Unifying theories in mathematics
Encyclopedia
There have been several attempts in history to reach a unified theory of mathematics. Some of the greatest mathematician
s have expressed views that the whole subject should be fitted into one theory.
For example, mechanics
and mathematical analysis
were commonly combined into one subject during the 18th century, united by the differential equation
concept; while algebra
and geometry
were considered largely distinct. Now we consider analysis, algebra, and geometry, but not mechanics, as parts of mathematics because they are primarily deductive formal science
s, while mechanics like physics
must proceed from observation. There is no major loss of content, with analytical mechanics
in the old sense now expressed in terms of symplectic topology
, based on the newer theory of manifold
s.
s, axiom
s, theorem
s, examples, and so on. (Examples include group theory
, Galois theory
, control theory
, and K-theory
.) In particular there is no connotation of hypothetical. Thus the term unifying theory is more like a sociological term used to study the actions of mathematicians. It may assume nothing conjectural that would be analogous to an undiscovered scientific link. There is really no cognate within mathematics to such concepts as Proto-World
in linguistics
or the Gaia hypothesis
.
Nonetheless there have been several episodes within the history of mathematics in which sets of individual theorems were found to be special cases of a single unifying result, or in which a single perspective about how to proceed when developing an area of mathematics could be applied fruitfully to multiple branches of the subject.
, which in the hands of mathematicians such as Descartes and Fermat showed that many theorems about curve
s and surface
s of special types could be stated in algebraic language (then new), each of which could then be proved using the same techniques. That is, the theorems were very similar algebraically, even if the geometrical interpretations were distinct.
At the end of the 19th century, Felix Klein
noted that the many branches of geometry which had been developed during that century (affine geometry
, projective geometry
, hyperbolic geometry
, etc.) could all be treated in a uniform way. He did this by considering the group
s under which the objects were invariant. This unification of geometry goes by the name of the Erlangen programme.
s", such as considered by the Quaternion Society, were put onto an axiomatic footing as branches of ring theory
(in this case, with the specific meaning of associative algebra
s over the field of complex numbers.) In this context, the quotient ring
concept is one of the most powerful unifiers.
This was a general change of methodology, since the needs of applications had up until then meant that much of mathematics was taught by means of algorithm
s (or processes close to being algorithmic). Arithmetic
is still taught that way. It was a parallel to the development of mathematical logic
as a stand-alone branch of mathematics. By the 1930s symbolic logic itself was adequately included within mathematics.
In most cases, mathematical objects under study can be defined (albeit non-canonically) as sets or, more informally, as sets with additional structure such as an addition operation. Set theory
now serves as a lingua franca
for the development of mathematical themes.
over commutative ring
s, and limiting to vector space
s over the real number
s only when absolutely necessary. The story proceeded in this fashion, even when the specializations were the theorems of primary interest.
In particular, this perspective placed little value on fields of mathematics (such as combinatorics
) whose objects of study are very often special, or found in situations which can only superficially be related to more axiomatic branches of the subject.
is a unifying theory of mathematics that was initially developed in the second half of the 20th century. In this respect it is an alternative and complement to set theory. A key theme from the "categorical" point of view is that mathematics requires not only certain kinds of objects (Lie group
s, Banach space
s, etc.) but also mappings between them that preserve their structure.
In particular, this clarifies exactly what it means for mathematical objects to be considered to be the same. (For example, are all equilateral triangles the same, or does size matter?) Saunders Mac Lane
proposed that any concept with enough 'ubiquity' (occurring in various branches of mathematics) deserved isolating and studying in its own right. Category theory is arguably better adapted to that end than any other current approach. The disadvantages of relying on so-called abstract nonsense
are a certain blandness and abstraction in the sense of breaking away from the roots in concrete problems. Nevertheless, the methods of category theory have steadily advanced in acceptance, in numerous areas (from D-module
s to categorical logic
).
thoroughly develops the connections between geometric objects (algebraic varieties, or more generally scheme
s) and algebraic ones (ideal
s); the touchstone result here is Hilbert's Nullstellensatz
which roughly speaking shows that there is a natural one-to-one correspondence between the two types of objects.
