Mark William Gross FRS (born 30 November 1965) is an American mathematician, specializing in differential geometry , algebraic geometry , and mirror symmetry .
80-623: Mathematics award Clay Research Award Awarded for Major breakthroughs in mathematical research Presented by Clay Mathematics Institute First awarded 1999 Last awarded 2024 Website www .claymath .org /research The Clay Research Award is an annual award given by the Oxford-based Clay Mathematics Institute to mathematicians to recognize their achievements in mathematical research. The following mathematicians have received
160-591: A set whose elements are unspecified, of operations acting on the elements of the set, and rules that these operations must follow. The scope of algebra thus grew to include the study of algebraic structures. This object of algebra was called modern algebra or abstract algebra , as established by the influence and works of Emmy Noether . Some types of algebraic structures have useful and often fundamental properties, in many areas of mathematics. Their study became autonomous parts of algebra, and include: The study of types of algebraic structures as mathematical objects
240-614: A foundation for all mathematics). Mathematics involves the description and manipulation of abstract objects that consist of either abstractions from nature or—in modern mathematics—purely abstract entities that are stipulated to have certain properties, called axioms . Mathematics uses pure reason to prove properties of objects, a proof consisting of a succession of applications of deductive rules to already established results. These results include previously proved theorems , axioms, and—in case of abstraction from nature—some basic properties that are considered true starting points of
320-637: A fruitful interaction between mathematics and science , to the benefit of both. Mathematical discoveries continue to be made to this very day. According to Mikhail B. Sevryuk, in the January ;2006 issue of the Bulletin of the American Mathematical Society , "The number of papers and books included in the Mathematical Reviews (MR) database since 1940 (the first year of operation of MR)
400-711: A full professor. He was a visiting professor at the University of Warwick in the academic year 2002–2003. Since 2013, he has been a professor at the University of Cambridge and since 2016, a Fellow of King's College, Cambridge . Gross works on complex geometry , algebraic geometry, and mirror symmetry. Gross and Bernd Siebert jointly developed a program (known as the Gross–Siebert Program) for studying mirror symmetry within algebraic geometry. The Gross–Siebert program builds on an earlier, differential-geometric, proposal of Strominger , Yau , and Zaslow , in which
480-479: A lifetime of achievement, especially for pointing the way to unify apparently disparate fields of mathematics and to discover their elegant simplicity through links with the physical world" "For establishing the existence of the scaling limit of two-dimensional percolation , and for verifying John Cardy 's conjectured relation" 2000 Alain Connes Laurent Lafforgue "For revolutionizing
560-404: A mathematical problem. In turn, the axiomatic method allows for the study of various geometries obtained either by changing the axioms or by considering properties that do not change under specific transformations of the space . Today's subareas of geometry include: Algebra is the art of manipulating equations and formulas. Diophantus (3rd century) and al-Khwarizmi (9th century) were
640-422: A mathematical statement that is taken to be true without need of proof. If a mathematical statement has yet to be proven (or disproven), it is termed a conjecture . Through a series of rigorous arguments employing deductive reasoning , a statement that is proven to be true becomes a theorem. A specialized theorem that is mainly used to prove another theorem is called a lemma . A proven instance that forms part of
720-402: A more general finding is termed a corollary . Numerous technical terms used in mathematics are neologisms , such as polynomial and homeomorphism . Other technical terms are words of the common language that are used in an accurate meaning that may differ slightly from their common meaning. For example, in mathematics, " or " means "one, the other or both", while, in common language, it
800-628: A novel geometric combinatorial method to study doubling properties of solutions to elliptic eigenvalue problems." "In recognition of their groundbreaking and conceptually novel work on the geometry of the Gaussian free field and its application to the solution of open problems in the theory of two-dimensional random structures." "In recognition of her groundbreaking work on sphere-packing problems in eight and twenty-four dimensions." 2016 Mark Gross and Bernd Siebert Geordie Williamson "In recognition of their groundbreaking contributions to
