Fields Medal

Not to be confused with Field's metal.

Fields Medal
The obverse of the Fields Medal
Awarded for	Outstanding contributions in mathematics attributed to young scientists
Presented by	International Mathematical Union
Reward(s)	CA$15,000
First awarded	1936 (1936)
Last awarded	2022 (2022)
Website	mathunion.org/imu-awards/fields-medal
The reverse of the medal

The Fields Medal is a prize awarded to two, three, or four mathematicians under 40 years of age at the International Congress of the International Mathematical Union (IMU), a meeting that takes place every four years. The name of the award honours the Canadian mathematician John Charles Fields.

The Fields Medal is regarded as one of the highest honors a mathematician can receive, and has been described as the Nobel Prize of Mathematics, although there are several major differences, including frequency of award, number of awards, age limits, monetary value, and award criteria. According to the annual Academic Excellence Survey by ARWU, the Fields Medal is consistently regarded as the top award in the field of mathematics worldwide, and in another reputation survey conducted by IREG in 2013–14, the Fields Medal came closely after the Abel Prize as the second most prestigious international award in mathematics.

The prize includes a monetary award which, since 2006, has been CA$15,000. Fields was instrumental in establishing the award, designing the medal himself, and funding the monetary component, though he died before it was established and his plan was overseen by John Lighton Synge.

The medal was first awarded in 1936 to Finnish mathematician Lars Ahlfors and American mathematician Jesse Douglas, and it has been awarded every four years since 1950. Its purpose is to give recognition and support to younger mathematical researchers who have made major contributions. In 2014, the Iranian mathematician Maryam Mirzakhani became the first female Fields Medalist. In total, 64 people have been awarded the Fields Medal.

The most recent group of Fields Medalists received their awards on 5 July 2022 in an online event which was live-streamed from Helsinki, Finland. It was originally meant to be held in Saint Petersburg, Russia, but was moved following the 2022 Russian invasion of Ukraine.

Conditions of the award

The Fields Medal has for a long time been regarded as the most prestigious award in the field of mathematics and is often described as the Nobel Prize of Mathematics. Unlike the Nobel Prize, the Fields Medal is only awarded every four years. The Fields Medal also has an age limit: a recipient must be under age 40 on 1 January of the year in which the medal is awarded. The under-40 rule is based on Fields's desire that "while it was in recognition of work already done, it was at the same time intended to be an encouragement for further achievement on the part of the recipients and a stimulus to renewed effort on the part of others." Moreover, an individual can only be awarded one Fields Medal; winners are ineligible to be awarded future medals.

First awarded in 1936, 64 people have won the medal as of 2022. With the exception of two PhD holders in physics (Edward Witten and Martin Hairer), only people with a PhD in mathematics have won the medal.

List of Fields medalists

In certain years, the Fields medalists have been officially cited for particular mathematical achievements, while in other years such specificities have not been given. However, in every year that the medal has been awarded, noted mathematicians have lectured at the International Congress of Mathematicians on each medalist's body of work. In the following table, official citations are quoted when possible (namely for the years 1958, 1998, and every year since 2006). For the other years through 1986, summaries of the ICM lectures, as written by Donald Albers, Gerald L. Alexanderson, and Constance Reid, are quoted. In the remaining years (1990, 1994, and 2002), part of the text of the ICM lecture itself has been quoted. The upcoming awarding of the Fields Medal at the 2026 International Congress of the International Mathematical Union is planned to take place in Philadelphia.