One may view other theorems in the same light. For example the fundamental theorem of Galois theory
asserts that there is a one-to-one correspondence between extensions of a field and subgroups of the field's Galois group
. The Taniyama-Shimura conjecture for elliptic curves (now proven) establishes a one-to-one correspondence between curves defined as modular forms and elliptic curve
s defined over the rational number
s. A research area sometimes nicknamed Monstrous Moonshine
developed connections between modular forms and the finite simple group known as the Monster
, starting solely with the surprise observation that in each of them the rather unusual number 196884 would arise very naturally. Another field, known as the Langlands program
, likewise starts with apparently haphazard similarities (in this case, between number-theoretical results and representations of certain groups) and looks for constructions from which both sets of results would be corollaries.
We illustrate the concept by discussing some of these themes in detail.
over the rational numbers can be translated into a modular form
(in such a way as to preserve the associated L-function
). There are difficulties in identifying this with an isomorphism, in any strict sense of the word. Certain curves had been known to be both elliptic curves (of genus
1) and modular curve
s, before the conjecture was formulated (about 1955). The surprising part of the conjecture was the extension to factors of Jacobian
s of modular curves of genus > 1. It had probably not seemed plausible that there would be 'enough' such rational factors, before the conjecture was enunciated; and in fact the numerical evidence was slight until around 1970, when tables began to confirm it. The case of elliptic curves with complex multiplication
was proved by Shimura in 1964. This conjecture stood for decades before being proved in generality.
In fact the Langlands program
(or philosophy) is much more like a web of unifying conjectures; it really does postulate that the general theory of automorphic form
s is regulated by the L-group
s introduced by Robert Langlands
. His principle of functoriality with respect to the L-group has a very large explanatory value with respect to known types of lifting of automorphic forms (now more broadly studied as automorphic representations). While this theory is in one sense closely linked with the Taniyama-Shimura conjecture, it should be understood that the conjecture actually operates in the opposite direction. It requires the existence of an automorphic form, starting with an object that (very abstractly) lies in a category of motives.
Another significant related point is that the Langlands approach stands apart from the whole development triggered by monstrous moonshine
(connections between elliptic modular functions as Fourier series
, and the group representation
s of the Monster group
and other sporadic group
s). The Langlands philosophy neither foreshadowed nor was able to include this line of research.
. The Baum-Connes conjecture, now a long-standing problem, has been joined by others in a group known as the isomorphism conjectures in K-theory. These include the Farrell-Jones conjecture and Bost conjecture.
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
s have expressed views that the whole subject should be fitted into one theory.
Historical perspective
The process of unification might be seen as helping to define what constitutes mathematics as a discipline.For example, mechanics
Mechanics
Mechanics is the branch of physics concerned with the behavior of physical bodies when subjected to forces or displacements, and the subsequent effects of the bodies on their environment....
and mathematical analysis
Mathematical analysis
Mathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of infinitesimal calculus. It is a branch of pure mathematics that includes the theories of differentiation, integration and measure, limits, infinite series, and analytic functions...
were commonly combined into one subject during the 18th century, united by the differential equation
Differential equation
A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders...
concept; while algebra
Algebra
Algebra is the branch of mathematics concerning the study of the rules of operations and relations, and the constructions and concepts arising from them, including terms, polynomials, equations and algebraic structures...
and geometry
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....
were considered largely distinct. Now we consider analysis, algebra, and geometry, but not mechanics, as parts of mathematics because they are primarily deductive formal science
Formal science
The formal sciences are the branches of knowledge that are concerned with formal systems, such as logic, mathematics, theoretical computer science, information theory, systems theory, decision theory, statistics, and some aspects of linguistics....
s, while mechanics like physics
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
must proceed from observation. There is no major loss of content, with analytical mechanics
Analytical mechanics
Analytical mechanics is a term used for a refined, mathematical form of classical mechanics, constructed from the 18th century onwards as a formulation of the subject as founded by Isaac Newton. Often the term vectorial mechanics is applied to the form based on Newton's work, to contrast it with...
in the old sense now expressed in terms of symplectic topology
Symplectic topology
Symplectic geometry is a branch of differential geometry and differential topology which studies symplectic manifolds; that is, differentiable manifolds equipped with a closed, nondegenerate 2-form...