880-535: A population mean with a given level of confidence. Because of its use of optimization , the mathematical theory of statistics overlaps with other decision sciences , such as operations research , control theory , and mathematical economics . Computational mathematics is the study of mathematical problems that are typically too large for human, numerical capacity. Numerical analysis studies methods for problems in analysis using functional analysis and approximation theory ; numerical analysis broadly includes
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#1732858062249960-411: A separate branch of mathematics until the seventeenth century. At the end of the 19th century, the foundational crisis in mathematics and the resulting systematization of the axiomatic method led to an explosion of new areas of mathematics. The 2020 Mathematics Subject Classification contains no less than sixty-three first-level areas. Some of these areas correspond to the older division, as
1040-424: A single unknown , which were called algebraic equations (a term still in use, although it may be ambiguous). During the 19th century, mathematicians began to use variables to represent things other than numbers (such as matrices , modular integers , and geometric transformations ), on which generalizations of arithmetic operations are often valid. The concept of algebraic structure addresses this, consisting of
1120-418: A statistical action, such as using a procedure in, for example, parameter estimation , hypothesis testing , and selecting the best . In these traditional areas of mathematical statistics , a statistical-decision problem is formulated by minimizing an objective function , like expected loss or cost , under specific constraints. For example, designing a survey often involves minimizing the cost of estimating
1200-596: A thesis on the surfaces in the four-dimensional Grassmannian . From 1990 to 1993 he was an assistant professor at the University of Michigan and spent the academic year 1992–1993 on leave as a postdoctoral researcher at the Mathematical Sciences Research Institute (MSRI) in Berkeley. He was at Cornell University in 1993–1997 an assistant professor and in 1997–2001 an associate professor and then at University of California, San Diego in 2001–2013
1280-477: A wide expansion of mathematical logic, with subareas such as model theory (modeling some logical theories inside other theories), proof theory , type theory , computability theory and computational complexity theory . Although these aspects of mathematical logic were introduced before the rise of computers , their use in compiler design, formal verification , program analysis , proof assistants and other aspects of computer science , contributed in turn to
1360-703: Is Fermat's Last Theorem . This conjecture was stated in 1637 by Pierre de Fermat, but it was proved only in 1994 by Andrew Wiles , who used tools including scheme theory from algebraic geometry , category theory , and homological algebra . Another example is Goldbach's conjecture , which asserts that every even integer greater than 2 is the sum of two prime numbers . Stated in 1742 by Christian Goldbach , it remains unproven despite considerable effort. Number theory includes several subareas, including analytic number theory , algebraic number theory , geometry of numbers (method oriented), diophantine equations , and transcendence theory (problem oriented). Geometry
1440-513: Is flat " and "a field is always a ring ". Mark Gross (mathematician) Mark William Gross was born on 30 November 1965 in Ithaca, New York , to Leonard Gross and Grazyna Gross. From 1982, he studied at Cornell University , graduating with a bachelor's degree in 1984. He gained a PhD in 1990 from the University of California, Berkeley , for research supervised by Robin Hartshorne with
1520-403: Is commonly used for advanced parts. Analysis is further subdivided into real analysis , where variables represent real numbers , and complex analysis , where variables represent complex numbers . Analysis includes many subareas shared by other areas of mathematics which include: Discrete mathematics, broadly speaking, is the study of individual, countable mathematical objects. An example
1600-509: Is defined by the set of all similar objects and the properties that these objects must have. For example, in Peano arithmetic , the natural numbers are defined by "zero is a number", "each number has a unique successor", "each number but zero has a unique predecessor", and some rules of reasoning. This mathematical abstraction from reality is embodied in the modern philosophy of formalism , as founded by David Hilbert around 1910. The "nature" of