Year	ICM location	Medalists		Affiliation (when awarded)	Affiliation (current/last)	Reasons
1936	Oslo, Norway		Lars Ahlfors	University of Helsinki, Finland	Harvard University, US	"Awarded medal for research on covering surfaces related to Riemann surfaces of inverse functions of entire and meromorphic functions. Opened up new fields of analysis."
			Jesse Douglas	Massachusetts Institute of Technology, US	City College of New York, US	"Did important work on the Plateau problem which is concerned with finding minimal surfaces connecting and determined by some fixed boundary."
1950	Cambridge, US		Laurent Schwartz	University of Nancy, France	University of Paris VII, France	"Developed the theory of distributions, a new notion of generalized function motivated by the Dirac delta-function of theoretical physics."
			Atle Selberg	Institute for Advanced Study, US	Institute for Advanced Study, US	"Developed generalizations of the sieve methods of Viggo Brun; achieved major results on zeros of the Riemann zeta function; gave an elementary proof of the prime number theorem (with P. Erdős), with a generalization to prime numbers in an arbitrary arithmetic progression."
1954	Amsterdam, Netherlands		Kunihiko Kodaira	Princeton University, US, University of Tokyo, Japan and Institute for Advanced Study, US	University of Tokyo, Japan	"Achieved major results in the theory of harmonic integrals and numerous applications to Kählerian and more specifically to algebraic varieties. He demonstrated, by sheaf cohomology, that such varieties are Hodge manifolds."
			Jean-Pierre Serre	University of Nancy, France	Collège de France, France	"Achieved major results on the homotopy groups of spheres, especially in his use of the method of spectral sequences. Reformulated and extended some of the main results of complex variable theory in terms of sheaves."
1958	Edinburgh, UK		Klaus Roth	University College London, UK	Imperial College London, UK	"for solving a famous problem of number theory, namely, the determination of the exact exponent in the Thue-Siegel inequality"
			René Thom	University of Strasbourg, France	Institut des Hautes Études Scientifiques, France	"for creating the theory of 'Cobordisme' which has, within the few years of its existence, led to the most penetrating insight into the topology of differentiable manifolds."
1962	Stockholm, Sweden		Lars Hörmander	University of Stockholm, Sweden	Lund University, Sweden	"Worked in partial differential equations. Specifically, contributed to the general theory of linear differential operators. The questions go back to one of Hilbert's problems at the 1900 congress."
			John Milnor	Princeton University, US	Stony Brook University, US	"Proved that a 7-dimensional sphere can have several differential structures; this led to the creation of the field of differential topology."
1966	Moscow, USSR		Michael Atiyah	University of Oxford, UK	University of Edinburgh, UK	"Did joint work with Hirzebruch in K-theory; proved jointly with Singer the index theorem of elliptic operators on complex manifolds; worked in collaboration with Bott to prove a fixed point theorem related to the 'Lefschetz formula'."
			Paul Cohen	Stanford University, US	Stanford University, US	"Used technique called "forcing" to prove the independence in set theory of the axiom of choice and of the generalized continuum hypothesis. The latter problem was the first of Hilbert's problems of the 1900 Congress."
			Alexander Grothendieck	Institut des Hautes Études Scientifiques, France	Centre National de la Recherche Scientifique, France	"Built on work of Weil and Zariski and effected fundamental advances in algebraic geometry. He introduced the idea of K-theory (the Grothendieck groups and rings). Revolutionized homological algebra in his celebrated ‘Tôhoku paper’."
			Stephen Smale	University of California, Berkeley, US	City University of Hong Kong, Hong Kong	"Worked in differential topology where he proved the generalized Poincaré conjecture in dimension $n\geq 5$: Every closed, n-dimensional manifold homotopy-equivalent to the n-dimensional sphere is homeomorphic to it. Introduced the method of handle-bodies to solve this and related problems."
1970	Nice, France		Alan Baker	University of Cambridge, UK	Trinity College, Cambridge, UK	"Generalized the Gelfond-Schneider theorem (the solution to Hilbert's seventh problem). From this work he generated transcendental numbers not previously identified."
			Heisuke Hironaka	Harvard University, US	Kyoto University, Japan	"Generalized work of Zariski who had proved for dimension ≤ 3 the theorem concerning the resolution of singularities on an algebraic variety. Hironaka proved the results in any dimension."