, based on the newer theory of manifold
Manifold
In mathematics , a manifold is a topological space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the manifold....
s.
Mathematical theories
The term theory is used informally within mathematics to mean a self-consistent body of definitionDefinition
A definition is a passage that explains the meaning of a term , or a type of thing. The term to be defined is the definiendum. A term may have many different senses or meanings...
s, axiom
Axiom
In traditional logic, an axiom or postulate is a proposition that is not proven or demonstrated but considered either to be self-evident or to define and delimit the realm of analysis. In other words, an axiom is a logical statement that is assumed to be true...
s, theorem
Theorem
In mathematics, a theorem is a statement that has been proven on the basis of previously established statements, such as other theorems, and previously accepted statements, such as axioms...
s, examples, and so on. (Examples include group theory
Group theory
In mathematics and abstract algebra, group theory studies the algebraic structures known as groups.The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces can all be seen as groups endowed with additional operations and...
, Galois theory
Galois theory
In mathematics, more specifically in abstract algebra, Galois theory, named after Évariste Galois, provides a connection between field theory and group theory...
, control theory
Control theory
Control theory is an interdisciplinary branch of engineering and mathematics that deals with the behavior of dynamical systems. The desired output of a system is called the reference...
, and K-theory
K-theory
In mathematics, K-theory originated as the study of a ring generated by vector bundles over a topological space or scheme. In algebraic topology, it is an extraordinary cohomology theory known as topological K-theory. In algebra and algebraic geometry, it is referred to as algebraic K-theory. It...
.) In particular there is no connotation of hypothetical. Thus the term unifying theory is more like a sociological term used to study the actions of mathematicians. It may assume nothing conjectural that would be analogous to an undiscovered scientific link. There is really no cognate within mathematics to such concepts as Proto-World
Proto-World language
The Proto-Human language is the hypothetical most recent common ancestor of all the world's languages.The concept of "Proto-Human" presupposes monogenesis of all recorded spoken human languages....
in linguistics
Linguistics
Linguistics is the scientific study of human language. Linguistics can be broadly broken into three categories or subfields of study: language form, language meaning, and language in context....
or the Gaia hypothesis
Gaia hypothesis
The Gaia hypothesis, also known as Gaia theory or Gaia principle, proposes that all organisms and their inorganic surroundings on Earth are closely integrated to form a single and self-regulating complex system, maintaining the conditions for life on the planet.The scientific investigation of the...
.
Nonetheless there have been several episodes within the history of mathematics in which sets of individual theorems were found to be special cases of a single unifying result, or in which a single perspective about how to proceed when developing an area of mathematics could be applied fruitfully to multiple branches of the subject.
Geometrical theories
A well-known example was the development of analytic geometryAnalytic geometry
Analytic geometry, or analytical geometry has two different meanings in mathematics. The modern and advanced meaning refers to the geometry of analytic varieties...
, which in the hands of mathematicians such as Descartes and Fermat showed that many theorems about curve
Curve
In mathematics, a curve is, generally speaking, an object similar to a line but which is not required to be straight...
s and surface
Surface
In mathematics, specifically in topology, a surface is a two-dimensional topological manifold. The most familiar examples are those that arise as the boundaries of solid objects in ordinary three-dimensional Euclidean space R3 — for example, the surface of a ball...
s of special types could be stated in algebraic language (then new), each of which could then be proved using the same techniques. That is, the theorems were very similar algebraically, even if the geometrical interpretations were distinct.
At the end of the 19th century, Felix Klein
Felix Klein
Christian Felix Klein was a German mathematician, known for his work in group theory, function theory, non-Euclidean geometry, and on the connections between geometry and group theory...
noted that the many branches of geometry which had been developed during that century (affine geometry
Affine geometry
In mathematics affine geometry is the study of geometric properties which remain unchanged by affine transformations, i.e. non-singular linear transformations and translations...
, projective geometry
Projective geometry
In mathematics, projective geometry is the study of geometric properties that are invariant under projective transformations. This means that, compared to elementary geometry, projective geometry has a different setting, projective space, and a selective set of basic geometric concepts...
, hyperbolic geometry
Hyperbolic geometry
In mathematics, hyperbolic geometry is a non-Euclidean geometry, meaning that the parallel postulate of Euclidean geometry is replaced...
