Hilbert's problems
Table of problems
Table:
Problem

Brief explanation

Status

Year Solved

1st

The continuum hypothesis (that is, there is no set whose cardinality is strictly between that of the integers and that of the real numbers)

Proven to be impossible to prove or disprove within Zermelo–Fraenkel set theory with or without the Axiom of Choice (provided Zermelo–Fraenkel set theory is consistent, i.e., it does not contain a contradiction). There is no consensus on whether this is a solution to the problem.

1940, 1963

2nd

Prove that the axioms of arithmetic are consistent.

There is no consensus on whether results of Gödel and Gentzen give a solution to the problem as stated by Hilbert. Gödel's second incompleteness theorem, proved in 1931, shows that no proof of its consistency can be carried out within arithmetic itself. Gentzen proved in 1936 that the consistency of arithmetic follows from the well-foundedness of the ordinal ε₀.

1931, 1936

3rd

Given any two polyhedra of equal volume, is it always possible to cut the first into finitely many polyhedral pieces that can be reassembled to yield the second?

Resolved. Result: No, proved using Dehn invariants.

1900

4th

Construct all metrics where lines are geodesics.

Too vague to be stated resolved or not.[h]

—

5th

Are continuous groups automatically differential groups?

Resolved by Andrew Gleason, depending on how the original statement is interpreted. If, however, it is understood as an equivalent of the Hilbert–Smith conjecture, it is still unsolved.

1953?

6th

Mathematical treatment of the axioms of physics
(a) axiomatic treatment of probability with limit theorems for foundation of statistical physics
(b) the rigorous theory of limiting processes "which lead from the atomistic view to the laws of motion of continua"

Partially resolved depending on how the original statement is interpreted.[9] Items (a) and (b) were two specific problems given by Hilbert in a later explanation.[1] Kolmogorov's axiomatics (1933) is now accepted as standard. There is some success on the way from the "atomistic view to the laws of motion of continua."[10]

1933–2002?

7th

Is ab transcendental, for algebraic a ≠ 0,1 and irrational algebraic b ?

Resolved. Result: Yes, illustrated by Gelfond's theorem or the Gelfond–Schneider theorem.

1934

8th

The Riemann hypothesis
("the real part of any non-trivial zero of the Riemann zeta function is ½")
and other prime number problems, among them Goldbach's conjecture and the twin prime conjecture

Unresolved.

—

9th

Find the most general law of the reciprocity theorem in any algebraic number field.

Partially resolved.[i]

—

10th

Find an algorithm to determine whether a given polynomial Diophantine equation with integer coefficients has an integer solution.

Resolved. Result: Impossible; Matiyasevich's theorem implies that there is no such algorithm.

1970

11th

Solving quadratic forms with algebraic numerical coefficients.

Partially resolved.[11]

—

12th

Extend the Kronecker–Weber theorem on Abelian extensions of the rational numbers to any base number field.

Unresolved.

—

13th

Solve 7th degree equation using algebraic (variant: continuous) functions of two parameters.

Unresolved. The continuous variant of this problem was solved by Vladimir Arnold in 1957 based on work by Andrei Kolmogorov, but the algebraic variant is unresolved.[j]

—

14th

Is the ring of invariants of an algebraic group acting on a polynomial ring always finitely generated?

Resolved. Result: No, a counterexample was constructed by Masayoshi Nagata.

1959

15th

Rigorous foundation of Schubert's enumerative calculus.

Partially resolved.[citation needed]

—

16th

Describe relative positions of ovals originating from a real algebraic curve and as limit cycles of a polynomial vector field on the plane.

Unresolved, even for algebraic curves of degree 8.

—

17th

Express a nonnegative rational function as quotient of sums of squares.

Resolved. Result: Yes, due to Emil Artin. Moreover, an upper limit was established for the number of square terms necessary.

1927

18th

(a) Is there a polyhedron that admits only an anisohedral tiling in three dimensions?
(b) What is the densest sphere packing?

(a) Resolved. Result: Yes (by Karl Reinhardt).
(b) Widely believed to be resolved, by computer-assisted proof (by Thomas Callister Hales). Result: Highest density achieved by close packings, each with density approximately 74%, such as face-centered cubic close packing and hexagonal close packing.[k]

(a) 1928
(b) 1998

19th

Are the solutions of regular problems in the calculus of variations always necessarily analytic?

Resolved. Result: Yes, proven by Ennio de Giorgi and, independently and using different methods, by John Forbes Nash.

1957

20th

Do all variational problems with certain boundary conditions have solutions?

Resolved. A significant topic of research throughout the 20th century, culminating in solutions for the non-linear case.

?

By analysing the given table and the context snippets listed below.

State the answer to the question: Which two people independently solved the same problem in 1957 ? Return your answer as JSON: {"answer": "YOUR ANSWER"}. Base your answer solely on the given information.

