"The grand unified theory of mathematics" (Edward Frenkel).
"A profound and far-reaching view of mathematics" (Kenneth Ribet).
"A program for the research of the future" (Stephen Gelbart).
The Langlands Program
Robert Langlands is currently professor emeritus of mathematics at the Institute for Advanced Study at Princeton. He occupies the same office that Einstein used.
The so-called "Laglands program" was born in January 1967, when Langlands, only 30 years old, sent a 17-page handwritten letter to André Weil asking for his opinion on some new mathematical ideas, ideas that were basically conjectures relating different mathematical fields. André Weil was a French mathematician one of the great figures of 20th century mathematics and one of the founding members of the Bourbaki group who made notable contributions to number theory and algebraic geometry.
The letter to Weil read as follows:
Professor Weil: In response to your invitation to give a talk, I enclose the following letter. After writing it I have realized that it hardly contains a single statement of which I am certain. If you read it as pure speculation I shall be grateful; if not, I am sure you will find a wastebasket handy. Yours sincerely, R. Langlans.
Bill Casselman - a Langlands&minus contributor; states that the letter contained "... a collection of far-reaching and extraordinary precise conjectures concerning number theory, automorphic forms, and representation theory. These have formed the nucleus of a program which is still in progress and which has to play a central role in all three subjects."
Weil obtained a typed version of the letter, and this version circulated widely among mathematicians interested in the topics discussed in the letter. Although Weil never replied to him, Langlands' letter became a kind of Magna Carta-dubbed "Langlands' Conjectures− leading to a new area of mathematical research that attempted to interrelate all fields of mathematics.
Langlands' program (or philosophy) is a set of conjectures aimed at achieving a unified theory relating all branches of mathematics: number theory, representation theory, geometry, algebra, analysis, and so on. In mathematics, a conjecture is a statement that is intuited to be true but has not yet been formally proved. Once a proof is found, the conjecture becomes a theorem.
This philosophy was already applied in analytic geometry, a discipline created or discovered by Descartes that relates algebra and geometry. For example, a circle and a line have algebraic equations and their points of intersection correspond to the solutions of both equations, which can have 0, 1 or 2 solutions.
The Langlands program is considered by many authors as "the great unified theory of mathematics" that will make it possible to relate all mathematics.
Langlands' program is currently a field of great activity, focusing mainly on number theory, harmonic analysis, geometry, representation theory and mathematical physics. It has earned Fields medals for several mathematicians. The Fields Medal is the highest award in mathematics, awarded every 4 years by the International Mathematical Union, and is considered the equivalent of the Nobel Prize.
The roots of the Langlands program
The Langlands program has its roots in symmetry theory, the foundations of which were laid by Évariste Galois with his notion of mathematical group structure. There are many types of groups, but the ones Langlands uses are precisely the Galois groups, the groups relating the solutions of polynomial equations.
Langlands' program initially focused on relating the representation theory of Galois groups to a mathematical field called harmonic analysis. The correspondence between these two fields would allow a concept, subject, or problem to be transferred from one field to the other. These two areas, which were apparently totally different, became closely related areas.
The union of these two branches of mathematics arose in part from efforts made to find patterns for decomposing natural numbers as sums of products of other integers. For example: 13 = 3•3 + 2•2 = 3•4 + 1 = 2•5 + 3.
Representation theory is a branch of mathematics that represents algebraic structures by transformations in a vector space. The vector space is called "representation space" and can be defined by natural numbers, real numbers, complex numbers, p-adic numbers, etc. The p-adic numbers are numbers written using a prime number as a base, for example, the number 35 is 100011 in base 2.
Galois group representations are like the "source code" of the number field that carries all the essential information about the numbers.
Harmonic analysis studies harmonics, which are waves whose frequencies are multiples of a fundamental frequency. A sound wave is a superposition of harmonics. Mathematically this means that a function can be expressed as the superposition of functions describing harmonics, such as the familiar sine and cosine functions.
Langlands conjectured that difficult questions in number theory could be solved more simply by using methods from harmonic analysis and, more specifically, with automorphic functions. According to Langlands, automorphic functions allow us to discover the "hidden harmony" existing in the apparent chaos of numbers, in the form of patterns full of symmetry and harmony.
Automorphic functions were discovered by Henri Poincaré −universalist mathematician and philosopher of science−, who named them "Fuchsian" (after the mathematician Lazarus Fuchs). An automorphic function is a complex function f(z) of complex variable z which is invariant under a numerable group of algebraic transformations of the type z' = (az + b)/(cz + d), also called Moebius transformations.