1680-579: Is different from Wikidata Webarchive template wayback links Mathematics Mathematics is a field of study that discovers and organizes methods, theories and theorems that are developed and proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), analysis (the study of continuous changes), and set theory (presently used as
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#17328580622491760-407: Is either ambiguous or means "one or the other but not both" (in mathematics, the latter is called " exclusive or "). Finally, many mathematical terms are common words that are used with a completely different meaning. This may lead to sentences that are correct and true mathematical assertions, but appear to be nonsense to people who do not have the required background. For example, "every free module
1840-487: Is in Babylonian mathematics that elementary arithmetic ( addition , subtraction , multiplication , and division ) first appear in the archaeological record. The Babylonians also possessed a place-value system and used a sexagesimal numeral system which is still in use today for measuring angles and time. In the 6th century BC, Greek mathematics began to emerge as a distinct discipline and some Ancient Greeks such as
1920-586: Is mostly used for numerical calculations . Number theory dates back to ancient Babylon and probably China . Two prominent early number theorists were Euclid of ancient Greece and Diophantus of Alexandria. The modern study of number theory in its abstract form is largely attributed to Pierre de Fermat and Leonhard Euler . The field came to full fruition with the contributions of Adrien-Marie Legendre and Carl Friedrich Gauss . Many easily stated number problems have solutions that require sophisticated methods, often from across mathematics. A prominent example
2000-404: Is not sufficient to verify by measurement that, say, two lengths are equal; their equality must be proven via reasoning from previously accepted results ( theorems ) and a few basic statements. The basic statements are not subject to proof because they are self-evident ( postulates ), or are part of the definition of the subject of study ( axioms ). This principle, foundational for all mathematics,
2080-1192: Is now more than 1.9 million, and more than 75 thousand items are added to the database each year. The overwhelming majority of works in this ocean contain new mathematical theorems and their proofs." Mathematical notation is widely used in science and engineering for representing complex concepts and properties in a concise, unambiguous, and accurate way. This notation consists of symbols used for representing operations , unspecified numbers, relations and any other mathematical objects, and then assembling them into expressions and formulas. More precisely, numbers and other mathematical objects are represented by symbols called variables, which are generally Latin or Greek letters, and often include subscripts . Operation and relations are generally represented by specific symbols or glyphs , such as + ( plus ), × ( multiplication ), ∫ {\textstyle \int } ( integral ), = ( equal ), and < ( less than ). All these symbols are generally grouped according to specific rules to form expressions and formulas. Normally, expressions and formulas do not appear alone, but are included in sentences of
2160-547: Is often held to be Archimedes ( c. 287 – c. 212 BC ) of Syracuse . He developed formulas for calculating the surface area and volume of solids of revolution and used the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series , in a manner not too dissimilar from modern calculus. Other notable achievements of Greek mathematics are conic sections ( Apollonius of Perga , 3rd century BC), trigonometry ( Hipparchus of Nicaea , 2nd century BC), and
2240-433: Is one of the oldest branches of mathematics. It started with empirical recipes concerning shapes, such as lines , angles and circles , which were developed mainly for the needs of surveying and architecture , but has since blossomed out into many other subfields. A fundamental innovation was the ancient Greeks' introduction of the concept of proofs , which require that every assertion must be proved . For example, it
2320-554: Is sometimes mistranslated as a condemnation of mathematicians. The apparent plural form in English goes back to the Latin neuter plural mathematica ( Cicero ), based on the Greek plural ta mathēmatiká ( τὰ μαθηματικά ) and means roughly "all things mathematical", although it is plausible that English borrowed only the adjective mathematic(al) and formed the noun mathematics anew, after