In mathematics , the continuum hypothesis ( abbreviated CH ) is a hypothesis about the possible sizes of infinite sets . It states :
In mathematics , the continuum hypothesis ( abbreviated CH ) is a hypothesis about the possible sizes of infinite sets . It states :
In mathematics , a set is a well-defined collection of distinct objects , considered as an object in its own right . For example , the numbers 2 , 4 , and 6 are distinct objects when considered separately , but when they are considered collectively they form a single set of size three , written { 2 , 4 , 6 } . The concept of a set is one of the most fundamental in mathematics . Developed at the end of the 19th century , set theory is now a ubiquitous part of mathematics , and can be used as a foundation from which nearly all of mathematics can be derived .
In mathematics , the cardinality of a set is a measure of the number of elements of the set . For example , the set A = { 2 , 4 , 6 } { \displaystyle A=\ { 2,4,6\ } } contains 3 elements , and therefore A { \displaystyle A } has a cardinality of 3 . Beginning in the late 19th century , this concept was generalized to infinite sets , allowing to distinguish several stages of infinity , and to perform arithmetic on them . There are two approaches to cardinality - one which compares sets directly using bijections and injections , and another which uses cardinal numbers . The cardinality of a set is also called its size , when no confusion with other notions of size is possible . The cardinality of a set A { \displaystyle A } is usually denoted | A | { \displaystyle |A| } , with a vertical bar on each side ; this is the same notation as absolute value and the meaning depends on context . Alternatively , the cardinality of a set A { \displaystyle A } may be denoted by n ( A ) { \displaystyle n ( A ) } , A { \displaystyle A } , card ⁡ ( A ) { \displaystyle \operatorname { card } ( A ) } , or # A { \displaystyle \ # A } .
An integer ( from the Latin integer meaning whole ) [ note 1 ] is a number that can be written without a fractional component . For example , 21 , 4 , 0 , and −2048 are integers , while 9.75 , 5+1/2 , and √2 are not . The set of integers consists of zero ( 0 ) , the positive natural numbers ( 1 , 2 , 3 , . ) , also called whole numbers or counting numbers , and their additive inverses ( the negative integers , i.e. , −1 , −2 , −3 , . ) . The set of integers is often denoted by a boldface Z ( Z ) or blackboard bold Z { \displaystyle \mathbb { Z } } ( Unicode U+2124 ℤ ) standing for the German word Zahlen ( [ ˈtsaːlən ] , numbers ) . Z is a subset of the set of all rational numbers Q , in turn a subset of the real numbers R. Like the natural numbers , Z is countably infinite . The integers form the smallest group and the smallest ring containing the natural numbers . In algebraic number theory , the integers are sometimes qualified as rational integers to distinguish them from the more general algebraic integers . In fact , the ( rational ) integers are the algebraic integers that are also rational numbers .
In mathematics , a real number is a value of a continuous quantity that can represent a distance along a line . The adjective real in this context was introduced in the 17th century by René Descartes , who distinguished between real and imaginary roots of polynomials . The real numbers include all the rational numbers , such as the integer −5 and the fraction 4/3 , and all the irrational numbers , such as √2 ( 1.41421356. , the square root of 2 , an irrational algebraic number ) . Included within the irrationals are the transcendental numbers , such as π ( 3.14159265. ) . In addition to measuring distance , real numbers can be used to measure quantities such as time , mass , energy , velocity , and many more . Real numbers can be thought of as points on an infinitely long line called the number line or real line , where the points corresponding to integers are equally spaced . Any real number can be determined by a possibly infinite decimal representation , such as that of 8.632 , where each consecutive digit is measured in units one tenth the size of the previous one . The real line can be thought of as a part of the complex plane , and complex numbers include real numbers .
In set theory , Zermelo-Fraenkel set theory , named after mathematicians Ernst Zermelo and Abraham Fraenkel , is an axiomatic system that was proposed in the early twentieth century in order to formulate a theory of sets free of paradoxes such as Russell 's paradox . Today , Zermelo-Fraenkel set theory , with the historically controversial axiom of choice ( AC ) included , is the standard form of axiomatic set theory and as such is the most common foundation of mathematics . Zermelo-Fraenkel set theory with the axiom of choice included is abbreviated ZFC , where C stands for choice , and ZF refers to the axioms of Zermelo-Fraenkel set theory with the axiom of choice excluded . Zermelo-Fraenkel set theory is intended to formalize a single primitive notion , that of a hereditary well-founded set , so that all entities in the universe of discourse are such sets . Thus the axioms of Zermelo-Fraenkel set theory refer only to pure sets and prevent its models from containing urelements ( elements of sets that are not themselves sets ) . Furthermore , proper classes ( collections of mathematical objects defined by a property shared by their members where the collections are too big to be sets ) can only be treated indirectly . Specifically , Zermelo-Fraenkel set theory does not allow for the existence of a universal set ( a set containing all sets ) nor for unrestricted comprehension , thereby avoiding Russell 's paradox . Von Neumann-Bernays-Gödel set theory ( NBG ) is a commonly used conservative extension of Zermelo-Fraenkel set theory that does allow explicit treatment of proper classes . There are many equivalent formulations of the axioms of Zermelo-Fraenkel set theory . Most of the axioms state the existence of particular sets defined from other sets . For example , the axiom of pairing says that given any two sets a { \displaystyle a } and b { \displaystyle b } there is a new set a , b { \displaystyle { a , b } } containing exactly a { \displaystyle a } and b { \displaystyle b } . Other axioms describe properties of set membership .
In mathematics , the axiom of choice , or AC , is an axiom of set theory equivalent to the statement that a Cartesian product of a collection of non-empty sets is non-empty . Informally put , the axiom of choice says that given any collection of bins , each containing at least one object , it is possible to make a selection of exactly one object from each bin , even if the collection is infinite . Formally , it states that for every indexed family ( S i ) i ∈ I { \displaystyle ( S_ { i } ) _ { i\in I } } of nonempty sets there exists an indexed family ( x i ) i ∈ I { \displaystyle ( x_ { i } ) _ { i\in I } } of elements such that x i ∈ S i { \displaystyle x_ { i } \in S_ { i } } for every i ∈ I { \displaystyle i\in I } . The axiom of choice was formulated in 1904 by Ernst Zermelo in order to formalize his proof of the well-ordering theorem . In many cases , such a selection can be made without invoking the axiom of choice ; this is in particular the case if the number of sets is finite , or if a selection rule is available - some distinguishing property that happens to hold for exactly one element in each set . An illustrative example is sets picked from the natural numbers . From such sets , one may always select the smallest number , e.g . in { { 4 , 5 , 6 } , { 10 , 12 } , { 1 , 400 , 617 , 8000 } } the smallest elements are { 4 , 10 , 1 } . In this case , select the smallest number is a choice function . Even if infinitely many sets were collected from the natural numbers , it will always be possible to choose the smallest element from each set to produce a set . That is , the choice function provides the set of chosen elements . However , no choice function is known for the collection of all non-empty subsets of the real numbers ( if there are non-constructible reals ) .
In classical deductive logic , a consistent theory is one that does not entail a contradiction . The lack of contradiction can be defined in either semantic or syntactic terms . The semantic definition states that a theory is consistent if it has a model , i.e. , there exists an interpretation under which all formulas in the theory are true . This is the sense used in traditional Aristotelian logic , although in contemporary mathematical logic the term satisfiable is used instead . The syntactic definition states a theory T { \displaystyle T } is consistent if there is no formula φ { \displaystyle \varphi } such that both φ { \displaystyle \varphi } and its negation ¬ φ { \displaystyle \lnot \varphi } are elements of the set of consequences of T { \displaystyle T } . Let A { \displaystyle A } be a set of closed sentences ( informally axioms ) and ⟨ A ⟩ { \displaystyle \langle A\rangle } the set of closed sentences provable from A { \displaystyle A } under some ( specified , possibly implicitly ) formal deductive system . The set of axioms A { \displaystyle A } is consistent when φ , ¬ φ ∈ ⟨ A ⟩ { \displaystyle \varphi , \lnot \varphi \in \langle A\rangle } for no formula φ { \displaystyle \varphi } . If there exists a deductive system for which these semantic and syntactic definitions are equivalent for any theory formulated in a particular deductive logic , the logic is called complete . [ citation needed ] The completeness of the sentential calculus was proved by Paul Bernays in 1918 [ citation needed ] and Emil Post in 1921 , while the completeness of predicate calculus was proved by Kurt Gödel in 1930 , and consistency proofs for arithmetics restricted with respect to the induction axiom schema were proved by Ackermann ( 1924 ) , von Neumann ( 1927 ) and Herbrand ( 1931 ) . Stronger logics , such as second-order logic , are not complete . A consistency proof is a mathematical proof that a particular theory is consistent . The early development of mathematical proof theory was driven by the desire to provide finitary consistency proofs for all of mathematics as part of Hilbert 's program .
In mathematics , Hilbert 's second problem was posed by David Hilbert in 1900 as one of his 23 problems . It asks for a proof that the arithmetic is consistent - free of any internal contradictions . Hilbert stated that the axioms he considered for arithmetic were the ones given in Hilbert ( 1900 ) , which include a second order completeness axiom . In the 1930s , Kurt Gödel and Gerhard Gentzen proved results that cast new light on the problem . Some feel that Gödel 's theorems give a negative solution to the problem , while others consider Gentzen 's proof as a partial positive solution .