, etc.) could all be treated in a uniform way. He did this by considering the group
Group theory
In mathematics and abstract algebra, group theory studies the algebraic structures known as groups.The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces can all be seen as groups endowed with additional operations and...
s under which the objects were invariant. This unification of geometry goes by the name of the Erlangen programme.
Through-axiomatisation
Early in the 20th century, many parts of mathematics began to be treated by delineating useful sets of axioms and then studying their consequences. Thus, for example, the studies of "hypercomplex numberHypercomplex number
In mathematics, a hypercomplex number is a traditional term for an element of an algebra over a field where the field is the real numbers or the complex numbers. In the nineteenth century number systems called quaternions, tessarines, coquaternions, biquaternions, and octonions became established...
s", such as considered by the Quaternion Society, were put onto an axiomatic footing as branches of ring theory
Ring theory
In abstract algebra, ring theory is the study of rings—algebraic structures in which addition and multiplication are defined and have similar properties to those familiar from the integers...
(in this case, with the specific meaning of associative algebra
Associative algebra
In mathematics, an associative algebra A is an associative ring that has a compatible structure of a vector space over a certain field K or, more generally, of a module over a commutative ring R...
s over the field of complex numbers.) In this context, the quotient ring
Quotient ring
In ring theory, a branch of modern algebra, a quotient ring, also known as factor ring or residue class ring, is a construction quite similar to the factor groups of group theory and the quotient spaces of linear algebra...
concept is one of the most powerful unifiers.
This was a general change of methodology, since the needs of applications had up until then meant that much of mathematics was taught by means of algorithm
Algorithm
In mathematics and computer science, an algorithm is an effective method expressed as a finite list of well-defined instructions for calculating a function. Algorithms are used for calculation, data processing, and automated reasoning...
s (or processes close to being algorithmic). Arithmetic
Arithmetic
Arithmetic or arithmetics is the oldest and most elementary branch of mathematics, used by almost everyone, for tasks ranging from simple day-to-day counting to advanced science and business calculations. It involves the study of quantity, especially as the result of combining numbers...
is still taught that way. It was a parallel to the development of mathematical logic
Mathematical logic
Mathematical logic is a subfield of mathematics with close connections to foundations of mathematics, theoretical computer science and philosophical logic. The field includes both the mathematical study of logic and the applications of formal logic to other areas of mathematics...
as a stand-alone branch of mathematics. By the 1930s symbolic logic itself was adequately included within mathematics.
In most cases, mathematical objects under study can be defined (albeit non-canonically) as sets or, more informally, as sets with additional structure such as an addition operation. Set theory
Set theory
Set theory is the branch of mathematics that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics...
now serves as a lingua franca
Lingua franca
A lingua franca is a language systematically used to make communication possible between people not sharing a mother tongue, in particular when it is a third language, distinct from both mother tongues.-Characteristics:"Lingua franca" is a functionally defined term, independent of the linguistic...
for the development of mathematical themes.
Bourbaki
The cause of axiomatic development was taken up in earnest by the Bourbaki group of mathematicians. Taken to its extreme, this attitude was thought to demand mathematics developed in its greatest generality. One started from the most general axioms, and then specialized, for example, by introducing modulesModule (mathematics)
In abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, wherein the corresponding scalars are allowed to lie in an arbitrary ring...
over commutative ring
Commutative ring
In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra....
s, and limiting to vector space
Vector space
A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
s over the real number
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
s only when absolutely necessary. The story proceeded in this fashion, even when the specializations were the theorems of primary interest.
In particular, this perspective placed little value on fields of mathematics (such as combinatorics
Combinatorics
Combinatorics is a branch of mathematics concerning the study of finite or countable discrete structures. Aspects of combinatorics include counting the structures of a given kind and size , deciding when certain criteria can be met, and constructing and analyzing objects meeting the criteria ,...
) whose objects of study are very often special, or found in situations which can only superficially be related to more axiomatic branches of the subject.