Automorphic functions are a generalization of trigonometric functions and elliptic functions or a generalization of periodic functions from Euclidean space into topological space. An elliptic function is a function defined on the complex plane and periodic in both directions, which have their origin in the calculation of the length of an arc of an ellipse.
A special case of automorphic functions are modular forms, mathematical entities of very high level of abstraction. Modular forms are analytic functions in the complex plane. A modular form is a symmetric object that remains unchanged in the face of particular transformations. It is defined by two axes of symmetry, where each axis has a real and an imaginary part. Modular shapes inhabit a four-dimensional space called "hyperbolic space". Modular forms appear in many areas, such as algebraic topology and string theory.
Langlands' program has "colonized" the traditional theory of automorphic functions. Many scattered topics of this theory have been "explained" a posteriori in terms of number theory. It has also invaded number theory, so that any study of number theory always "stumbles" into Langlands' program.
The conjectures of Langlands' program
The most salient conjectures of the Langlands program are:
The Shimura-Taniyama-Weil conjecture, which relates two seemingly distinct types of mathematical objects: elliptic curves and modular forms. The conjecture is: "To every elliptic curve with rational coefficients corresponds a modular form". This is usually simplified by saying that "All elliptic curves are modular". More precisely: to every elliptic curve in Euclidean geometry corresponds a modular form in hyperbolic geometry.
The name "elliptic curve" has nothing to do with ellipses. Nor does it have anything to do with elliptic functions, which are functions that were developed to study the perimeter of an ellipse.
The conjecture was finally proved in 2001. Today this conjecture is called the "modularity theorem". In 1986, Kenneth Ribet proved that Fermat's last theorem is a consequence of the Shimura-Taniyama-Weil conjecture. This conjecture was used by Andrew Wiles as the basis for the proof of Fermat's last theorem. This theorem states that there are no three integers x, y, z such that xn+yn = zn para n>2.
The union of two seemingly different branches of mathematics: representations of Galois groups and automorphic forms. This was the origin of Langlands' program.
Langlands' program has been extended to continuous groups or Lie groups (Galois groups are discrete). These are the so-called "dual Langlands groups". These are Lie groups that are dual to each other, one being the representation of the other.
The conjecture that all Riemann zeta functions arising in number theory are particular cases of L functions, which are a type of automorphic functions. In other words, the L functions are a generalization of the Riemann zeta function, or that the Riemann zeta function is the most representative example of the L functions.
The conjecture that there are relations between harmonic analysis and Riemann surfaces.
A Riemann surface −also known as a "complex curve"− is a differentiable variety (manifold) complex of a complex variable that allows to measure distances and angles, and which is a conformal structure: a correspondence of the complex plane on itself that preserves angles. For example, the Mercator map is a conformal map of the Earth's surface.
A variety is a geometric object that generalizes the intuitive notion of curve (1-variety) and surface (2-variety). A variety is a topological space that behaves locally (intrinsically) like a Euclidean space. Each variety has a dimension, which is the number of coordinates (or parameters) needed in the local coordinate system. For example, a 3D sphere (3-sphere) is a 2-dimensional variety (2-variety). A Riemannian manifold of dimension 4 was the one used by Einstein to model space-time in the theory of relativity.
A simple example of a Riemann surface is a sphere. Other examples are closed surfaces with nholes (n>>0). The number of holes in a closed surface is called the "genus" of that surface.
On Riemann surfaces there are also group structures: the so-called "fundamental groups", which are continuous groups (Lie groups). The concept of a fundamental group of a Riemann surface is one of the most important concepts in topology. It is a set of closed paths (or closed curves) on the Riemann surface that begin and end at the same point. The operation defined on two closed paths is the "sum of paths" which consists of traversing the first path n times and then traversing the second path m times. You thus get a new path that also starts and ends at the same point.
There is a deep analogy between the Galois groups and the fundamental groups of the Riemann surface. But in this case one has to replace the automorphic functions by other more sophisticated mathematical objects called "bundles" (sheafs). This is called the "geometric Langlands program" and was proposed by Vladimir Drinfeld in the 1980s, who won the Fields medal in 1990.
André Weil's analogical view of mathematics
André Weil, while in prison for not obeying his "military obligations", wrote on March 26, 1940 a 14-page letter to his younger sister Simone, a famous philosopher, humanist and activist. In his letter he explained in overly technical terms (which his sister could not understand) the important role to be played by analogy in mathematics, especially between number theory, function theory and geometry.
Weil spoke in the letter of "the Rosetta stone" of mathematics constituted by three fields that he tried to relate:
Number theory.
Curves over finite fields.
A finite field is a finite set of numbers {0, 1, 2,..., p}, where p is a prime number with the operations of sum and product modulo p.
Riemann surfaces.