			Sergei Novikov	Moscow State University, USSR	Steklov Mathematical Institute, Russia Moscow State University, Russia University of Maryland-College Park, US	"Made important advances in topology, the most well-known being his proof of the topological invariance of the Pontryagin classes of the differentiable manifold. His work included a study of the cohomology and homotopy of Thom spaces."
			John G. Thompson	University of Cambridge, UK	University of Cambridge, UK University of Florida, US	"Proved jointly with W. Feit that all non-cyclic finite simple groups have even order. The extension of this work by Thompson determined the minimal simple finite groups, that is, the simple finite groups whose proper subgroups are solvable."
1974	Vancouver, Canada		Enrico Bombieri	University of Pisa, Italy	Institute for Advanced Study, US	"Major contributions in the primes, in univalent functions and the local Bieberbach conjecture, in theory of functions of several complex variables, and in theory of partial differential equations and minimal surfaces – in particular, to the solution of Bernstein's problem in higher dimensions."
			David Mumford	Harvard University, US	Brown University, US	"Contributed to problems of the existence and structure of varieties of moduli, varieties whose points parametrize isomorphism classes of some type of geometric object. Also made several important contributions to the theory of algebraic surfaces."
1978	Helsinki, Finland		Pierre Deligne	Institut des Hautes Études Scientifiques, France	Institute for Advanced Study, US	"Gave solution of the three Weil conjectures concerning generalizations of the Riemann hypothesis to finite fields. His work did much to unify algebraic geometry and algebraic number theory."
			Charles Fefferman	Princeton University, US	Princeton University, US	"Contributed several innovations that revised the study of multidimensional complex analysis by finding correct generalizations of classical (low-dimensional) results."
			Grigory Margulis	Moscow State University, USSR	Yale University, US	"Provided innovative analysis of the structure of Lie groups. His work belongs to combinatorics, differential geometry, ergodic theory, dynamical systems, and Lie groups."
			Daniel Quillen	Massachusetts Institute of Technology, US	University of Oxford, UK	"The prime architect of the higher algebraic K-theory, a new tool that successfully employed geometric and topological methods and ideas to formulate and solve major problems in algebra, particularly ring theory and module theory."
1982	Warsaw, Poland		Alain Connes	Institut des Hautes Études Scientifiques, France	Institut des Hautes Études Scientifiques, France Collège de France, France Ohio State University, US	"Contributed to the theory of operator algebras, particularly the general classification and structure theorem of factors of type III, classification of automorphisms of the hyperfinite factor, classification of injective factors, and applications of the theory of C*-algebras to foliations and differential geometry in general."
			William Thurston	Princeton University, US	Cornell University, US	"Revolutionized study of topology in 2 and 3 dimensions, showing interplay between analysis, topology, and geometry. Contributed idea that a very large class of closed 3-manifolds carry a hyperbolic structure."
			Shing-Tung Yau	Institute for Advanced Study, US	Tsinghua University, China	"Made contributions in differential equations, also to the Calabi conjecture in algebraic geometry, to the positive mass conjecture of general relativity theory, and to real and complex Monge–Ampère equations."
1986	Berkeley, US		Simon Donaldson	University of Oxford, UK	Imperial College London, UK Stony Brook University, US	"Received medal primarily for his work on topology of four-manifolds, especially for showing that there is a differential structure on euclidian four-space which is different from the usual structure."
			Gerd Faltings	Princeton University, US	Max Planck Institute for Mathematics, Germany	"Using methods of arithmetic algebraic geometry, he received medal primarily for his proof of the Mordell Conjecture."
			Michael Freedman	University of California, San Diego, US	Microsoft Station Q, US	"Developed new methods for topological analysis of four-manifolds. One of his results is a proof of the four-dimensional Poincaré Conjecture."
1990	Kyoto, Japan		Vladimir Drinfeld	B Verkin Institute for Low Temperature Physics and Engineering, USSR	University of Chicago, US	"Drinfeld's main preoccupation in the last decade [are] Langlands' program and quantum groups. In both domains, Drinfeld's work constituted a decisive breakthrough and prompted a wealth of research."
			Vaughan Jones	University of California, Berkeley, US	University of California, Berkeley, US Vanderbilt University, US	"Jones discovered an astonishing relationship between von Neumann algebras and geometric topology. As a result, he found a new polynomial invariant for knots and links in 3-space."