Category theory as a rival
Category theoryCategory theory
Category theory is an area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of objects and arrows , where these collections satisfy certain basic conditions...
is a unifying theory of mathematics that was initially developed in the second half of the 20th century. In this respect it is an alternative and complement to set theory. A key theme from the "categorical" point of view is that mathematics requires not only certain kinds of objects (Lie group
Lie group
In mathematics, a Lie group is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure...
s, Banach space
Banach space
In mathematics, Banach spaces is the name for complete normed vector spaces, one of the central objects of study in functional analysis. A complete normed vector space is a vector space V with a norm ||·|| such that every Cauchy sequence in V has a limit in V In mathematics, Banach spaces is the...
s, etc.) but also mappings between them that preserve their structure.
In particular, this clarifies exactly what it means for mathematical objects to be considered to be the same. (For example, are all equilateral triangles the same, or does size matter?) Saunders Mac Lane
Saunders Mac Lane
Saunders Mac Lane was an American mathematician who cofounded category theory with Samuel Eilenberg.-Career:...
proposed that any concept with enough 'ubiquity' (occurring in various branches of mathematics) deserved isolating and studying in its own right. Category theory is arguably better adapted to that end than any other current approach. The disadvantages of relying on so-called abstract nonsense
Abstract nonsense
In mathematics, abstract nonsense, general abstract nonsense, and general nonsense are terms used facetiously by some mathematicians to describe certain kinds of arguments and methods related to category theory. roughly speaking, category theory is the study of the general form of mathematical...
are a certain blandness and abstraction in the sense of breaking away from the roots in concrete problems. Nevertheless, the methods of category theory have steadily advanced in acceptance, in numerous areas (from D-module
D-module
In mathematics, a D-module is a module over a ring D of differential operators. The major interest of such D-modules is as an approach to the theory of linear partial differential equations...
s to categorical logic
Categorical logic
Categorical logic is a branch of category theory within mathematics, adjacent to mathematical logic but more notable for its connections to theoretical computer science. In broad terms, categorical logic represents both syntax and semantics by a category, and an interpretation by a functor...
).
Uniting theories
On a less grandiose scale, there are frequent instances in which it appears that sets of results in two different branches of mathematics are similar, and one might ask whether there is a unifying framework which clarifies the connections. We have already noted the example of analytic geometry, and more generally the field of algebraic geometryAlgebraic geometry
Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...
thoroughly develops the connections between geometric objects (algebraic varieties, or more generally scheme
Scheme (mathematics)
In mathematics, a scheme is an important concept connecting the fields of algebraic geometry, commutative algebra and number theory. Schemes were introduced by Alexander Grothendieck so as to broaden the notion of algebraic variety; some consider schemes to be the basic object of study of modern...
s) and algebraic ones (ideal
Ideal (ring theory)
In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring. The ideal concept allows the generalization in an appropriate way of some important properties of integers like "even number" or "multiple of 3"....
s); the touchstone result here is Hilbert's Nullstellensatz
Hilbert's Nullstellensatz
Hilbert's Nullstellensatz is a theorem which establishes a fundamental relationship between geometry and algebra. This relationship is the basis of algebraic geometry, an important branch of mathematics. It relates algebraic sets to ideals in polynomial rings over algebraically closed fields...
which roughly speaking shows that there is a natural one-to-one correspondence between the two types of objects.
One may view other theorems in the same light. For example the fundamental theorem of Galois theory
Fundamental theorem of Galois theory
In mathematics, the fundamental theorem of Galois theory is a result that describes the structure of certain types of field extensions.In its most basic form, the theorem asserts that given a field extension E /F which is finite and Galois, there is a one-to-one correspondence between its...
asserts that there is a one-to-one correspondence between extensions of a field and subgroups of the field's Galois group
Galois group
In mathematics, more specifically in the area of modern algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension...
. The Taniyama-Shimura conjecture for elliptic curves (now proven) establishes a one-to-one correspondence between curves defined as modular forms and elliptic curve
Elliptic curve
In mathematics, an elliptic curve is a smooth, projective algebraic curve of genus one, on which there is a specified point O. An elliptic curve is in fact an abelian variety — that is, it has a multiplication defined algebraically with respect to which it is a group — and O serves as the identity...
s defined over the rational number
Rational number
In mathematics, a rational number is any number that can be expressed as the quotient or fraction a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a rational number...
s. A research area sometimes nicknamed Monstrous Moonshine
Monstrous moonshine
In mathematics, monstrous moonshine, or moonshine theory, is a term devised by John Horton Conway and Simon P. Norton in 1979, used to describe the connection between the monster group M and modular functions .- History :Specifically, Conway and Norton, following an initial observationby John...
developed connections between modular forms and the finite simple group known as the Monster
Monster group
In the mathematical field of group theory, the Monster group M or F1 is a group of finite order:...