An axiom or postulate is a statement that is taken to be true , to serve as a premise or starting point for further reasoning and arguments . The word comes from the Greek axíōma ( ἀξίωμα ) 'that which is thought worthy or fit ' or 'that which commends itself as evident . ' The term has subtle differences in definition when used in the context of different fields of study . As defined in classic philosophy , an axiom is a statement that is so evident or well-established , that it is accepted without controversy or question . As used in modern logic , an axiom is a premise or starting point for reasoning . As used in mathematics , the term axiom is used in two related but distinguishable senses : logical axioms and non-logical axioms . Logical axioms are usually statements that are taken to be true within the system of logic they define and are often shown in symbolic form ( e.g. , ( A and B ) implies A ) , while non-logical axioms ( e.g. , a + b = b + a ) are actually substantive assertions about the elements of the domain of a specific mathematical theory ( such as arithmetic ) . When used in the latter sense , axiom , postulate , and assumption may be used interchangeably . In most cases , a non-logical axiom is simply a formal logical expression used in deduction to build a mathematical theory , and might or might not be self-evident in nature ( e.g. , parallel postulate in Euclidean geometry ) . To axiomatize a system of knowledge is to show that its claims can be derived from a small , well-understood set of sentences ( the axioms ) , and there may be multiple ways to axiomatize a given mathematical domain . Any axiom is a statement that serves as a starting point from which other statements are logically derived . Whether it is meaningful ( and , if so , what it means ) for an axiom to be true is a subject of debate in the philosophy of mathematics .
Arithmetic ( from the Greek ἀριθμός arithmos , number and τική [ τέχνη ] , tiké [ téchne ] , art ) is a branch of mathematics that consists of the study of numbers , especially the properties of the traditional operations on them - addition , subtraction , multiplication and division . Arithmetic is an elementary part of number theory , and number theory is considered to be one of the top-level divisions of modern mathematics , along with algebra , geometry , and analysis . The terms arithmetic and higher arithmetic were used until the beginning of the 20th century as synonyms for number theory and are sometimes still used to refer to a wider part of number theory .
Kurt Friedrich Gödel ( /ˈɡɜːrdəl/ ; German : [ ˈkʊɐ̯t ˈɡøːdl̩ ] ( listen ) ; April 28 , 1906 - January 14 , 1978 ) was an Austro-Hungarian-born Austrian logician , mathematician , and analytic philosopher . Considered along with Aristotle and Gottlob Frege to be one of the most significant logicians in history , Gödel had an immense effect upon scientific and philosophical thinking in the 20th century , a time when others such as Bertrand Russell , Alfred North Whitehead , and David Hilbert were analyzing the use of logic and set theory to understand the foundations of mathematics pioneered by Georg Cantor . Gödel published his two incompleteness theorems in 1931 when he was 25 years old , one year after finishing his doctorate at the University of Vienna . The first incompleteness theorem states that for any self-consistent recursive axiomatic system powerful enough to describe the arithmetic of the natural numbers ( for example Peano arithmetic ) , there are true propositions about the naturals that can not be proved from the axioms . To prove this theorem , Gödel developed a technique now known as Gödel numbering , which codes formal expressions as natural numbers . He also showed that neither the axiom of choice nor the continuum hypothesis can be disproved from the accepted axioms of set theory , assuming these axioms are consistent . The former result opened the door for mathematicians to assume the axiom of choice in their proofs . He also made important contributions to proof theory by clarifying the connections between classical logic , intuitionistic logic , and modal logic .
Gerhard Karl Erich Gentzen ( November 24 , 1909 - August 4 , 1945 ) was a German mathematician and logician . He made major contributions to the foundations of mathematics , proof theory , especially on natural deduction and sequent calculus . He died of starvation in a Soviet prison camp in Prague in 1945 , having been interned as a German national after the Second World War .
Gödel 's incompleteness theorems are two theorems of mathematical logic that demonstrate the inherent limitations of every formal axiomatic system capable of modelling basic arithmetic . These results , published by Kurt Gödel in 1931 , are important both in mathematical logic and in the philosophy of mathematics . The theorems are widely , but not universally , interpreted as showing that Hilbert 's program to find a complete and consistent set of axioms for all mathematics is impossible . The first incompleteness theorem states that no consistent system of axioms whose theorems can be listed by an effective procedure ( i.e. , an algorithm ) is capable of proving all truths about the arithmetic of natural numbers . For any such consistent formal system , there will always be statements about natural numbers that are true , but that are unprovable within the system . The second incompleteness theorem , an extension of the first , shows that the system can not demonstrate its own consistency . Employing a diagonal argument , Gödel 's incompleteness theorems were the first of several closely related theorems on the limitations of formal systems . They were followed by Tarski 's undefinability theorem on the formal undefinability of truth , Church 's proof that Hilbert 's Entscheidungsproblem is unsolvable , and Turing 's theorem that there is no algorithm to solve the halting problem .
In mathematics , a binary relation R is called well-founded ( or wellfounded ) on a class X if every non-empty subset S ⊆ X has a minimal element with respect to R , that is , an element m not related by sRm ( for instance , s is not smaller than m ) for any s ∈ S. In other words , a relation is well founded if
In mathematics , the epsilon numbers are a collection of transfinite numbers whose defining property is that they are fixed points of an exponential map . Consequently , they are not reachable from 0 via a finite series of applications of the chosen exponential map and of weaker operations like addition and multiplication . The original epsilon numbers were introduced by Georg Cantor in the context of ordinal arithmetic ; they are the ordinal numbers ε that satisfy the equation
The third of Hilbert 's list of mathematical problems , presented in 1900 , was the first to be solved . The problem is related to the following question : given any two polyhedra of equal volume , is it always possible to cut the first into finitely many polyhedral pieces which can be reassembled to yield the second ? Based on earlier writings by Gauss , Hilbert conjectured that this is not always possible . This was confirmed within the year by his student Max Dehn , who proved that the answer in general is no by producing a counterexample . The answer for the analogous question about polygons in 2 dimensions is yes and had been known for a long time ; this is the Wallace-Bolyai-Gerwien theorem . Unknown to Hilbert and Dehn , Hilbert 's third problem was also proposed independently by Władysław Kretkowski for a math contest of 1882 by the Academy of Arts and Sciences of Kraków , and was solved by Ludwik Antoni Birkenmajer with a different method than Dehn . Birkenmajer did not publish the result , and the original manuscript containing his solution was rediscovered years later .
In geometry , a polyhedron ( plural polyhedra or polyhedrons ) is a solid in three dimensions with flat polygonal faces , straight edges and sharp corners or vertices . The word polyhedron comes from the Classical Greek πολύεδρον , as poly- ( stem of πολύς , many ) + -hedron ( form of ἕδρα , base or seat ) . A convex polyhedron is the convex hull of finitely many points , not all on the same plane . Cubes and pyramids are examples of convex polyhedra . A polyhedron is a 3-dimensional example of the more general polytope in any number of dimensions .
In geometry , the Dehn invariant of a polyhedron is a value used to determine whether polyhedra can be dissected into each other or whether they can tile space . It is named after Max Dehn , who used it to solve Hilbert 's third problem on whether all polyhedra with equal volume could be dissected into each other . Two polyhedra have a dissection into polyhedral pieces that can be reassembled into either one , if and only if their volumes and Dehn invariants are equal . A polyhedron can be cut up and reassembled to tile space if and only if its Dehn invariant is zero , so having Dehn invariant zero is a necessary condition for being a space-filling polyhedron . The Dehn invariant of a self-intersection free flexible polyhedron is invariant as it flexes . The Dehn invariant is zero for the cube but nonzero for the other Platonic solids , implying that the other solids can not tile space and that they can not be dissected into a cube . All of the Archimedean solids have Dehn invariants that are rational combinations of the invariants for the Platonic solids . In particular , the truncated octahedron also tiles space and has Dehn invariant zero like the cube . The Dehn invariants of polyhedra are elements of an infinite-dimensional vector space . As an abelian group , this space is part of an exact sequence involving group homology . Similar invariants can also be defined for some other dissection puzzles , including the problem of dissecting rectilinear polygons into each other by axis-parallel cuts and translations .
In mathematics , Hilbert 's fourth problem in the 1900 Hilbert problems is a foundational question in geometry . In one statement derived from the original , it was to find - up to an isomorphism - all geometries that have an axiomatic system of the classical geometry ( Euclidean , hyperbolic and elliptic ) , with those axioms of congruence that involve the concept of the angle dropped , and ` triangle inequality ' , regarded as an axiom , added . If one assumes the continuity axiom in addition , then , in the case of the Euclidean plane , we come to the problem posed by Darboux : To determine all the calculus of variation problems in the plane whose solutions are all the plane straight lines . There are several interpretations of the original statement of Hilbert . Nevertheless , a solution was sought with the German mathematician Georg Hamel being the first who contribute to the solution of Hilbert 's fourth problem . A recognized solution was given by Ukrainian mathematician Aleksei Pogorelov in 1973 . In 1976 , Armenian mathematician Rouben V. Ambartzumian proposed another proof of Hilbert 's fourth problem .
In mathematics , a metric space is a set together with a metric on the set . The metric is a function that defines a concept of distance between any two members of the set , which are usually called points . The metric satisfies a few simple properties . Informally :
In differential geometry , a geodesic ( /ˌdʒiːəˈdɛsɪk , ˌdʒiːoʊ- , -ˈdiː- , -zɪk/ ) is a curve representing in some sense the shortest path between two points in a surface , or more generally in a Riemannian manifold . It is a generalization of the notion of a straight line to a more general setting . The term geodesic comes from geodesy , the science of measuring the size and shape of Earth . In the original sense , a geodesic was the shortest route between two points on the Earth 's surface . For a spherical Earth , it is a segment of a great circle . The term has been generalized to include measurements in much more general mathematical spaces ; for example , in graph theory , one might consider a geodesic between two vertices/nodes of a graph . In a Riemannian manifold or submanifold geodesics are characterised by the property of having vanishing geodesic curvature . More generally , in the presence of an affine connection , a geodesic is defined to be a curve whose tangent vectors remain parallel if they are transported along it . Applying this to the Levi-Civita connection of a Riemannian metric recovers the previous notion . Geodesics are of particular importance in general relativity . Timelike geodesics in general relativity describe the motion of free falling test particles .