			Shigefumi Mori	Kyoto University, Japan	Kyoto University, Japan	"The most profound and exciting development in algebraic geometry during the last decade or so was [...] Mori's Program in connection with the classification problems of algebraic varieties of dimension three." "Early in 1979, Mori brought to algebraic geometry a completely new excitement, that was his proof of Hartshorne's conjecture."
			Edward Witten	Institute for Advanced Study, US	Institute for Advanced Study, US	"Time and again he has surprised the mathematical community by a brilliant application of physical insight leading to new and deep mathematical theorems."
1994	Zürich, Switzerland		Jean Bourgain	Institut des Hautes Études Scientifiques, France	Institute for Advanced Study, US	"Bourgain's work touches on several central topics of mathematical analysis: the geometry of Banach spaces, convexity in high dimensions, harmonic analysis, ergodic theory, and finally, nonlinear partial differential equations from mathematical physics."
			Pierre-Louis Lions	University of Paris 9, France	Collège de France, France École polytechnique, France	"His contributions cover a variety of areas, from probability theory to partial differential equations (PDEs). Within the PDE area he has done several beautiful things in nonlinear equations. The choice of his problems have always been motivated by applications."
			Jean-Christophe Yoccoz	Paris-Sud 11 University, France	Collège de France, France	"Yoccoz obtained a very enlightening proof of Bruno's theorem, and he was able to prove the converse [...] Palis and Yoccoz obtained a complete system of C∞ conjugation invariants for Morse-Smale diffeomorphisms."
			Efim Zelmanov	University of Wisconsin-Madison University of Chicago, US	Steklov Mathematical Institute, Russia, University of California, San Diego, US	"For the solution of the restricted Burnside problem."
1998	Berlin, Germany		Richard Borcherds	University of California, Berkeley, US University of Cambridge, UK	University of California, Berkeley, US	"For his contributions to algebra, the theory of automorphic forms, and mathematical physics, including the introduction of vertex algebras and Borcherds' Lie algebras, the proof of the Conway–Norton moonshine conjecture and the discovery of a new class of automorphic infinite products."
			Timothy Gowers	University of Cambridge, UK	University of Cambridge, UK	"For his contributions to functional analysis and combinatorics, developing a new vision of infinite-dimensional geometry, including the solution of two of Banach's problems and the discovery of the so called Gowers' dichotomy: every infinite dimensional Banach space contains either a subspace with many symmetries (technically, with an unconditional basis) or a subspace every operator on which is Fredholm of index zero."
			Maxim Kontsevich	Institut des Hautes Études Scientifiques, France Rutgers University, US	Institut des Hautes Études Scientifiques, France Rutgers University, US	"For his contributions to algebraic geometry, topology, and mathematical physics, including the proof of Witten's conjecture of intersection numbers in moduli spaces of stable curves, construction of the universal Vassiliev invariant of knots, and formal quantization of Poisson manifolds."
			Curtis T. McMullen	Harvard University, US	Harvard University, US	"For his contributions to the theory of holomorphic dynamics and geometrization of three-manifolds, including proofs of Bers' conjecture on the density of cusp points in the boundary of the Teichmüller space, and Kra's theta-function conjecture."
2002	Beijing, China		Laurent Lafforgue	Institut des Hautes Études Scientifiques, France	Institut des Hautes Études Scientifiques, France	"Laurent Lafforgue has been awarded the Fields Medal for his proof of the Langlands correspondence for the full linear groups GLr (r≥1) over function fields of positive characteristic."
			Vladimir Voevodsky	Institute for Advanced Study, US	Institute for Advanced Study, US	"He defined and developed motivic cohomology and the A1-homotopy theory, provided a framework for describing many new cohomology theories for algebraic varieties; he proved the Milnor conjectures on the K-theory of fields."
2006	Madrid, Spain		Andrei Okounkov	Princeton University, US	Columbia University, US University of California, Berkeley, US	"For his contributions bridging probability, representation theory and algebraic geometry."
			Grigori Perelman (declined)	None	St. Petersburg Department of Steklov Institute of Mathematics of Russian Academy of Sciences, Russia	"For his contributions to geometry and his revolutionary insights into the analytical and geometric structure of the Ricci flow."