2400-418: Is the purpose of universal algebra and category theory . The latter applies to every mathematical structure (not only algebraic ones). At its origin, it was introduced, together with homological algebra for allowing the algebraic study of non-algebraic objects such as topological spaces ; this particular area of application is called algebraic topology . Calculus, formerly called infinitesimal calculus,
2480-405: Is the set of all integers. Because the objects of study here are discrete, the methods of calculus and mathematical analysis do not directly apply. Algorithms —especially their implementation and computational complexity —play a major role in discrete mathematics. The four color theorem and optimal sphere packing were two major problems of discrete mathematics solved in the second half of
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2560-508: Is true regarding number theory (the modern name for higher arithmetic ) and geometry. Several other first-level areas have "geometry" in their names or are otherwise commonly considered part of geometry. Algebra and calculus do not appear as first-level areas but are respectively split into several first-level areas. Other first-level areas emerged during the 20th century or had not previously been considered as mathematics, such as mathematical logic and foundations . Number theory began with
2640-544: The André-Oort Conjecture in the case of products of modular curves " 2010 not awarded 2009 Jean-Loup Waldspurger Ian Agol , Danny Calegari and David Gabai "For his work in p-adic harmonic analysis , particularly his contributions to the transfer conjecture and the fundamental lemma " "For their solutions of the Marden Tameness Conjecture , and, by implication through
2720-574: The Babylonians and Egyptians began using arithmetic, algebra, and geometry for taxation and other financial calculations, for building and construction, and for astronomy. The oldest mathematical texts from Mesopotamia and Egypt are from 2000 to 1800 BC. Many early texts mention Pythagorean triples and so, by inference, the Pythagorean theorem seems to be the most ancient and widespread mathematical concept after basic arithmetic and geometry. It
2800-527: The Calabi–Yau manifold is fibred by special Lagrangian tori, and the mirror by dual tori. The program's central idea is to translate this into an algebro-geometric construction in an appropriate limit, involving combinatorial data associated with a degenerating family of Calabi–Yau manifolds. It draws on many areas of geometry, analysis and combinatorics and has made a deep impact on fields such as tropical and non-archimedean geometry , logarithmic geometry ,
2880-753: The Golden Age of Islam , especially during the 9th and 10th centuries, mathematics saw many important innovations building on Greek mathematics. The most notable achievement of Islamic mathematics was the development of algebra . Other achievements of the Islamic period include advances in spherical trigonometry and the addition of the decimal point to the Arabic numeral system. Many notable mathematicians from this period were Persian, such as Al-Khwarizmi , Omar Khayyam and Sharaf al-Dīn al-Ṭūsī . The Greek and Arabic mathematical texts were in turn translated to Latin during
2960-505: The Pythagoreans appeared to have considered it a subject in its own right. Around 300 BC, Euclid organized mathematical knowledge by way of postulates and first principles, which evolved into the axiomatic method that is used in mathematics today, consisting of definition, axiom, theorem, and proof. His book, Elements , is widely considered the most successful and influential textbook of all time. The greatest mathematician of antiquity
3040-524: The Renaissance , mathematics was divided into two main areas: arithmetic , regarding the manipulation of numbers, and geometry , regarding the study of shapes. Some types of pseudoscience , such as numerology and astrology , were not then clearly distinguished from mathematics. During the Renaissance, two more areas appeared. Mathematical notation led to algebra which, roughly speaking, consists of
3120-609: The Ricci Flow Equation and its development into one of the most powerful tools of geometric analysis " "For his ground-breaking work in analysis, notably his optimal restriction theorems in Fourier analysis , his work on the wave map equation , his global existence theorems for KdV type equations , as well as significant work in quite distant areas of mathematics" 2002 Oded Schramm Manindra Agrawal "For his work in combining analytic power with geometric insight in