Hilbert 's fifth problem is the fifth mathematical problem from the problem list publicized in 1900 by mathematician David Hilbert , and concerns the characterization of Lie groups . The theory of Lie groups describes continuous symmetry in mathematics ; its importance there and in theoretical physics ( for example quark theory ) grew steadily in the twentieth century . In rough terms , Lie group theory is the common ground of group theory and the theory of topological manifolds . The question Hilbert asked was an acute one of making this precise : is there any difference if a restriction to smooth manifolds is imposed ? The expected answer was in the negative ( the classical groups , the most central examples in Lie group theory , are smooth manifolds ) . This was eventually confirmed in the early 1950s . Since the precise notion of manifold was not available to Hilbert , there is room for some debate about the formulation of the problem in contemporary mathematical language .
In mathematics , a group is a set equipped with a binary operation that combines any two elements to form a third element in such a way that four conditions called group axioms are satisfied , namely closure , associativity , identity and invertibility . One of the most familiar examples of a group is the set of integers together with the addition operation , but groups are encountered in numerous areas within and outside mathematics , and help focusing on essential structural aspects , by detaching them from the concrete nature of the subject of the study . Groups share a fundamental kinship with the notion of symmetry . For example , a symmetry group encodes symmetry features of a geometrical object : the group consists of the set of transformations that leave the object unchanged and the operation of combining two such transformations by performing one after the other . Lie groups are the symmetry groups used in the Standard Model of particle physics ; Poincaré groups , which are also Lie groups , can express the physical symmetry underlying special relativity ; and point groups are used to help understand symmetry phenomena in molecular chemistry . The concept of a group arose from the study of polynomial equations , starting with Évariste Galois in the 1830s , who introduced the term of group ( groupe , in French ) for the symmetry group of the roots of an equation , now called a Galois group . After contributions from other fields such as number theory and geometry , the group notion was generalized and firmly established around 1870 . Modern group theory - an active mathematical discipline - studies groups in their own right . [ a ] To explore groups , mathematicians have devised various notions to break groups into smaller , better-understandable pieces , such as subgroups , quotient groups and simple groups . In addition to their abstract properties , group theorists also study the different ways in which a group can be expressed concretely , both from a point of view of representation theory ( that is , through the representations of the group ) and of computational group theory . A theory has been developed for finite groups , which culminated with the classification of finite simple groups , completed in 2004 . [ aa ] Since the mid-1980s , geometric group theory , which studies finitely generated groups as geometric objects , has become an active area in group theory .
In mathematics , a Lie group ( pronounced /liː/ Lee ) is a group whose elements are organized continuously and smoothly , as opposed to discrete groups , where the elements are separated - this makes Lie groups differentiable manifolds . Lie groups are named after Norwegian mathematician Sophus Lie , who laid the foundations of the theory of continuous transformation groups . In rough terms , a Lie group is a continuous group : it is a group whose elements are described by several real parameters . As such , Lie groups provide a natural model for the concept of continuous symmetry , such as rotational symmetry in three dimensions . Lie groups are widely used in many parts of modern mathematics and physics . Lie 's original motivation for introducing Lie groups was to model the continuous symmetries of differential equations , in much the same way that finite groups are used in Galois theory to model the discrete symmetries of algebraic equations .
Andrew Mattei Gleason ( 1921-2008 ) was an American mathematician who made fundamental contributions to widely varied areas of mathematics , including the solution of Hilbert 's fifth problem , and was a leader in reform and innovation in math­e­mat­ics teaching at all levels . Gleason 's theorem in quantum logic and the Greenwood-Gleason graph , an important example in Ramsey theory , are named for him . As a young World War II naval officer , Gleason broke German and Japanese military codes . After the war he spent his entire academic career at Harvard University , from which he retired in 1992 . His numerous academic and scholarly leadership posts included chairmanship of the Harvard Mathematics Department and the Harvard Society of Fellows , and presidency of the American Mathematical Society . He continued to advise the United States government on cryptographic security , and the Commonwealth of Massachusetts on math­e­mat­ics education for children , almost until the end of his life . Gleason won the Newcomb Cleveland Prize in 1952 and the Gung-Hu Distinguished Service Award of the American Mathematical Society in 1996 . He was a member of the National Academy of Sciences and of the American Philosophical Society , and held the Hollis Chair of Mathematics and Natural Philosophy at Harvard . He was fond of saying that math­e­mat­ic­al proofs really are n't there to convince you that something is true‍ - ‌they 're there to show you why it is true . The Notices of the American Mathematical Society called him one of the quiet giants of twentieth-century mathematics , the consummate professor dedicated to scholarship , teaching , and service in equal measure .
In mathematics , the Hilbert-Smith conjecture is concerned with the transformation groups of manifolds ; and in particular with the limitations on topological groups G that can act effectively ( faithfully ) on a ( topological ) manifold M. Restricting to G which are locally compact and have a continuous , faithful group action on M , it states that G must be a Lie group . Because of known structural results on G , it is enough to deal with the case where G is the additive group Zp of p-adic integers , for some prime number p. An equivalent form of the conjecture is that Zp has no faithful group action on a topological manifold . The naming of the conjecture is for David Hilbert , and the American topologist Paul A. Smith . It is considered by some to be a better formulation of Hilbert 's fifth problem , than the characterisation in the category of topological groups of the Lie groups often cited as a solution . In 1997 , Dušan Repovš and Evgenij Ščepin proved the Hilbert-Smith conjecture for groups acting by Lipschitz maps on a Riemannian manifold using the covering , fractal and cohomological dimension theory . In 1999 , Gaven Martin extended their dimension-theoretic argument to quasiconformal actions on a Riemannian manifold and gave applications concerning unique analytic continuation for Beltrami systems . In 2013 , John Pardon proved the three-dimensional case of the Hilbert-Smith conjecture .
Hilbert 's sixth problem is to axiomatize those branches of physics in which mathematics is prevalent . It occurs on the widely cited list of Hilbert 's problems in mathematics that he presented in the year 1900 . In its common English translation , the explicit statement reads :
Physics ( from Ancient Greek : φυσική ( ἐπιστήμη ) , romanized : physikḗ ( epistḗmē ) , lit . 'knowledge of nature ' , from φύσις phýsis 'nature ' ) is the natural science that studies matter , its motion and behavior through space and time , and the related entities of energy and force . Physics is one of the most fundamental scientific disciplines , and its main goal is to understand how the universe behaves . [ a ] Physics is one of the oldest academic disciplines and , through its inclusion of astronomy , perhaps the oldest . Over much of the past two millennia , physics , chemistry , biology , and certain branches of mathematics were a part of natural philosophy , but during the Scientific Revolution in the 17th century these natural sciences emerged as unique research endeavors in their own right . [ b ] Physics intersects with many interdisciplinary areas of research , such as biophysics and quantum chemistry , and the boundaries of physics are not rigidly defined . New ideas in physics often explain the fundamental mechanisms studied by other sciences and suggest new avenues of research in academic disciplines such as mathematics and philosophy . Advances in physics often enable advances in new technologies . For example , advances in the understanding of electromagnetism , solid-state physics , and nuclear physics led directly to the development of new products that have dramatically transformed modern-day society , such as television , computers , domestic appliances , and nuclear weapons ; advances in thermodynamics led to the development of industrialization ; and advances in mechanics inspired the development of calculus .
Statistical physics is a branch of physics that uses methods of probability theory and statistics , and particularly the mathematical tools for dealing with large populations and approximations , in solving physical problems . It can describe a wide variety of fields with an inherently stochastic nature . Its applications include many problems in the fields of physics , biology , chemistry , neuroscience , and even some social sciences , such as sociology and linguistics . Its main purpose is to clarify the properties of matter in aggregate , in terms of physical laws governing atomic motion . In particular , statistical mechanics develops the phenomenological results of thermodynamics from a probabilistic examination of the underlying microscopic systems . Historically , one of the first topics in physics where statistical methods were applied was the field of mechanics , which is concerned with the motion of particles or objects when subjected to a force .
The Kolmogorov Axioms are the foundations of Probability Theory introduced by Andrey Kolmogorov in 1933 . These axioms remain central and have direct contributions to mathematics , the physical sciences , and real-world probability cases . An alternative approach to formalising probability , favoured by some Bayesians , is given by Cox 's theorem .
Hilbert 's seventh problem is one of David Hilbert 's list of open mathematical problems posed in 1900 . It concerns the irrationality and transcendence of certain numbers ( Irrationalität und Transzendenz bestimmter Zahlen ) .