			Terence Tao	University of California, Los Angeles, US	University of California, Los Angeles, US	"For his contributions to partial differential equations, combinatorics, harmonic analysis and additive number theory."
			Wendelin Werner	Paris-Sud 11 University, France	ETH Zurich, Switzerland	"For his contributions to the development of stochastic Loewner evolution, the geometry of two-dimensional Brownian motion, and conformal field theory."
2010	Hyderabad, India		Elon Lindenstrauss	Hebrew University of Jerusalem, Israel Princeton University, US	Hebrew University of Jerusalem, Israel	"For his results on measure rigidity in ergodic theory, and their applications to number theory."
			Ngô Bảo Châu	Paris-Sud 11 University, France Institute for Advanced Study, US	University of Chicago, US Institute for Advanced Study, US	"For his proof of the Fundamental Lemma in the theory of automorphic forms through the introduction of new algebro-geometric methods."
			Stanislav Smirnov	University of Geneva, Switzerland	University of Geneva, Switzerland St. Petersburg State University, Russia	"For the proof of conformal invariance of percolation and the planar Ising model in statistical physics."
			Cédric Villani	École Normale Supérieure de Lyon, France Institut Henri Poincaré, France	Lyon University, France Institut Henri Poincaré, France	"For his proofs of nonlinear Landau damping and convergence to equilibrium for the Boltzmann equation."
2014	Seoul, South Korea		Artur Avila	University of Paris VII, France CNRS, France Instituto Nacional de Matemática Pura e Aplicada, Brazil	University of Zurich, Switzerland Instituto Nacional de Matemática Pura e Aplicada, Brazil	"For his profound contributions to dynamical systems theory, which have changed the face of the field, using the powerful idea of renormalization as a unifying principle."
			Manjul Bhargava	Princeton University, US	Princeton University, US	"For developing powerful new methods in the geometry of numbers, which he applied to count rings of small rank and to bound the average rank of elliptic curves."
			Martin Hairer	University of Warwick, UK	Imperial College London, UK	"For his outstanding contributions to the theory of stochastic partial differential equations, and in particular for the creation of a theory of regularity structures for such equations."
			Maryam Mirzakhani	Stanford University, US	Stanford University, US	"For her outstanding contributions to the dynamics and geometry of Riemann surfaces and their moduli spaces."
2018	Rio de Janeiro, Brazil		Caucher Birkar	University of Cambridge, UK	University of Cambridge, UK	"For the proof of the boundedness of Fano varieties and for contributions to the minimal model program."
			Alessio Figalli	Swiss Federal Institute of Technology Zurich, Switzerland	Swiss Federal Institute of Technology Zurich, Switzerland	"For contributions to the theory of optimal transport and its applications in partial differential equations, metric geometry and probability."
			Peter Scholze	University of Bonn, Germany	University of Bonn, Germany	"For having transformed arithmetic algebraic geometry over p-adic fields."
			Akshay Venkatesh	Stanford University, US	Institute for Advanced Study, US	"For his synthesis of analytic number theory, homogeneous dynamics, topology, and representation theory, which has resolved long-standing problems in areas such as the equidistribution of arithmetic objects."
2022	Helsinki, Finland		Hugo Duminil-Copin	Institut des Hautes Études Scientifiques, France University of Geneva, Switzerland	Institut des Hautes Études Scientifiques, France University of Geneva, Switzerland	"For solving longstanding problems in the probabilistic theory of phase transitions in statistical physics, especially in dimensions three and four."
			June Huh	Princeton University, US	Princeton University, US	"For bringing the ideas of Hodge theory to combinatorics, the proof of the Dowling–Wilson conjecture for geometric lattices, the proof of the Heron–Rota–Welsh conjecture for matroids, the development of the theory of Lorentzian polynomials, and the proof of the strong Mason conjecture."
			James Maynard	University of Oxford, UK	University of Oxford, UK	"For contributions to analytic number theory, which have led to major advances in the understanding of the structure of prime numbers and in Diophantine approximation."
			Maryna Viazovska	École Polytechnique Fédérale de Lausanne, Switzerland	École Polytechnique Fédérale de Lausanne, Switzerland	"For the proof that the $E_{8}$ lattice provides the densest packing of identical spheres in 8 dimensions, and further contributions to related extremal problems and interpolation problems in Fourier analysis."