3200-580: The Wayback Machine 2019 Clay Research Awards 2021 Clay Research Award 2022 Clay Research Award 2023 Clay Research Award Retrieved from " https://en.wikipedia.org/w/index.php?title=Clay_Research_Award&oldid=1222208922 " Categories : Mathematics awards Awards established in 1999 Research awards 1999 establishments in England Hidden categories: Articles with short description Short description
3280-446: The controversy over Cantor's set theory . In the same period, various areas of mathematics concluded the former intuitive definitions of the basic mathematical objects were insufficient for ensuring mathematical rigour . This became the foundational crisis of mathematics. It was eventually solved in mainstream mathematics by systematizing the axiomatic method inside a formalized set theory . Roughly speaking, each mathematical object
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3360-400: The 17th century, when René Descartes introduced what is now called Cartesian coordinates . This constituted a major change of paradigm : Instead of defining real numbers as lengths of line segments (see number line ), it allowed the representation of points using their coordinates , which are numbers. Algebra (and later, calculus) can thus be used to solve geometrical problems. Geometry
3440-405: The 19th century, mathematicians discovered non-Euclidean geometries , which do not follow the parallel postulate . By questioning that postulate's truth, this discovery has been viewed as joining Russell's paradox in revealing the foundational crisis of mathematics . This aspect of the crisis was solved by systematizing the axiomatic method, and adopting that the truth of the chosen axioms is not
3520-532: The 20th century. The P versus NP problem , which remains open to this day, is also important for discrete mathematics, since its solution would potentially impact a large number of computationally difficult problems. Discrete mathematics includes: The two subjects of mathematical logic and set theory have belonged to mathematics since the end of the 19th century. Before this period, sets were not considered to be mathematical objects, and logic , although used for mathematical proofs, belonged to philosophy and
3600-456: The MNOP conjecture that he formulated with Maulik, Okounkov and Nekrasov" 2012 Jeremy Kahn and Vladimir Markovic "For their work in hyperbolic geometry " 2011 Yves Benoist and Jean-François Quint Jonathan Pila "For their spectacular work on stationary measures and orbit closures for actions of non-abelian groups on homogeneous spaces " "For his resolution of
3680-620: The Middle Ages and made available in Europe. During the early modern period , mathematics began to develop at an accelerating pace in Western Europe , with innovations that revolutionized mathematics, such as the introduction of variables and symbolic notation by François Viète (1540–1603), the introduction of logarithms by John Napier in 1614, which greatly simplified numerical calculations, especially for astronomy and marine navigation ,
3760-494: The average size of ideal class groups " "For his complete resolution of a conjecture made by F. Treves and L. Nirenberg in 1970" 2004 Ben Green Gérard Laumon and Ngô Bảo Châu "For his joint work with Terry Tao on arithmetic progressions of prime numbers " "For their proof of the Fundamental Lemma for unitary groups " 2003 Richard S. Hamilton Terence Tao "For his discovery of
3840-406: The award: Year Winner Citation 2024 Paul Nelson James Newton and Jack Thorne "in recognition of his groundbreaking contributions to the analytic theory of automorphic forms. His work has resulted in the first convexity-breaking bounds for a large class of L-functions on the critical line (including all the standard ones of GL(n))." "for their remarkable proof of
3920-574: The beginnings of algebra (Diophantus, 3rd century AD). The Hindu–Arabic numeral system and the rules for the use of its operations, in use throughout the world today, evolved over the course of the first millennium AD in India and were transmitted to the Western world via Islamic mathematics . Other notable developments of Indian mathematics include the modern definition and approximation of sine and cosine , and an early form of infinite series . During
4000-680: The calculation of Gromov–Witten invariants , the theory of cluster algebras and combinatorial representation theory . Gross was an Invited Speaker, jointly with Siebert, with talk Local mirror symmetry in the tropics at the International Congress of Mathematicians in Seoul 2014. In 2016 Gross and Siebert jointly received the Clay Research Award . Gross was elected a Fellow of the Royal Society in 2017. “All text published under
