and direct links between different branches of mathematics. Numbers theory (the field based on arithmetic, “sums”), algebraic representation theory and another field called “automorphic forms”, (which is related to harmonic analysis and assists, for example, in the physical study of waves and frequencies). By means of a new geometrical object he developed, which is now called “Drinfeld Chtoucas”, Drinfeld succeeded in proving some of the connections that had been indicated by the Langlands Program. In the eighties he invented the concept of algebraic “Quantum Group”, which led to a profusion of developments and innovations not only in pure mathematics but also in mathematical physics (for example in statistical mechanics).

Drinfeld and Beilinson, together created a geometric model of algebraic theory that plays a key role in both field theory and physical string theory, thereby further strengthening the connections between abstract modern mathematics and physics. In 2004 they jointly published their work in a book that describes important algebraic structures used in quantum field theory, which is the theoretical basis for the particle physics of today. This publication has since become the basic reference book on this complex subject.

Alexander Beilinson

Wolf Prize Laureate in Mathematics 2018

Alexander Beilinson

 

Affiliation at the time of the award:

University of Chicago, USA

 

Award citation:

“for their ground-breaking work in algebraic geometry (a field that integrates abstract algebra with geometry), in mathematical physics and in presentation theory, a field which helps to understand complex algebraic structures”.

 

Prize Share:

Alexander Beilinson 

Vladimir Drinfeld

 

An “algebraic structure” is a set of objects, including the actions that can be performed on those objectas, that obey certain axioms. One of the roles of modern algebra is to research, in the most general and abstract way possible, the properties of various algebraic structures (including their objects), many of which are amazingly complicated

Alexander Beilinson was born in Moscow (1957), won the Ostrowski Prize (together with a colleague) in 1999, for extraordinary achievements in mathematics, and in 2017 he was elected to the National Academy of Sciences of the United States. His outstanding achievements include proofs of the Kashdan-Lustig and Jantzen conjectures, which play a key role in the representation theory, the development of important conjectures (“Beilinson’s Conjectures”) in algebraic geometry, and a significant contribution to the interface between geometry and mathematical physics. The joint work of Beilinson and Vladimir Drinfeld on the Langlands Program – a woven fabric of theorems and conjectures designed to link key areas of mathematics – has led to impressive progress in implementing the program in important areas of physics, such as quantum field theory and string theory.

Drinfeld and Beilinson, together created a geometric model of algebraic theory that plays a key role in both field theory and physical string theory, thereby further strengthening the connections between abstract modern mathematics and physics. In 2004 they jointly published their work in a book that describes important algebraic structures used in quantum field theory, which is the theoretical basis for the particle physics of today. This publication has since become the basic reference book on this complex subject.