Concinnitas Artists

Mathematics is both an Art and a Science and beauty plays an essential role, a fact recognized by all mathematicians. The great German mathematician Hermann Weyl, one of my heroes, put it well when he said: “My work has always tried to unite the true with the beautiful and when I had to choose one or the other, I usually chose the beautiful.”

Since mathematics is the most precise of the sciences and is devoted to finding out the truth, Weyl’s statement might appear bizarre and even provocative – a tongue-in-cheek remark. But I believe Weyl was quite serious. The apparent paradox in Weyl’s dictum is that objective truth is what we all search for but, at any stage, it is uncertain and provisional. But beauty, which is a subjective experience “in the eye of the beholder,” is the light we follow in the hope that it is leading us to truth. But what is beauty in mathematics, and is it similar to beauty in art, music or poetry? Karl Weierstrass, who was outwardly an austere analyst once said that: “It is not possible to be a complete mathematician without having the soul of a poet.” Such statements are hard for outsiders to understand, though I have argued that the famous equation of Euler, e2πi = 1, is in its brevity and depth, the equivalent of Hamlet’s famous question, “To be or not to be…”

But perhaps architecture is the art most comparable to mathematics, where there is a grand vision full of delicate detail, where solid foundations and functional utility are all essential components. The equation I have chosen to epitomize beauty in my own work has the grandeur that comes from a rich history and multiple connections to different branches of mathematics: topology, geometry, analysis. But the delicate details of the arguments, whose simplicity deceptively covers hidden depths, are only apparent to the craftsmen who can appreciate them. Like a building with three towers the equation has three terms, from different parts of mathematics, linked together in a striking manner. As with great architecture it has features that trace their roots far back in time, while at the same time embodying the latest techniques and point towards the future.

The ancestry of the equation connects it to many royal families: to Euler crossing the Königsberg bridges, to Riemann counting prime numbers and to Gauss surveying the earth. There is poetry in this story, but the future is as important as the past. Many royal families die out, only a few survive. My equation is about forty years old and since that time it has found fascinating and totally unexpected applications in fundamental physics, which Weyl would have both understood and appreciated. In fact, many of the key ideas can be traced back to Weyls own work.

Finally, on a more personal note, my equation embodies extensive collaboration with many of my colleagues: Fritz Hirzebruch from Bonn, Raoul Bott from Harvard, Is Singer from MIT and Vijay Patodi from Bombay, who like many talented poets died at a tragically early age. Beauty is a human experience and is best shared with friends.

Does beauty exist in mathematics? The question concerns mathematical objects and their relations, the real subject of verifiable proofs. Mathematicians generally agree that beauty does exist in the structural beauty of theorems and proofs, even if most of the time it is largely visible only to mathematicians themselves.

The concept of group beautifully expresses symmetry in mathematics. What is a group? Consider any object, concrete or abstract. A symmetry of the object –mathematically, an automorphism– is a mapping of the object onto itself that preserves all its properties. The product of two symmetries, one followed by the other, also is a symmetry, and every symmetry has an inverse that undoes it. Mathematicians consider continuous Lie groups, such as the rotations of a circle or of a sphere, to be a beautiful foundation for a great portion of mathematics, and for physics as well. Besides continuous Lie groups there are noncontinuous finite and discrete groups; some are obtainable from Lie groups by reduction to a finite or discrete setting.

Groups can be extremely complicated. Given a group, it may happen that there is a mapping of it to another group, preserving the product structure. A group is simple if the image of such a mapping is always either a copy of the first group or just one element, the identity. Simple groups are the basic building blocks of all groups, so knowing all simple groups is essential in the study of arbitrary groups. General finite groups of symmetries appeared for the first time in the work of Évariste Galois on the subject of algebraic equations. Galois, at the age of only eighteen, was able to prove that the general equation of degree 5 is insoluble by means of algebraic operations by showing that the group A5 of even permutations (that is, permutations consisting of an even number of pair exchanges) on the five letters a,b,c,d,e is a simple group. This group is the smallest non-commutative simple group and turns out to be the group of symmetries of the icosahedron, a very nice geometry! It was conceivable that simple groups could be described as symmetries of special geometric objects, but the difficulty of studying an abstract, hypothetical, simple group consisted precisely in building a rich geometry out of its internal properties. As of today, the complete proof of the classification theorem that lists all finite simple groups runs over three thousand pages and took over forty years of the collective efforts of more than one hundred mathematicians.