Precisely the original Langlands program was developed in the first two fields. Weil intuited that there was a third field (geometry), a subject which −as we have remarked− was later developed by Vladimir Drinfeld.
Weil realized that an algebraic equation, depending on the domain under consideration, gives rise to a set of points, a curve, or a surface. But in all cases the expression is the same. So a result in one of the domains can be transferred to the other two. Weil spoke of establishing "bridges" as a result of analogies between the fields. In the letter, Weil said, "My job is to decipher a trilingual text of which I have only scattered fragments."
MENTAL vs. Langlands program
The Langlands program has many limitations:
It claims to relate almost all fields of mathematics, but in principle its current network of conjectures contemplates only a few fields: number theory, harmonic analysis, group representations, and complex algebraic geometry.
It is not really a unified theory based on universal principles. Langlands' strategy is a horizontal one, bridging different fields, rather than seeking a common foundation from deep within that gives rise (vertically) to all of mathematics. There are no what we can call "universal conjectures" that relate all fields. Moreover, it is necessary to "adapt" the conjectures to each of the fields.
It is based on a set of conjectures to try to link mathematical fields, conjectures that may or may not be true. It is also a set of ad hoc methods for solving particular questions.
Some conjectures are ambiguous or depend on objects whose existence has not been proved.
Conjectures have been evolving since they were established in 1967.
The proofs of the conjectures are very complex. Some mathematicians think it would take centuries to complete the program. For example, the Shimura-Taniyama-Weil conjecture took many years to prove. And there are, in theory, many similar conjectures in difficulty.
The field of automorphic functions is of great complexity and from complexity it is very difficult to unify. Unification can only be done from simplicity.
The conjectures are excessively technical, not very conceptual, very far from what is called "humanistic mathematics".
It does not provide a mathematical language, which is the key to the unification of mathematics.
It is only a theory. It does not contemplate practice.
On the other hand, with MENTAL:
Everything is born already unified, for all are manifestations of the primary archetypes, in a process of descending causality, from the great principles, the universal principles to the particular truths and expressions. All fields of mathematics are related from the deep and fundamental level. The boundaries between the different fields of mathematics are thus blurred.
It is more difficult to relate fields at the horizontal level than at the top-down vertical level (from the generic to the specific).
There is no need to "translate" a result from one mathematical field to another. What we have to do is to change the interpretation of expressions at a superficial level because at a deep level the semantics is unique (primary). With a universal language it is easier to see the connections between the different fields by using the same primary or fundamental concepts.
It provides a universal model theory that leads to the establishment of universal laws. The theory of models was born with the axiomatic systems, but with MENTAL it is universalized: every expression admits interpretations.
It is a universal formal language that integrates theory and practice.
It is a new way of thinking, more universal, humanistic, natural and simple, beyond technical virtuosity. Simple approaches are the only way to unification and awareness of the whole.
It is the supreme conceptual abstraction. In fact, it transcends mathematics itself. Mathematics is a manifestation of MENTAL, as it is of the other formal sciences (computer science, artificial intelligence, linguistics, etc.). MENTAL provides a common foundation for these disciplines.
It unites mathematics and meta-mathematics.
Everything is simpler. The complex must be built from the simple. Presumably, with MENTAL, the proofs of conjectures and of mathematics in general are simplified.
The key concept of Langlands' program is symmetry, a very important aspect of mathematical entities. In MENTAL, the key concept is the general or universal union of opposites, the key mechanism of consciousness. This concept is essential for establishing a model of internal and external reality.
Perhaps the attempt to relate number theory to automorphic functions is due to a more or less conscious desire to unite the two modes of consciousness: the analytic of number theory and the synthetic of automorphic functions.
Langlands' program tries to relate the specific (numbers and their relations) to the generic (functions). Automorphic functions make it possible to refer to infinitely many numbers with a single expression, which facilitates the discovery of patterns and laws in the numerical field. But this can also be done in MENTAL, in a simpler way, by means of parameterized generic expressions.
MENTAL, a universal conjecture
MENTAL can be considered a universal conjecture:
It is not based on the primitives as individual conjectures because the primitives are degrees of freedom. It is based on the set of all of them.
Unlike the particular conjectures, the universal conjecture of MENTAL cannot be proved. More than a conjecture, it is a universal thesis referring to the profound nature of internal and external reality.
Particular conjectures proceed from intuition. The universal conjecture of MENTAL comes from primary intuitions, which are the dimensions of consciousness.
Another way of considering the universal conjecture is that there are no degrees of freedom other than the primitives of MENTAL.
The unification of mathematics has to come from the archetypes of consciousness. MENTAL is the grand unified theory of mathematics.