4080-605: The compressible Euler and Navier-Stokes equations." 2022 Søren Galatius and Oscar Randal-Williams John Pardon "for their profound contributions to the understanding of high dimensional manifolds and their diffeomorphism groups; they have transformed and reinvigorated the subject." "in recognition of his wide-ranging and transformative work in geometry and topology, particularly his groundbreaking achievements in symplectic topology." 2021 Bhargav Bhatt "For his groundbreaking achievements in commutative algebra, arithmetic algebraic geometry, and topology in
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#17328580622494160-503: The concept of a proof and its associated mathematical rigour first appeared in Greek mathematics , most notably in Euclid 's Elements . Since its beginning, mathematics was primarily divided into geometry and arithmetic (the manipulation of natural numbers and fractions ), until the 16th and 17th centuries, when algebra and infinitesimal calculus were introduced as new fields. Since then,
4240-399: The current language, where expressions play the role of noun phrases and formulas play the role of clauses . Mathematics has developed a rich terminology covering a broad range of fields that study the properties of various abstract, idealized objects and how they interact. It is based on rigorous definitions that provide a standard foundation for communication. An axiom or postulate is
4320-553: The derived expression mathēmatikḗ tékhnē ( μαθηματικὴ τέχνη ), meaning ' mathematical science ' . It entered the English language during the Late Middle English period through French and Latin. Similarly, one of the two main schools of thought in Pythagoreanism was known as the mathēmatikoi (μαθηματικοί)—which at the time meant "learners" rather than "mathematicians" in the modern sense. The Pythagoreans were likely
4400-461: The existence of the symmetric power functorial lift for Hilbert modular forms." 2023 Frank Merle , Pierre Raphaël , Igor Rodnianski and Jérémie Szeftel "for their groundbreaking advances in the understanding of singular solutions to the fundamental equations of fluids dynamics, including their construction of smooth self-similar solutions for the compressible Euler equation, and families of finite-energy blow-up solutions for both
4480-428: The expansion of these logical theories. The field of statistics is a mathematical application that is employed for the collection and processing of data samples, using procedures based on mathematical methods especially probability theory . Statisticians generate data with random sampling or randomized experiments . Statistical theory studies decision problems such as minimizing the risk ( expected loss ) of
4560-587: The field of operator algebras , for inventing modern non-commutative geometry , and for discovering that these ideas appear everywhere, including the foundations of theoretical physics" "For his work on the Langlands program " 1999 Andrew Wiles "For his role in the development of number theory " See also [ edit ] List of mathematics awards External links [ edit ] Official web page 2014 Clay Research Awards 2017 Clay Research Awards Archived 2018-01-11 at
4640-440: The field of random walks , percolation , and probability theory in general, especially for formulating stochastic Loewner evolution " "For finding an algorithm that solves a modern version of a problem going back to the ancient Chinese and Greeks about how one can determine whether a number is prime in a time that increases polynomially with the size of the number" 2001 Edward Witten Stanislav Smirnov "For
4720-567: The first to constrain the use of the word to just the study of arithmetic and geometry. By the time of Aristotle (384–322 BC) this meaning was fully established. In Latin and English, until around 1700, the term mathematics more commonly meant " astrology " (or sometimes " astronomy ") rather than "mathematics"; the meaning gradually changed to its present one from about 1500 to 1800. This change has resulted in several mistranslations: For example, Saint Augustine 's warning that Christians should beware of mathematici , meaning "astrologers",
4800-491: The interaction between mathematical innovations and scientific discoveries has led to a correlated increase in the development of both. At the end of the 19th century, the foundational crisis of mathematics led to the systematization of the axiomatic method , which heralded a dramatic increase in the number of mathematical areas and their fields of application. The contemporary Mathematics Subject Classification lists more than sixty first-level areas of mathematics. Before
4880-400: The introduction of coordinates by René Descartes (1596–1650) for reducing geometry to algebra, and the development of calculus by Isaac Newton (1643–1727) and Gottfried Leibniz (1646–1716). Leonhard Euler (1707–1783), the most notable mathematician of the 18th century, unified these innovations into a single corpus with a standardized terminology, and completed them with the discovery and