, starting solely with the surprise observation that in each of them the rather unusual number 196884 would arise very naturally. Another field, known as the Langlands program
Langlands program
The Langlands program is a web of far-reaching and influential conjectures that relate Galois groups in algebraic number theory to automorphic forms and representation theory of algebraic groups over local fields and adeles. It was proposed by ....
, likewise starts with apparently haphazard similarities (in this case, between number-theoretical results and representations of certain groups) and looks for constructions from which both sets of results would be corollaries.
Reference list of major unifying concepts
A short list of these theories might include:- Cartesian geometry
- CalculusCalculusCalculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modern mathematics education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental theorem...
- Complex analysisComplex analysisComplex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is useful in many branches of mathematics, including number theory and applied mathematics; as well as in physics,...
- Galois theoryGalois theoryIn mathematics, more specifically in abstract algebra, Galois theory, named after Évariste Galois, provides a connection between field theory and group theory...
- Erlangen programme
- Lie groupLie groupIn mathematics, a Lie group is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure...
- Set theorySet theorySet theory is the branch of mathematics that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics...
- Hilbert spaceHilbert spaceThe mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions...
- Computable functionComputable functionComputable functions are the basic objects of study in computability theory. Computable functions are the formalized analogue of the intuitive notion of algorithm. They are used to discuss computability without referring to any concrete model of computation such as Turing machines or register...
- Characteristic classCharacteristic classIn mathematics, a characteristic class is a way of associating to each principal bundle on a topological space X a cohomology class of X. The cohomology class measures the extent to which the bundle is "twisted" — particularly, whether it possesses sections or not...
es - Homological algebraHomological algebraHomological algebra is the branch of mathematics which studies homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology and abstract algebra at the end of the 19th century, chiefly by Henri Poincaré and...
- Homotopy theory
- Grothendieck's schemesScheme (mathematics)In mathematics, a scheme is an important concept connecting the fields of algebraic geometry, commutative algebra and number theory. Schemes were introduced by Alexander Grothendieck so as to broaden the notion of algebraic variety; some consider schemes to be the basic object of study of modern...
- Langlands programLanglands programThe Langlands program is a web of far-reaching and influential conjectures that relate Galois groups in algebraic number theory to automorphic forms and representation theory of algebraic groups over local fields and adeles. It was proposed by ....
- Non-commutative geometry.
We illustrate the concept by discussing some of these themes in detail.
Recent developments in relation with modular theory
A well-known example is the Taniyama-Shimura conjecture, now the modularity theorem, which proposed that each elliptic curveElliptic curve
In mathematics, an elliptic curve is a smooth, projective algebraic curve of genus one, on which there is a specified point O. An elliptic curve is in fact an abelian variety — that is, it has a multiplication defined algebraically with respect to which it is a group — and O serves as the identity...
over the rational numbers can be translated into a modular form
Modular form
In mathematics, a modular form is a analytic function on the upper half-plane satisfying a certain kind of functional equation and growth condition. The theory of modular forms therefore belongs to complex analysis but the main importance of the theory has traditionally been in its connections...
(in such a way as to preserve the associated L-function
L-function
The theory of L-functions has become a very substantial, and still largely conjectural, part of contemporary analytic number theory. In it, broad generalisations of the Riemann zeta function and the L-series for a Dirichlet character are constructed, and their general properties, in most cases...
). There are difficulties in identifying this with an isomorphism, in any strict sense of the word. Certain curves had been known to be both elliptic curves (of genus
Genus (mathematics)
In mathematics, genus has a few different, but closely related, meanings:-Orientable surface:The genus of a connected, orientable surface is an integer representing the maximum number of cuttings along non-intersecting closed simple curves without rendering the resultant manifold disconnected. It...