With MENTAL one can achieve the motto inscribed on the obverse of the Fields Medal: "Transire suum pectus mundoque potiri" (Overcome human limitations and become masters of the universe). To go beyond oneself is to connect with the primary archetypes, the archetypes of consciousness. From that deep level you can truly master all things, the inner world and the outer world, the real world and the possible worlds. MENTAL is the revolution of simplicity, a theoretical-practical revolution that can forever change our view of mathematics.
Addenda
The Langlands program on the Internet
In 1995 Langlands began a collaboration with Bill Casselman (of the University of British Columbia), with the goal of publishing all of his writings −including publications, preprints and correspondence− on the Internet. The correspondence includes a copy of Langlands' famous letter to Weil. Currently all of Langlands' publications are on the Internet under the title "The Works of Robert Langlands".
Famous Conjectures
Conjectures have played a fundamental role in mathematics and have opened up new areas to try to solve them. Among the famous conjectures are:
Fermat's Last Theorem (1637). It was proved by Andrew Wiles in 1995.
Goldbach's conjecture (1742): every even number greater than 2 is the sum of two prime numbers (equal or different). It has not yet been proved.
The theorem of the 4 colors that are sufficient to color a map. Conjectured by Francis Guthrie in 1852, it was proved with the aid of a computer by Kenneth Appel and Wolfgang Haken in 1976.
Riemann's hypothesis (1859). It is a conjecture about the distribution of the zeros of the Riemann zeta function corresponding to prime numbers. It has not yet been proved.
Poincaré's conjecture (1904): every closed simply connected manifold is homeomorphic (isomorphic or topologically equivalent) to the 3D sphere.
A closed variety is one that is compact and has no edges (boundaries).
A compact variety is one in which every loop or closed circle can be transformed (compactified) into a point. This property allows to know intrinsically a variety without considering the space in which it is immersed.
A variety is simply connected when for each pair of points all the paths connecting them are homologous to each other (there is only one kind of homotopy).
The conjecture P ≠ NP: problems P (those that are solved in polynomial time) are different from problems of type NP (those whose solution can be verified in polynomial time). It has not yet been proved.
The Clay Institute selected "8 problems of the millennium" in 2000, 100 years after Hilbert listed 23 mathematical problems at the 1900 International Congress of Mathematicians in Paris. Among these problems are the Riemann hypothesis, the P ≠ NP conjecture, and the Poincaré conjecture. The latter was proved by Grigori Perelman in 2003.
Bibliography
Bernstein, Joseph & Gelbart, Steve (eds.). An introduction to the Langlands Program. Birkhäuser, 2004. (Un buena introducción al programa de Langlands.)
Bump, Daniel. Automorphic Forms and Representations. Cambridge Studies in Advanced Mathematics. Cambrige University Press, 1998.
Casseman, Bill (recopilador). The Works of Robert Langlands. Internet. (Colección completa de artículos y escritos de Robert Langlands.)
Frenkel, Edward. Love and Math: the heart of hidden reality. Basic Books, 2013. Disponible en Internet. (Explica en términos comprensibles en qué consiste el programa de Langlands, así como sus ramificaciones en geometría y física.)
Kieger, Martin H. Doing Mathematics: Convention, Subject, Calculatio, Analogy. World Scientific, 2003.
Frenkel, Edward. Lectures on the Langlands Program and conformal field theory. En Frontiers in number theory, physics, and geometry, vol. II, pp. 387-533. Springer, 2007.
Frenkel, Edward. Langlands Correspondence for Loop Groups. Cambridge University Press, 2007. Disponible online.
Institute for Advanced Study. The work of Robert Langlands. Internet.
Lorenzini, Dino. An Invitation to Arithmetic Geometry. University of Georgia, 1996.
O'Connor, John J.; Robertson, Edmund F. Robert Langlands, MacTutor History of Mathematics archive, University of St Andrews.
Pickover, Clifford A. El programa de Langlands. En “El Libro de las Matemáticas”, pp. 434-5. Librero, 2011.
Poincaré, Henri. Ciencia y Método. Espasa, 1965. (Describe las funciones automorfas.)
Poincaré, Henri. Papers in Automorphic Forms. Springer Verlag, 1981.
Ramos Martínez, Alberto. El programa de Langlands. Investigación y Ciencia, Agosto 2014, pp. 92-94. Disponible en Internet.
Timón García-Longaria, Ágata; Fernández, David. El horizonte visible de las matemáticas. Investigación y Ciencia, Noviembre 2014, pp. 12-14.
Weil, André. A 1940 Letter of André Weil on Analogy in Mathematics. Notices of AMS, vol. 52, no. 3, pp. 334-341, March 2005. Disponible online. También en Kriger, 2003, pp. 293-305.