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#17328580622494960-409: The manipulation of numbers , that is, natural numbers ( N ) , {\displaystyle (\mathbb {N} ),} and later expanded to integers ( Z ) {\displaystyle (\mathbb {Z} )} and rational numbers ( Q ) . {\displaystyle (\mathbb {Q} ).} Number theory was once called arithmetic, but nowadays this term
5040-400: The natural numbers, there are theorems that are true (that is provable in a stronger system), but not provable inside the system. This approach to the foundations of mathematics was challenged during the first half of the 20th century by mathematicians led by Brouwer , who promoted intuitionistic logic , which explicitly lacks the law of excluded middle . These problems and debates led to
5120-536: The objects defined this way is a philosophical problem that mathematicians leave to philosophers, even if many mathematicians have opinions on this nature, and use their opinion—sometimes called "intuition"—to guide their study and proofs. The approach allows considering "logics" (that is, sets of allowed deducing rules), theorems, proofs, etc. as mathematical objects, and to prove theorems about them. For example, Gödel's incompleteness theorems assert, roughly speaking that, in every consistent formal system that contains
5200-677: The p-adic setting." 2020 not awarded 2019 Wei Zhang Tristan Buckmaster , Philip Isett and Vlad Vicol "In recognition of his ground-breaking work in arithmetic geometry and arithmetic aspects of automorphic forms." "In recognition of the profound contributions that each of them has made to the analysis of partial differential equations, particularly the Navier-Stokes and Euler equations." 2018 not awarded 2017 Aleksandr Logunov and Eugenia Malinnikova Jason Miller and Scott Sheffield Maryna Viazovska "In recognition of their introduction of
5280-514: The pattern of physics and metaphysics , inherited from Greek. In English, the noun mathematics takes a singular verb. It is often shortened to maths or, in North America, math . In addition to recognizing how to count physical objects, prehistoric peoples may have also known how to count abstract quantities, like time—days, seasons, or years. Evidence for more complex mathematics does not appear until around 3000 BC , when
5360-419: The proof of an analogue of Ratner's theorem on unipotent flows for moduli of flat surfaces." "For his many and significant contributions to arithmetic algebraic geometry , particularly in the development and applications of the theory of perfectoid spaces" 2013 Rahul Pandharipande "For his recent outstanding work in enumerative geometry , specifically for his proof in a large class of cases of
5440-654: The proof of numerous theorems. Perhaps the foremost mathematician of the 19th century was the German mathematician Carl Gauss , who made numerous contributions to fields such as algebra, analysis, differential geometry , matrix theory , number theory, and statistics . In the early 20th century, Kurt Gödel transformed mathematics by publishing his incompleteness theorems , which show in part that any consistent axiomatic system—if powerful enough to describe arithmetic—will contain true propositions that cannot be proved. Mathematics has since been greatly extended, and there has been
5520-697: The quasi-isometric rigidity of sol" "For their work in advancing our understanding of the birational geometry of algebraic varieties in dimension greater than three, in particular, for their inductive proof of the existence of flips " "For their work on local and global Galois representations , partly in collaboration with Clozel and Shepherd-Barron, culminating in the solution of the Sato-Tate conjecture for elliptic curves with non-integral j-invariants " 2006 not awarded 2005 Manjul Bhargava Nils Dencker "For his discovery of new composition laws for quadratic forms , and for his work on
5600-657: The study and the manipulation of formulas . Calculus , consisting of the two subfields differential calculus and integral calculus , is the study of continuous functions , which model the typically nonlinear relationships between varying quantities, as represented by variables . This division into four main areas—arithmetic, geometry, algebra, and calculus —endured until the end of the 19th century. Areas such as celestial mechanics and solid mechanics were then studied by mathematicians, but now are considered as belonging to physics. The subject of combinatorics has been studied for much of recorded history, yet did not become