In mathematics , a transcendental number is a complex number that is not an algebraic number - that is , not a root ( i.e. , solution ) of a nonzero polynomial equation with integer coefficients . The best-known transcendental numbers are π and e. Though only a few classes of transcendental numbers are known , in part because it can be extremely difficult to show that a given number is transcendental , transcendental numbers are not rare . Indeed , almost all real and complex numbers are transcendental , since the algebraic numbers compose a countable set , while the set of real numbers and the set of complex numbers are both uncountable sets , and therefore larger than any countable set . All real transcendental numbers are irrational numbers , since all rational numbers are algebraic . The converse is not true : not all irrational numbers are transcendental . For example , the square root of 2 is an irrational number , but it is not a transcendental number as it is a root of the polynomial equation x2 − 2 = 0 . The golden ratio ( denoted φ { \displaystyle \varphi } or ϕ { \displaystyle \phi } ) is another irrational number that is not transcendental , as it is a root of the polynomial equation x2 − x − 1 = 0 .
An algebraic number is any complex number ( including real numbers ) that is a root of a non-zero polynomial ( that is , a value which causes the polynomial to equal 0 ) in one variable with rational coefficients ( or equivalently - by clearing denominators - with integer coefficients ) . All integers and rational numbers are algebraic , as are all roots of integers . There are real and complex numbers that are not algebraic , such as π and e. These numbers are called transcendental numbers . While the set of complex numbers is uncountable , the set of algebraic numbers is countable and has measure zero in the Lebesgue measure as a subset of the complex numbers , and in this sense almost all complex numbers are transcendental .
In mathematics , the irrational numbers are all the real numbers which are not rational numbers , the latter being the numbers constructed from ratios ( or fractions ) of integers . When the ratio of lengths of two line segments is an irrational number , the line segments are also described as being incommensurable , meaning that they share no measure in common , that is , there is no length ( the measure ) , no matter how short , that could be used to express the lengths of both of the two given segments as integer multiples of itself . Among irrational numbers are the ratio π of a circle 's circumference to its diameter , Euler 's number e , the golden ratio φ , and the square root of two ; in fact all square roots of natural numbers , other than of perfect squares , are irrational . Like all real numbers , irrational numbers can be expressed in positional notation , notably as a decimal number . In the case of irrational numbers , the decimal expansion does not terminate , nor end with a repeating sequence . For example , the decimal representation of π starts with 3.14159 , but no finite number of digits can represent π exactly , nor does it repeat . Conversely , a decimal expansion that terminates or repeats must be a rational number . These are provable properties of rational numbers and positional number systems , and are not used as definitions in mathematics . Irrational numbers can also be expressed as non-terminating continued fractions and many other ways . As a consequence of Cantor 's proof that the real numbers are uncountable and the rationals countable , it follows that almost all real numbers are irrational .
In mathematics , the Gelfond-Schneider theorem establishes the transcendence of a large class of numbers .
In mathematics , the Gelfond-Schneider theorem establishes the transcendence of a large class of numbers .
Hilbert 's eighth problem is one of David Hilbert 's list of open mathematical problems posed in 1900 . It concerns number theory , and in particular the Riemann hypothesis , although it is also concerned with the Goldbach Conjecture . The problem as stated asked for more work on the distribution of primes and generalizations of Riemann hypothesis to other rings where prime ideals take the place of primes . This problem has yet to be resolved .
In mathematics , the Riemann hypothesis is a conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part 1/2 . Many consider it to be the most important unsolved problem in pure mathematics ( Bombieri 2000 ) . It is of great interest in number theory because it implies results about the distribution of prime numbers . It was proposed by Bernhard Riemann ( 1859 ) , after whom it is named . The Riemann hypothesis and some of its generalizations , along with Goldbach 's conjecture and the twin prime conjecture , comprise Hilbert 's eighth problem in David Hilbert 's list of 23 unsolved problems ; it is also one of the Clay Mathematics Institute 's Millennium Prize Problems . The name is also used for some closely related analogues , such as the Riemann hypothesis for curves over finite fields . The Riemann zeta function ζ ( s ) is a function whose argument s may be any complex number other than 1 , and whose values are also complex . It has zeros at the negative even integers ; that is , ζ ( s ) = 0 when s is one of −2 , −4 , −6 , . These are called its trivial zeros . However , the negative even integers are not the only values for which the zeta function is zero . The other ones are called non-trivial zeros . The Riemann hypothesis is concerned with the locations of these non-trivial zeros , and states that :
In mathematics , the adjective trivial is often used to refer to a claim or a case which can be readily obtained from context , or an object which possesses a simple structure ( e.g. , groups , topological spaces ) . The noun triviality usually refers to a simple technical aspect of some proof or definition . The origin of the term in mathematical language comes from the medieval trivium curriculum , which distinguishes from the more difficult quadrivium curriculum . The opposite of trivial is nontrivial , which is commonly used to indicate that an example or a solution is not simple , or that a statement or a theorem is not easy to prove .
In mathematics , a zero ( also sometimes called a root ) of a real- , complex- , or generally vector-valued function f { \displaystyle f } , is a member x { \displaystyle x } of the domain of f { \displaystyle f } such that f ( x ) { \displaystyle f ( x ) } vanishes at x { \displaystyle x } ; that is , the function f { \displaystyle f } attains the value of 0 at x { \displaystyle x } , or equivalently , x { \displaystyle x } is the solution to the equation f ( x ) = 0 { \displaystyle f ( x ) =0 } . A zero of a function is thus an input value that produces an output of 0 { \displaystyle 0 } . A root of a polynomial is a zero of the corresponding polynomial function . The fundamental theorem of algebra shows that any non-zero polynomial has a number of roots at most equal to its degree , and that the number of roots and the degree are equal when one considers the complex roots ( or more generally , the roots in an algebraically closed extension ) counted with their multiplicities . For example , the polynomial f { \displaystyle f } of degree two , defined by
The Riemann zeta function or Euler-Riemann zeta function , ζ ( s ) , is a function of a complex variable s that analytically continues the sum of the Dirichlet series
Goldbach 's conjecture is one of the oldest and best-known unsolved problems in number theory and all of mathematics . It states :
A twin prime is a prime number that is either 2 less or 2 more than another prime number - for example , either member of the twin prime pair ( 41 , 43 ) . In other words , a twin prime is a prime that has a prime gap of two . Sometimes the term twin prime is used for a pair of twin primes ; an alternative name for this is prime twin or prime pair . Twin primes become increasingly rare as one examines larger ranges , in keeping with the general tendency of gaps between adjacent primes to become larger as the numbers themselves get larger . However , it is unknown whether there are infinitely many twin primes or there is a largest pair . The work of Yitang Zhang in 2013 , as well as work by James Maynard , Terence Tao and others , has made substantial progress towards proving that there are infinitely many twin primes , but at present this remains unsolved .
Hilbert 's ninth problem , from the list of 23 Hilbert 's problems ( 1900 ) , asked to find the most general reciprocity law for the norm residues of k-th order in a general algebraic number field , where k is a power of a prime .
In number theory , the law of quadratic reciprocity is a theorem about modular arithmetic that gives conditions for the solvability of quadratic equations modulo prime numbers . Due to its subtlety , it has many formulations , but the most standard statement is :
Algebra ( from Arabic : الجبر ( al-jabr , meaning reunion of broken parts and bonesetting ) is one of the broad parts of mathematics , together with number theory , geometry and analysis . In its most general form , algebra is the study of mathematical symbols and the rules for manipulating these symbols ; it is a unifying thread of almost all of mathematics . It includes everything from elementary equation solving to the study of abstractions such as groups , rings , and fields . The more basic parts of algebra are called elementary algebra ; the more abstract parts are called abstract algebra or modern algebra . Elementary algebra is generally considered to be essential for any study of mathematics , science , or engineering , as well as such applications as medicine and economics . Abstract algebra is a major area in advanced mathematics , studied primarily by professional mathematicians . Elementary algebra differs from arithmetic in the use of abstractions , such as using letters to stand for numbers that are either unknown or allowed to take on many values . For example , in x + 2 = 5 { \displaystyle x+2=5 } the letter x { \displaystyle x } is unknown , but applying additive inverses can reveal its value : x = 3 { \displaystyle x=3 } . In E = mc2 , the letters E { \displaystyle E } and m { \displaystyle m } are variables , and the letter c { \displaystyle c } is a constant , the speed of light in a vacuum . Algebra gives methods for writing formulas and solving equations that are much clearer and easier than the older method of writing everything out in words . The word algebra is also used in certain specialized ways . A special kind of mathematical object in abstract algebra is called an algebra , and the word is used , for example , in the phrases linear algebra and algebraic topology .
In mathematics , an algebraic number field ( or simply number field ) F is a finite degree ( and hence algebraic ) field extension of the field of rational numbers Q . Thus F is a field that contains Q and has finite dimension when considered as a vector space over Q . The study of algebraic number fields , and , more generally , of algebraic extensions of the field of rational numbers , is the central topic of algebraic number theory .
Hilbert 's tenth problem is the tenth on the list of mathematical problems that the German mathematician David Hilbert posed in 1900 . It is the challenge to provide a general algorithm which , for any given Diophantine equation ( a polynomial equation with integer coefficients and a finite number of unknowns ) , can decide whether the equation has a solution with all unknowns taking integer values . For example , the Diophantine equation 3 x 2 − 2 x y − y 2 z − 7 = 0 { \displaystyle 3x^ { 2 } -2xy-y^ { 2 } z-7=0 } has an integer solution : x = 1 , y = 2 , z = − 2 { \displaystyle x=1 , \ y=2 , \ z=-2 } . By contrast , the Diophantine equation x 2 + y 2 + 1 = 0 { \displaystyle x^ { 2 } +y^ { 2 } +1=0 } has no such solution . Hilbert 's tenth problem has been solved , and it has a negative answer : such a general algorithm does not exist . This is the result of combined work of Martin Davis , Yuri Matiyasevich , Hilary Putnam and Julia Robinson which spans 21 years , with Matiyasevich completing the theorem in 1970 . The theorem is now known as Matiyasevich 's theorem or the MRDP theorem .