Landmarks

The medal was first awarded in 1936 to the Finnish mathematician Lars Ahlfors and the American mathematician Jesse Douglas, and it has been awarded every four years since 1950. Its purpose is to give recognition and support to younger mathematical researchers who have made major contributions.

In 1954, Jean-Pierre Serre became the youngest winner of the Fields Medal, at 27. He retains this distinction.

In 1966, Alexander Grothendieck boycotted the ICM, held in Moscow, to protest Soviet military actions taking place in Eastern Europe. Léon Motchane, founder and director of the Institut des Hautes Études Scientifiques, attended and accepted Grothendieck's Fields Medal on his behalf.

In 1970, Sergei Novikov, because of restrictions placed on him by the Soviet government, was unable to travel to the congress in Nice to receive his medal.

In 1978, Grigory Margulis, because of restrictions placed on him by the Soviet government, was unable to travel to the congress in Helsinki to receive his medal. The award was accepted on his behalf by Jacques Tits, who said in his address: "I cannot but express my deep disappointment—no doubt shared by many people here—in the absence of Margulis from this ceremony. In view of the symbolic meaning of this city of Helsinki, I had indeed grounds to hope that I would have a chance at last to meet a mathematician whom I know only through his work and for whom I have the greatest respect and admiration."

In 1982, the congress was due to be held in Warsaw but had to be rescheduled to the next year, because of martial law introduced in Poland on 13 December 1981. The awards were announced at the ninth General Assembly of the IMU earlier in the year and awarded at the 1983 Warsaw congress.

In 1990, Edward Witten became the first physicist to win the award.

In 1998, at the ICM, Andrew Wiles was presented by the chair of the Fields Medal Committee, Yuri I. Manin, with the first-ever IMU silver plaque in recognition of his proof of Fermat's Last Theorem. Don Zagier referred to the plaque as a "quantized Fields Medal". Accounts of this award frequently make reference that at the time of the award Wiles was over the age limit for the Fields medal. Although Wiles was slightly over the age limit in 1994, he was thought to be a favorite to win the medal; however, a gap (later resolved by Taylor and Wiles) in the proof was found in 1993.

In 2006, Grigori Perelman, who proved the Poincaré conjecture, refused his Fields Medal and did not attend the congress.

In 2014, Maryam Mirzakhani became the first Iranian as well as the first woman to win the Fields Medal, and Artur Avila became the first South American and Manjul Bhargava became the first person of Indian origin to do so.

In 2022, Maryna Viazovska became the first Ukrainian to win the Fields Medal, and June Huh became the first person of Korean ancestry to do so.

Medal

The reverse of the Fields Medal

The medal was designed by Canadian sculptor R. Tait McKenzie. It is made of 14KT gold, has a diameter of 63.5mm, and weighs 169g.

• On the obverse is Archimedes and a quote attributed to 1st century AD poet Manilius, which reads in Latin: Transire suum pectus mundoque potiri ("To surpass one's understanding and master the world"). The year number 1933 is written in Roman numerals and contains an error (MCNXXXIII rather than MCMXXXIII). In capital Greek letters the word Ἀρχιμηδους, or "of Archimedes," is inscribed.
• On the reverse is the inscription:

Congregati
ex toto orbe
mathematici
ob scripta insignia
tribuere

Translation: "Mathematicians gathered from the entire world have awarded [understood but not written: 'this prize'] for outstanding writings."

In the background, there is the representation of Archimedes' tomb, with the carving illustrating his theorem On the Sphere and Cylinder, behind an olive branch. (This is the mathematical result of which Archimedes was reportedly most proud: Given a sphere and a circumscribed cylinder of the same height and diameter, the ratio between their volumes is equal to 2⁄3.)

The rim bears the name of the prizewinner.

Female recipients

The Fields Medal has had two female recipients, Maryam Mirzakhani from Iran in 2014, and Maryna Viazovska from Ukraine in 2022.

In popular culture

The Fields Medal gained some recognition in popular culture due to references in the 1997 film, Good Will Hunting. In the movie, Gerald Lambeau (Stellan Skarsgård) is an MIT professor who won the award prior to the events of the story. Throughout the film, references made to the award are meant to convey its prestige in the field.

See also

• Abel Prize
• Kyoto Prize
• List of prizes known as the Nobel or the highest honors of a field
• List of mathematics awards
• Nevanlinna Prize
• Rolf Schock Prizes
• Turing Award
• Wolf Prize in Mathematics

Further reading

• McKinnon Riehm, Elaine; Hoffman, Frances (2011). Turbulent Times in Mathematics: The Life of J.C. Fields and the History of the Fields Medal. Providence, RI: American Mathematical Society. ISBN 978-0-8218-6914-7.
• Monastyrsky, Michael (1998). Modern Mathematics in the Light of the Fields Medal. Wellesley, MA: A. K. Peters. ISBN 1-56881-083-0.
• Tropp, Henry S. (1976). "The Origins and History of the Fields Medal". Historia Mathematica. 3 (2): 167–181. doi:10.1016/0315-0860(76)90033-1..

External links

• Official website
• Overview at britannica.com