5680-561: The study of approximation and discretization with special focus on rounding errors . Numerical analysis and, more broadly, scientific computing also study non-analytic topics of mathematical science, especially algorithmic- matrix -and- graph theory . Other areas of computational mathematics include computer algebra and symbolic computation . The word mathematics comes from the Ancient Greek word máthēma ( μάθημα ), meaning ' something learned, knowledge, mathematics ' , and
5760-672: The theory under consideration. Mathematics is essential in the natural sciences , engineering , medicine , finance , computer science , and the social sciences . Although mathematics is extensively used for modeling phenomena, the fundamental truths of mathematics are independent of any scientific experimentation. Some areas of mathematics, such as statistics and game theory , are developed in close correlation with their applications and are often grouped under applied mathematics . Other areas are developed independently from any application (and are therefore called pure mathematics ) but often later find practical applications. Historically,
5840-487: The title of his main treatise . Algebra became an area in its own right only with François Viète (1540–1603), who introduced the use of variables for representing unknown or unspecified numbers. Variables allow mathematicians to describe the operations that have to be done on the numbers represented using mathematical formulas . Until the 19th century, algebra consisted mainly of the study of linear equations (presently linear algebra ), and polynomial equations in
5920-504: The two main precursors of algebra. Diophantus solved some equations involving unknown natural numbers by deducing new relations until he obtained the solution. Al-Khwarizmi introduced systematic methods for transforming equations, such as moving a term from one side of an equation into the other side. The term algebra is derived from the Arabic word al-jabr meaning 'the reunion of broken parts' that he used for naming one of these methods in
6000-578: The understanding of mirror symmetry , in joint work generally known as the ‘Gross-Siebert Program’" "In recognition of his groundbreaking work in representation theory and related fields" 2015 Larry Guth and Nets Katz "For their solution of the Erdős distance problem and for other joint and separate contributions to combinatorial incidence geometry " 2014 Maryam Mirzakhani Peter Scholze "For her many and significant contributions to geometry and ergodic theory , in particular to
6080-792: The work of Thurston and Canary, of the Ahlfors Measure Conjecture " 2008 Clifford Taubes Claire Voisin "For his proof of the Weinstein conjecture in dimension three" "For her disproof of the Kodaira conjecture" 2007 Alex Eskin Christopher Hacon and James McKernan Michael Harris and Richard Taylor "For his work on rational billiards and geometric group theory , in particular, his crucial contribution to joint work with David Fisher and Kevin Whyte establishing
6160-457: Was first elaborated for geometry, and was systematized by Euclid around 300 BC in his book Elements . The resulting Euclidean geometry is the study of shapes and their arrangements constructed from lines, planes and circles in the Euclidean plane ( plane geometry ) and the three-dimensional Euclidean space . Euclidean geometry was developed without change of methods or scope until
6240-414: Was introduced independently and simultaneously by 17th-century mathematicians Newton and Leibniz . It is fundamentally the study of the relationship of variables that depend on each other. Calculus was expanded in the 18th century by Euler with the introduction of the concept of a function and many other results. Presently, "calculus" refers mainly to the elementary part of this theory, and "analysis"
6320-437: Was not specifically studied by mathematicians. Before Cantor 's study of infinite sets , mathematicians were reluctant to consider actually infinite collections, and considered infinity to be the result of endless enumeration . Cantor's work offended many mathematicians not only by considering actually infinite sets but by showing that this implies different sizes of infinity, per Cantor's diagonal argument . This led to
6400-571: Was split into two new subfields: synthetic geometry , which uses purely geometrical methods, and analytic geometry , which uses coordinates systemically. Analytic geometry allows the study of curves unrelated to circles and lines. Such curves can be defined as the graph of functions , the study of which led to differential geometry . They can also be defined as implicit equations , often polynomial equations (which spawned algebraic geometry ). Analytic geometry also makes it possible to consider Euclidean spaces of higher than three dimensions. In
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