The families of simple finite groups arising from Lie groups were found early, with three exceptions. These families arise by working not over the real or complex numbers, but instead over finite fields of characteristic p, where p is a prime number. There, one can still do the ordinary operations of arithmetic, but multiplication by p always yields 0. Everything went smoothly, if not easily, except for the discovery by the mathematician Ree that the Lie groups B2 and F4 in characteristic 2, and G2 in characteristic 3, also admitted an extra symmetry which could be used to obtain new families of simple groups, nowadays called the twisted Ree groups; the twisted B2 groups and their uniqueness had been obtained earlier by Suzuki using entirely different methods. Uniqueness in the F4 case was also found, but in the G2 case it turned out to be elusive.

After a great effort by Thompson, the uniqueness problem for G2 was reduced to proving that a certain transformation σ of a finite field of characteristic 3, satisfying a very complicated set of equations in many variables, had the property that its square σ2 on x was the same as the cube x3, in other words σ2 = 3. Unfortunately, ordinary algebra for eliminating variables quickly led to equations with a number of terms so big that all computers in the world could not store the formulas in their memory banks. What to do? Already in 1973 Thompson got me interested in the problem, but I got nowhere in the maze of formulas. In 1979, when the work on classification was in full swing, I looked again at Thompson’s equations. I asked myself whether it was necessary to write down these ‘impossible formulas’, perhaps there was a way around. By a strange trick, it turned out that one small, but useful additional piece of information could be extracted from the elimination. By redoing the elimination together with the trick and the new piece of information, the additional information was refined. By repeating three times this refinement process, the sought for equation σ2 = 3 was obtained, except possibly for a few cases checkable by computer. Thus, the uniqueness problem was solved and another brick was added to the proof of the classification of finite simple groups.

The print is done as a writing in white chalk on a dark slate-blue blackboard, starting at the left with the Thompson equations and with the double arrow pointing to σ2 = 3, indicating that indeed the equations on the left imply the uniqueness of the twisted Ree groups. The problem was beautiful, the expected answer was also simple, hence beautiful, and the Thompson equations had an inner secret beauty because they reflected the properties of a group. To the expert, the solution obtained by avoiding brute force also had its own beauty. Indeed, mathematicians, sometimes involuntarily, in their search for truth also look for beauty as a guide. As the poet Keats wrote, beauty is truth, truth beauty.

*Thanks to Sarah Jones Nelson

Much of my research involves the interaction between certain topics in differential geometry, related to Mathematical Physics, and the topology of four-dimensional spaces. This blackboard represents some of these ideas, partly by analogy in three dimensions.

The main theme of the blackboard is Ampère’s Law in electromagnetism, and much of the board is similar to what one sees in standard physics texts. The picture in the upper-left side depicts a current j flowing in a closed wire, indicated by the thick black curve. The current produces a magnetic field B, indicated by the small arrows. In a two-dimensional situation this corresponds to the pattern you would see by scattering iron filings on a sheet of paper. The magnetic field is defined everywhere, so really, we should imagine a little arrow at each point, but it is only practical to draw a few of them. The basic physical phenomenon, stated in ordinary language, is that the magnetic field “goes around” the wire, and Ampère’s Law amounts to a precise quantitative expression of this idea.

The notion of a “vector field” such as the magnetic field (or the current, which is a vector field confined to the interior of the wire) was a crucial conceptual advance in early 19th century Mathematical Physics. It gives a common framework for describing electricity, magnetism, gravity and much else. One important concept is the “flux” of a vector field through a surface. The mathematical definition involves the process of integration over the surface, but the idea can be made intuitive by imagining that the vector field represents the velocity of a fluid: then the flux is the rate of flow of fluid through the surface.

One formulation of Ampère’s Law – the “integral form” – states that the total circulation of the current around the boundary of a piece of surface is equal to the flux of the current through the surface. This is illustrated by the disc cutting across the wire in the middle of the blackboard. Another formulation – the “differential form” – is the set of the equations at the lower-right of the board, expressing the components of the current in terms of derivatives in the three space directions x, y, z of the components of the magnetic field. This board attempts to convey, or at least hint at, several broad aspects of mathematics which I find beautiful. On the left we see a picture and words: in the lower right, a set of equations. These are different descriptions of the same thing, and they stimulate different ways of thought: pictorial and symbolic. Further, the picture can represent an actual object – a copper wire carrying a current – in the real world, but a mathematician will often draw such a picture with a schematic meaning: for example, we might imagine that rather than a one-dimensional wire in three-dimensional space we are looking at a three-dimensional object in seven-dimensional space, or even objects in an infinite-dimensional space. Such extensions of our physical intuition to more abstract situations can be remarkably effective. The interactions between the intuitive, the pictorial, the symbolic and the abstract are beautiful and delight the mind.