In mathematics , a Diophantine equation is a polynomial equation , usually in two or more unknowns , such that only the integer solutions are sought or studied ( an integer solution is such that all the unknowns take integer values ) . A linear Diophantine equation equates the sum of two or more monomials , each of degree 1 in one of the variables , to a constant . An exponential Diophantine equation is one in which exponents on terms can be unknowns . Diophantine problems have fewer equations than unknown variables and involve finding integers that work correctly for all equations . In more technical language , they define an algebraic curve , algebraic surface , or more general object , and ask about the lattice points on it . The word Diophantine refers to the Hellenistic mathematician of the 3rd century , Diophantus of Alexandria , who made a study of such equations and was one of the first mathematicians to introduce symbolism into algebra . The mathematical study of Diophantine problems that Diophantus initiated is now called Diophantine analysis . While individual equations present a kind of puzzle and have been considered throughout history , the formulation of general theories of Diophantine equations ( beyond the theory of quadratic forms ) was an achievement of the twentieth century .
In mathematics , a Diophantine equation is an equation of the form P ( x1 , . , xj , y1 , . , yk ) = 0 ( usually abbreviated P ( x , y ) = 0 ) where P ( x , y ) is a polynomial with integer coefficients . A Diophantine set is a subset S of Nj so that for some Diophantine equation P ( x , y ) = 0 ,
Hilbert 's eleventh problem is one of David Hilbert 's list of open mathematical problems posed at the Second International Congress of Mathematicians in Paris in 1900 . A furthering of the theory of quadratic forms , he stated the problem as follows :
In mathematics , a quadratic form is a polynomial with terms all of degree two . For example ,
In mathematics , a coefficient is a multiplicative factor in some term of a polynomial , a series , or any expression ; it is usually a number , but may be any expression . In the latter case , the variables appearing in the coefficients are often called parameters , and must be clearly distinguished from the other variables . For example , in
Kronecker 's Jugendtraum or Hilbert 's twelfth problem , of the 23 mathematical Hilbert problems , is the extension of the Kronecker-Weber theorem on abelian extensions of the rational numbers , to any base number field . That is , it asks for analogues of the roots of unity , as complex numbers that are particular values of the exponential function ; the requirement is that such numbers should generate a whole family of further number fields that are analogues of the cyclotomic fields and their subfields . The classical theory of complex multiplication , now often known as the Kronecker Jugendtraum , does this for the case of any imaginary quadratic field , by using modular functions and elliptic functions chosen with a particular period lattice related to the field in question . Goro Shimura extended this to CM fields . The general case is still open as of 2014 [ update ] . Leopold Kronecker described the complex multiplication issue as his liebster Jugendtraum or dearest dream of his youth .
In algebraic number theory , it can be shown that every cyclotomic field is an abelian extension of the rational number field Q , having Galois group of the form ( Z / n Z ) × { \displaystyle ( \mathbb { Z } /n\mathbb { Z } ) ^ { \times } } . The Kronecker-Weber theorem provides a partial converse : every finite abelian extension of Q is contained within some cyclotomic field . In other words , every algebraic integer whose Galois group is abelian can be expressed as a sum of roots of unity with rational coefficients . For example ,
In mathematics , a rational number is a number that can be expressed as the quotient or fraction p/q of two integers , a numerator p and a non-zero denominator q . Since q may be equal to 1 , every integer is a rational number . The set of all rational numbers , often referred to as the rationals , the field of rationals or the field of rational numbers is usually denoted by a boldface Q ( or blackboard bold Q { \displaystyle \mathbb { Q } } , Unicode ℚ ) ; it was thus denoted in 1895 by Giuseppe Peano after quoziente , Italian for quotient . The decimal expansion of a rational number always either terminates after a finite number of digits or begins to repeat the same finite sequence of digits over and over . Moreover , any repeating or terminating decimal represents a rational number . These statements hold true not just for base 10 , but also for any other integer base ( e.g . binary , hexadecimal ) . A real number that is not rational is called irrational . Irrational numbers include √2 , π , e , and φ . The decimal expansion of an irrational number continues without repeating . Since the set of rational numbers is countable , and the set of real numbers is uncountable , almost all real numbers are irrational . Rational numbers can be formally defined as equivalence classes of pairs of integers ( p , q ) such that q ≠ 0 , for the equivalence relation defined by ( p1 , q1 ) ~ ( p2 , q2 ) if , and only if p1q2 = p2q1 .
Hilbert 's thirteenth problem is one of the 23 Hilbert problems set out in a celebrated list compiled in 1900 by David Hilbert . It entails proving whether a solution exists for all 7th-degree equations using algebraic ( variant : continuous ) functions of two arguments . It was first presented in the context of nomography , and in particular nomographic construction - a process whereby a function of several variables is constructed using functions of two variables . The variant for continuous functions was resolved in 1957 by Vladimir Arnold when he proved the Kolmogorov-Arnold representation theorem , but the variant for algebraic functions remains unresolved .
In algebra , a septic equation is an equation of the form
In mathematics , a function [ note 1 ] is a relation between sets that associates to every element of a first set exactly one element of the second set . Typical examples are functions from integers to integers or from the real numbers to real numbers . Functions were originally the idealization of how a varying quantity depends on another quantity . For example , the position of a planet is a function of time . Historically , the concept was elaborated with the infinitesimal calculus at the end of the 17th century , and , until the 19th century , the functions that were considered were differentiable ( that is , they had a high degree of regularity ) . The concept of function was formalized at the end of the 19th century in terms of set theory , and this greatly enlarged the domains of application of the concept . A function is a process or a relation that associates each element x of a set X , the domain of the function , to a single element y of another set Y ( possibly the same set ) , the codomain of the function . If the function is called f , this relation is denoted y = f ( x ) ( which is spoken aloud as f of x ) , the element x is the argument or input of the function , and y is the value of the function , the output , or the image of x by f. The symbol that is used for representing the input is the variable of the function ( one often says that f is a function of the variable x ) . A function is uniquely represented by the set of all pairs ( x , f ( x ) ) , called the graph of the function . [ note 2 ] When the domain and the codomain are sets of real numbers , each such pair may be considered as the Cartesian coordinates of a point in the plane . The set of these points is called the graph of the function ; it is a popular means to illustrate the function . Functions are widely used in science , and in most fields of mathematics .
A parameter ( from the Ancient Greek παρά , para : beside , subsidiary ; and μέτρον , metron : measure ) , generally , is any characteristic that can help in defining or classifying a particular system ( meaning an event , project , object , situation , etc. ) . That is , a parameter is an element of a system that is useful , or critical , when identifying the system , or when evaluating its performance , status , condition , etc . Parameter has more specific meanings within various disciplines , including mathematics , computer programming , engineering , statistics , logic and linguistics .
Vladimir Igorevich Arnold ( alternative spelling Arnol 'd , Russian : Влади́мир И́горевич Арно́льд , 12 June 1937 - 3 June 2010 ) was a Soviet and Russian mathematician . While he is best known for the Kolmogorov-Arnold-Moser theorem regarding the stability of integrable systems , he made important contributions in several areas including dynamical systems theory , algebra , catastrophe theory , topology , algebraic geometry , symplectic geometry , differential equations , classical mechanics , hydrodynamics and singularity theory , including posing the ADE classification problem , since his first main result - the solution of Hilbert 's thirteenth problem in 1957 at the age of 19 . He co-founded two new branches of mathematics - KAM theory , and topological Galois theory ( this , with his student Askold Khovanskii ) . Arnold was also known as a popularizer of mathematics . Through his lectures , seminars , and as the author of several textbooks ( such as the famous Mathematical Methods of Classical Mechanics , and Lectures on Partial Differential Equations ) and popular mathematics books , he influenced many mathematicians and physicists . Many of his books were translated into English . His views on education were particularly anti-Bourbaki .
Andrey Nikolaevich Kolmogorov ( Russian : Андре́й Никола́евич Колмого́ров , IPA : [ ɐnˈdrʲej nʲɪkɐˈlajɪvʲɪtɕ kəlmɐˈɡorəf ] ( listen ) , 25 April 1903 - 20 October 1987 ) was a Soviet mathematician who made significant contributions to the mathematics of probability theory , topology , intuitionistic logic , turbulence , classical mechanics , algorithmic information theory and computational complexity .
In mathematics , Hilbert 's fourteenth problem , that is , number 14 of Hilbert 's problems proposed in 1900 , asks whether certain algebras are finitely generated . The setting is as follows : Assume that k is a field and let K be a subfield of the field of rational functions in n variables ,
Invariant theory is a branch of abstract algebra dealing with actions of groups on algebraic varieties , such as vector spaces , from the point of view of their effect on functions . Classically , the theory dealt with the question of explicit description of polynomial functions that do not change , or are invariant , under the transformations from a given linear group . For example , if we consider the action of the special linear group SLn on the space of n by n matrices by left multiplication , then the determinant is an invariant of this action because the determinant of A X equals the determinant of X , when A is in SLn .