1) and modular curve
Modular curve
In number theory and algebraic geometry, a modular curve Y is a Riemann surface, or the corresponding algebraic curve, constructed as a quotient of the complex upper half-plane H by the action of a congruence subgroup Γ of the modular group of integral 2×2 matrices SL...
s, before the conjecture was formulated (about 1955). The surprising part of the conjecture was the extension to factors of Jacobian
Jacobian variety
In mathematics, the Jacobian variety J of a non-singular algebraic curve C of genus g is the moduli space of degree 0 line bundles...
s of modular curves of genus > 1. It had probably not seemed plausible that there would be 'enough' such rational factors, before the conjecture was enunciated; and in fact the numerical evidence was slight until around 1970, when tables began to confirm it. The case of elliptic curves with complex multiplication
Complex multiplication
In mathematics, complex multiplication is the theory of elliptic curves E that have an endomorphism ring larger than the integers; and also the theory in higher dimensions of abelian varieties A having enough endomorphisms in a certain precise sense In mathematics, complex multiplication is the...
was proved by Shimura in 1964. This conjecture stood for decades before being proved in generality.
In fact the Langlands program
Langlands program
The Langlands program is a web of far-reaching and influential conjectures that relate Galois groups in algebraic number theory to automorphic forms and representation theory of algebraic groups over local fields and adeles. It was proposed by ....
(or philosophy) is much more like a web of unifying conjectures; it really does postulate that the general theory of automorphic form
Automorphic form
In mathematics, the general notion of automorphic form is the extension to analytic functions, perhaps of several complex variables, of the theory of modular forms...
s is regulated by the L-group
L-group
In mathematics, L-group may refer to the following groups:* The Langlands dual, LG, of a reductive algebraic group G* A group in L-theory, L...
s introduced by Robert Langlands
Robert Langlands
Robert Phelan Langlands is a mathematician, best known as the founder of the Langlands program. He is an emeritus professor at the Institute for Advanced Study...
. His principle of functoriality with respect to the L-group has a very large explanatory value with respect to known types of lifting of automorphic forms (now more broadly studied as automorphic representations). While this theory is in one sense closely linked with the Taniyama-Shimura conjecture, it should be understood that the conjecture actually operates in the opposite direction. It requires the existence of an automorphic form, starting with an object that (very abstractly) lies in a category of motives.
Another significant related point is that the Langlands approach stands apart from the whole development triggered by monstrous moonshine
Monstrous moonshine
In mathematics, monstrous moonshine, or moonshine theory, is a term devised by John Horton Conway and Simon P. Norton in 1979, used to describe the connection between the monster group M and modular functions .- History :Specifically, Conway and Norton, following an initial observationby John...
(connections between elliptic modular functions as Fourier series
Fourier series
In mathematics, a Fourier series decomposes periodic functions or periodic signals into the sum of a set of simple oscillating functions, namely sines and cosines...
, and the group representation
Group representation
In the mathematical field of representation theory, group representations describe abstract groups in terms of linear transformations of vector spaces; in particular, they can be used to represent group elements as matrices so that the group operation can be represented by matrix multiplication...
s of the Monster group
Monster group
In the mathematical field of group theory, the Monster group M or F1 is a group of finite order:...
and other sporadic group
Sporadic group
In the mathematical field of group theory, a sporadic group is one of the 26 exceptional groups in the classification of finite simple groups. A simple group is a group G that does not have any normal subgroups except for the subgroup consisting only of the identity element, and G itself...
s). The Langlands philosophy neither foreshadowed nor was able to include this line of research.
Isomorphism conjectures in K-theory
Another case, which so far is less well-developed but covers a wide range of mathematics, is the conjectural basis of some parts of K-theoryK-theory
In mathematics, K-theory originated as the study of a ring generated by vector bundles over a topological space or scheme. In algebraic topology, it is an extraordinary cohomology theory known as topological K-theory. In algebra and algebraic geometry, it is referred to as algebraic K-theory. It...
. The Baum-Connes conjecture, now a long-standing problem, has been joined by others in a group known as the isomorphism conjectures in K-theory. These include the Farrell-Jones conjecture and Bost conjecture.