In algebraic geometry , an algebraic group ( or group variety ) is a group that is an algebraic variety , such that the multiplication and inversion operations are given by regular maps on the variety . In terms of category theory , an algebraic group is a group object in the category of algebraic varieties .
In mathematics , especially in the field of algebra , a polynomial ring or polynomial algebra is a ring ( which is also a commutative algebra ) formed from the set of polynomials in one or more indeterminates ( traditionally also called variables ) with coefficients in another ring , often a field . Polynomial rings occur in many parts of mathematics , and the study of their properties was among the main motivations for the development of commutative algebra and ring theory . Polynomial rings and their ideals are fundamental in algebraic geometry . Many classes of rings , such as unique factorization domains , regular rings , group rings , rings of formal power series , Ore polynomials , graded rings , are generalizations of polynomial rings . A closely related notion is that of the ring of polynomial functions on a vector space , and , more generally , ring of regular functions on an algebraic variety .
In mathematics , a finitely generated algebra ( also called an algebra of finite type ) is a commutative associative algebra A over a field K where there exists a finite set of elements a1 , . , an of A such that every element of A can be expressed as a polynomial in a1 , . , an , with coefficients in K. Equivalently , there exist elements a 1 , … , a n ∈ A { \displaystyle a_ { 1 } , \dots , a_ { n } \in A } s.t . the evaluation homomorphism at a = ( a 1 , … , a n ) { \displaystyle { \bf { a } } = ( a_ { 1 } , \dots , a_ { n } ) }
Masayoshi Nagata ( Japanese : 永田 雅宜 Nagata Masayoshi ; February 9 , 1927 - August 27 , 2008 ) was a Japanese mathematician , known for his work in the field of commutative algebra .
Hilbert 's fifteenth problem is one of the 23 Hilbert problems set out in a celebrated list compiled in 1900 by David Hilbert . The problem is to put Schubert 's enumerative calculus on a rigorous foundation .
In mathematics , Schubert calculus is a branch of algebraic geometry introduced in the nineteenth century by Hermann Schubert , in order to solve various counting problems of projective geometry ( part of enumerative geometry ) . It was a precursor of several more modern theories , for example characteristic classes , and in particular its algorithmic aspects are still of current interest . The phrase Schubert calculus is sometimes used to mean the enumerative geometry of linear subspaces , roughly equivalent to describing the cohomology ring of Grassmannians , and sometimes used to mean the more general enumerative geometry of nonlinear varieties . Even more generally , Schubert calculus is often understood to encompass the study of analogous questions in generalized cohomology theories . The objects introduced by Schubert are the Schubert cells , which are locally closed sets in a Grassmannian defined by conditions of incidence of a linear subspace in projective space with a given flag . For details see Schubert variety . The intersection theory of these cells , which can be seen as the product structure in the cohomology ring of the Grassmannian of associated cohomology classes , in principle allows the prediction of the cases where intersections of cells results in a finite set of points , which are potentially concrete answers to enumerative questions . A supporting theoretical result is that the Schubert cells ( or rather , their classes ) span the whole cohomology ring . In detailed calculations the combinatorial aspects enter as soon as the cells have to be indexed . Lifted from the Grassmannian , which is a homogeneous space , to the general linear group that acts on it , similar questions are involved in the Bruhat decomposition and classification of parabolic subgroups ( by block matrix ) . Putting Schubert 's system on a rigorous footing is Hilbert 's fifteenth problem .
Hilbert 's 16th problem was posed by David Hilbert at the Paris conference of the International Congress of Mathematicians in 1900 , as part of his list of 23 problems in mathematics . The original problem was posed as the Problem of the topology of algebraic curves and surfaces ( Problem der Topologie algebraischer Kurven und Flächen ) . Actually the problem consists of two similar problems in different branches of mathematics :
In mathematics , a real number is a value of a continuous quantity that can represent a distance along a line . The adjective real in this context was introduced in the 17th century by René Descartes , who distinguished between real and imaginary roots of polynomials . The real numbers include all the rational numbers , such as the integer −5 and the fraction 4/3 , and all the irrational numbers , such as √2 ( 1.41421356. , the square root of 2 , an irrational algebraic number ) . Included within the irrationals are the transcendental numbers , such as π ( 3.14159265. ) . In addition to measuring distance , real numbers can be used to measure quantities such as time , mass , energy , velocity , and many more . Real numbers can be thought of as points on an infinitely long line called the number line or real line , where the points corresponding to integers are equally spaced . Any real number can be determined by a possibly infinite decimal representation , such as that of 8.632 , where each consecutive digit is measured in units one tenth the size of the previous one . The real line can be thought of as a part of the complex plane , and complex numbers include real numbers .
In mathematics , an affine algebraic plane curve is the zero set of a polynomial in two variables . A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables . An affine algebraic plane curve can be completed in a projective algebraic plane curve by homogenizing its defining polynomial . Conversely , a projective algebraic plane curve of homogeneous equation h ( x , y , t ) = 0 can be restricted to the affine algebraic plane curve of equation h ( x , y , 1 ) = 0 . These two operations are each inverse to the other ; therefore , the phrase algebraic plane curve is often used without specifying explicitly whether it is the affine or the projective case that is considered . More generally , an algebraic curve is an algebraic variety of dimension one . Equivalently , an algebraic curve is an algebraic variety that is birationally equivalent to an algebraic plane curve . If the curve is contained in an affine space or a projective space , one can take a projection for such a birational equivalence . These birational equivalences reduce most of the study of algebraic curves to the study of algebraic plane curves . However , some properties are not kept under birational equivalence and must be studied on non-plane curves . This is , in particular , the case for the degree and smoothness . For example , there exist smooth curves of genus 0 and degree greater than two , but any plane projection of such curves has singular points ( see Genus-degree formula ) .
In mathematics , in the study of dynamical systems with two-dimensional phase space , a limit cycle is a closed trajectory in phase space having the property that at least one other trajectory spirals into it either as time approaches infinity or as time approaches negative infinity . Such behavior is exhibited in some nonlinear systems . Limit cycles have been used to model the behavior of a great many real world oscillatory systems . The study of limit cycles was initiated by Henri Poincaré ( 1854-1912 ) .
In vector calculus and physics , a vector field is an assignment of a vector to each point in a subset of space . A vector field in the plane ( for instance ) , can be visualised as a collection of arrows with a given magnitude and direction , each attached to a point in the plane . Vector fields are often used to model , for example , the speed and direction of a moving fluid throughout space , or the strength and direction of some force , such as the magnetic or gravitational force , as it changes from one point to another point . The elements of differential and integral calculus extend naturally to vector fields . When a vector field represents force , the line integral of a vector field represents the work done by a force moving along a path , and under this interpretation conservation of energy is exhibited as a special case of the fundamental theorem of calculus . Vector fields can usefully be thought of as representing the velocity of a moving flow in space , and this physical intuition leads to notions such as the divergence ( which represents the rate of change of volume of a flow ) and curl ( which represents the rotation of a flow ) . In coordinates , a vector field on a domain in n-dimensional Euclidean space can be represented as a vector-valued function that associates an n-tuple of real numbers to each point of the domain . This representation of a vector field depends on the coordinate system , and there is a well-defined transformation law in passing from one coordinate system to the other . Vector fields are often discussed on open subsets of Euclidean space , but also make sense on other subsets such as surfaces , where they associate an arrow tangent to the surface at each point ( a tangent vector ) . More generally , vector fields are defined on differentiable manifolds , which are spaces that look like Euclidean space on small scales , but may have more complicated structure on larger scales . In this setting , a vector field gives a tangent vector at each point of the manifold ( that is , a section of the tangent bundle to the manifold ) . Vector fields are one kind of tensor field .
Hilbert 's seventeenth problem is one of the 23 Hilbert problems set out in a celebrated list compiled in 1900 by David Hilbert . It concerns the expression of positive definite rational functions as sums of quotients of squares . The original question may be reformulated as :
In mathematics , a rational function is any function which can be defined by a rational fraction , i.e . an algebraic fraction such that both the numerator and the denominator are polynomials . The coefficients of the polynomials need not be rational numbers ; they may be taken in any field K. In this case , one speaks of a rational function and a rational fraction over K. The values of the variables may be taken in any field L containing K. Then the domain of the function is the set of the values of the variables for which the denominator is not zero and the codomain is L. The set of rational functions over a field K is a field , the field of fractions of the ring of the polynomial functions over K .
In arithmetic , a quotient ( from Latin : quotiens how many times , pronounced /ˈkwoʊʃənt/ ) is the quantity produced by the division of two numbers . The quotient has widespread use throughout mathematics , and is commonly referred to as a fraction or a ratio . For example , when dividing twenty ( the dividend ) by three ( the divisor ) , the quotient is six and two thirds . In this sense , a quotient is the ratio of a dividend to its divisor .
In mathematics , a square is the result of multiplying a number by itself . The verb to square is used to denote this operation . Squaring is the same as raising to the power 2 , and is denoted by a superscript 2 ; for instance , the square of 3 may be written as 32 , which is the number 9 . In some cases when superscripts are not available , as for instance in programming languages or plain text files , the notations x^2 or x**2 may be used in place of x2 . The adjective which corresponds to squaring is quadratic . The square of an integer may also be called a square number or a perfect square . In algebra , the operation of squaring is often generalized to polynomials , other expressions , or values in systems of mathematical values other than the numbers . For instance , the square of the linear polynomial x + 1 is the quadratic polynomial ( x+1 ) 2 = x2 + 2x + 1 . One of the important properties of squaring , for numbers as well as in many other mathematical systems , is that ( for all numbers x ) , the square of x is the same as the square of its additive inverse −x . That is , the square function satisfies the identity x2 = ( −x ) 2 . This can also be expressed by saying that the square function is an even function .
Emil Artin ( German : [ ˈaɐ̯tiːn ] ; March 3 , 1898 - December 20 , 1962 ) was an Austrian mathematician of Armenian descent . Artin was one of the leading mathematicians of the twentieth century . He is best known for his work on algebraic number theory , contributing largely to class field theory and a new construction of L-functions . He also contributed to the pure theories of rings , groups and fields .
Hilbert 's eighteenth problem is one of the 23 Hilbert problems set out in a celebrated list compiled in 1900 by mathematician David Hilbert . It asks three separate questions about lattices and sphere packing in Euclidean space .
In geometry , a shape is said to be anisohedral if it admits a tiling , but no such tiling is isohedral ( tile-transitive ) ; that is , in any tiling by that shape there are two tiles that are not equivalent under any symmetry of the tiling . A tiling by an anisohedral tile is referred to as an anisohedral tiling .
In geometry , a sphere packing is an arrangement of non-overlapping spheres within a containing space . The spheres considered are usually all of identical size , and the space is usually three-dimensional Euclidean space . However , sphere packing problems can be generalised to consider unequal spheres , spaces of other dimensions ( where the problem becomes circle packing in two dimensions , or hypersphere packing in higher dimensions ) or to non-Euclidean spaces such as hyperbolic space . A typical sphere packing problem is to find an arrangement in which the spheres fill as much of the space as possible . The proportion of space filled by the spheres is called the density of the arrangement . As the local density of a packing in an infinite space can vary depending on the volume over which it is measured , the problem is usually to maximise the average or asymptotic density , measured over a large enough volume . For equal spheres in three dimensions , the densest packing uses approximately 74% of the volume . A random packing of equal spheres generally has a density around 64% .
Karl August Reinhardt ( 27 January 1895 Frankfurt am Main - 27 April 1941 Berlin ) was a German mathematician who discovered the 5 tile-transitive pentagon tilings , solved the odd case of the biggest little polygon problem , and constructed the smoothed octagon conjectured to be the worst-packing point-symmetric planar convex shape . He also gave a partial solution to Hilbert 's eighteenth problem by discovering an anisohedral tiling in three dimensions .
A computer-assisted proof is a mathematical proof that has been at least partially generated by computer . Most computer-aided proofs to date have been implementations of large proofs-by-exhaustion of a mathematical theorem . The idea is to use a computer program to perform lengthy computations , and to provide a proof that the result of these computations implies the given theorem . In 1976 , the four color theorem was the first major theorem to be verified using a computer program . Attempts have also been made in the area of artificial intelligence research to create smaller , explicit , new proofs of mathematical theorems from the bottom up using machine reasoning techniques such as heuristic search . Such automated theorem provers have proved a number of new results and found new proofs for known theorems . [ citation needed ] Additionally , interactive proof assistants allow mathematicians to develop human-readable proofs which are nonetheless formally verified for correctness . Since these proofs are generally human-surveyable ( albeit with difficulty , as with the proof of the Robbins conjecture ) they do not share the controversial implications of computer-aided proofs-by-exhaustion .
Thomas Callister Hales ( born June 4 , 1958 ) is an American mathematician working in the areas of representation theory , discrete geometry , and formal verification . In representation theory he is known for his work on the Langlands program and the proof of the fundamental lemma over the group Sp ( 4 ) ( many of his ideas were incorporated into the final proof , due to Ngô Bảo Châu ) . In discrete geometry , he settled the Kepler conjecture on the density of sphere packings and the honeycomb conjecture . In 2014 , he announced the completion of the Flyspeck Project , which formally verified the correctness of his proof of the Kepler conjecture .
In geometry , close-packing of equal spheres is a dense arrangement of congruent spheres in an infinite , regular arrangement ( or lattice ) . Carl Friedrich Gauss proved that the highest average density - that is , the greatest fraction of space occupied by spheres - that can be achieved by a lattice packing is
Hilbert 's nineteenth problem is one of the 23 Hilbert problems , set out in a list compiled in 1900 by David Hilbert . It asks whether the solutions of regular problems in the calculus of variations are always analytic . Informally , and perhaps less directly , since Hilbert 's concept of a regular variational problem identifies precisely a variational problem whose Euler-Lagrange equation is an elliptic partial differential equation with analytic coefficients , Hilbert 's nineteenth problem , despite its seemingly technical statement , simply asks whether , in this class of partial differential equations , any solution function inherits the relatively simple and well understood structure from the solved equation . Hilbert 's nineteenth problem was solved independently in the late 1950s by Ennio De Giorgi and John Forbes Nash , Jr .
The calculus of variations is a field of mathematical analysis that uses variations , which are small changes in functions and functionals , to find maxima and minima of functionals : mappings from a set of functions to the real numbers . [ a ] Functionals are often expressed as definite integrals involving functions and their derivatives . Functions that maximize or minimize functionals may be found using the Euler-Lagrange equation of the calculus of variations . A simple example of such a problem is to find the curve of shortest length connecting two points . If there are no constraints , the solution is a straight line between the points . However , if the curve is constrained to lie on a surface in space , then the solution is less obvious , and possibly many solutions may exist . Such solutions are known as geodesics . A related problem is posed by Fermat 's principle : light follows the path of shortest optical length connecting two points , where the optical length depends upon the material of the medium . One corresponding concept in mechanics is the principle of least/stationary action . Many important problems involve functions of several variables . Solutions of boundary value problems for the Laplace equation satisfy the Dirichlet principle . Plateau 's problem requires finding a surface of minimal area that spans a given contour in space : a solution can often be found by dipping a frame in a solution of soap suds .
In mathematics , an analytic function is a function that is locally given by a convergent power series . There exist both real analytic functions and complex analytic functions , categories that are similar in some ways , but different in others . Functions of each type are infinitely differentiable , but complex analytic functions exhibit properties that do not hold generally for real analytic functions . A function is analytic if and only if its Taylor series about x0 converges to the function in some neighborhood for every x0 in its domain .
Ennio De Giorgi ( 8 February 1928 - 25 October 1996 ) was an Italian mathematician , member of the House of Giorgi , who worked on partial differential equations and the foundations of mathematics .
John Forbes Nash Jr. ( June 13 , 1928 - May 23 , 2015 ) was an American mathematician who made fundamental contributions to game theory , differential geometry , and the study of partial differential equations . Nash 's work has provided insight into the factors that govern chance and decision-making inside complex systems found in everyday life . His theories are widely used in economics . Serving as a senior research mathematician at Princeton University during the later part of his life , he shared the 1994 Nobel Memorial Prize in Economic Sciences with game theorists Reinhard Selten and John Harsanyi . In 2015 , he also shared the Abel Prize with Louis Nirenberg for his work on nonlinear partial differential equations . John Nash is the only person to be awarded both the Nobel Memorial Prize in Economic Sciences and the Abel Prize . In 1959 , Nash began showing clear signs of mental illness , and spent several years at psychiatric hospitals being treated for paranoid schizophrenia . After 1970 , his condition slowly improved , allowing him to return to academic work by the mid-1980s . His struggles with his illness and his recovery became the basis for Sylvia Nasar 's biography , A Beautiful Mind , as well as a film of the same name starring Russell Crowe as Nash . On May 23 , 2015 , Nash and his wife Alicia died in a car crash while riding in a taxi on the New Jersey Turnpike .
Hilbert 's twentieth problem is one of the 23 Hilbert problems set out in a celebrated list compiled in 1900 by David Hilbert . It asks whether all boundary value problems can be solved ( that is , do variational problems with certain boundary conditions have solutions ) .
In mathematics , in the field of differential equations , a boundary value problem is a differential equation together with a set of additional constraints , called the boundary conditions . A solution to a boundary value problem is a solution to the differential equation which also satisfies the boundary conditions . Boundary value problems arise in several branches of physics as any physical differential equation will have them . Problems involving the wave equation , such as the determination of normal modes , are often stated as boundary value problems . A large class of important boundary value problems are the Sturm-Liouville problems . The analysis of these problems involves the eigenfunctions of a differential operator . To be useful in applications , a boundary value problem should be well posed . This means that given the input to the problem there exists a unique solution , which depends continuously on the input . Much theoretical work in the field of partial differential equations is devoted to proving that boundary value problems arising from scientific and engineering applications are in fact well-posed . Among the earliest boundary value problems to be studied is the Dirichlet problem , of finding the harmonic functions ( solutions to Laplace 's equation ) ; the solution was given by the Dirichlet 's principle .