What does this have to do with topology – the study of phenomena independent of continuous deformations? This is indicated schematically by the fact that the wire loop is knotted – it cannot be deformed into a standard circle without cutting and re-joining. This is something which is not so easy to demonstrate mathematically but which we understand from our experience. Further, we understand that such knots could be arbitrarily complicated, providing an intuitive demonstration that topology can be a subtle matter. At a more detailed level, there are precise connections between knots and four-dimensional spaces: a knot encodes instructions for building a four-dimensional space by gluing together some standard building blocks.

The mathematics which this board alludes to is represented in spirit rather than precision. The idea which it seeks to convey is that doing “something like” studying the magnetic field generated by a knotted current could have “something to do” with the topology of knots and four-dimensional spaces. Over the past three decades there have indeed been many developments in this spirit, although the details are somewhat different. For example, these developments involve generalisations of electromagnetism to “Yang-Mills fields” and are also bound up with Quantum Mechanics and Quantum Field Theory. These last are represented in the lower left of the board, where we consider the flux of the magnetic field through a small disc. This quantity does not have any meaning in classical electromagnetism (as far as the writer knows) but is central in the quantum theory of the interaction of the magnetic field with the “wave function” of an electron.

What is special about three and four dimensions? In topology, this is a profound question. It turns out that spaces of dimension bigger than four are in many respects easier to understand. Even without knowing any of their detailed meaning, one can see some aspect of this dimension-specificity from the displayed formulae. They are given by permuting in cyclic order the three coordinates x, y, z. The fact that one can write down these equations depends on the fact that there are exactly three pairs (xy), (yz), (zx) of three objects x, y, z. One can generalise electromagnetic theory to higher dimensions but then the magnetic field is no longer a vector field but a more complicated kind of object. Three dimensions are special because the magnetic field is a vector field, just like the electric field. This extends to similar phenomena in four dimensions which are somehow bound up with the special topological features. Understanding all of this, at a fundamental level, is a fascinating problem and we only see at the moment some shadow of the ultimate truth. Here we find other aspects of the beauty of mathematics: surprising but mysterious connections between different fields, and the intermingling of the seemingly simple and well-understood with the completely unknown.

The MacDonald Equation is the most beautiful thing that I ever discovered. It belongs to the theory of numbers, the most useless and ancient branch of mathematics. My friend Ian MacDonald had the joy of discovering it first, and I had the almost equal joy of discovering it second. Neither of us knew that the other was working on it. We had daughters in the same class at school, so we talked about our daughters and not about mathematics. We discovered an equation for the “Tau-function” (written τ(n) in the equation), an object explored by the Indian genius Srinivasa Ramanujan four years before he died at age thirty-two. Here I wrote down MacDonald’s equation for the Tau-function. The MacDonald equation has an amazing five-fold symmetry that Ramanujan missed. You can see the five-fold symmetry in the ten differences multiplied together on the right-hand side of the equation. We are grateful to Ramanujan, not only for the many beautiful things that he discovered, but also for the beautiful things that he left for other people to discover.

To explain how the MacDonald equation works, let us look at the first three cases, n=1, 2, 3. The sum is over sets of five integers a, b, c, d, e with sum zero and with the sum of their squares equal to 10n. The “(mod 5)” statement means that a is of the form 5j+1, b is of the form 5k+2, and so on up to e of the form 5p+5, where j, k, and p are positive or negative integers. The exclamation marks in the equation mean 1!=1, 2!=1x2=2, 3!=1x2x3=6, 4!=1x2x3x4=24. So when n=1, the only choice for a, b, c, d, e is 1, 2, -2, -1, 0, and we find tau(1)=1. When n=2, the only choice is 1, -3 ,3 ,-1 , 0, and we find tau(2)=-24. When n=3, there are two choices, 1, -3, -2, 4, 0 and -4, 2, 3, -1, 0, which give equal contributions, and we find tau(3)=252. It is easy to check that these three values of tau(n) agree with the values given by Ramanujan’s equation.

The MacDonald equation is a special case of a much deeper connection that Ian MacDonald discovered between two kinds of symmetry which we call modular and affine. The two kinds of symmetry were originally found in separate parts of science, modular in pure mathematics and affine in physics. Modular symmetry is displayed for everyone to see in the drawings of flying angels and devils by the artist Mauritz Escher. Escher understood the mathematics and got the details right. Affine symmetry is displayed in the peculiar groupings of particles created by physicists with high-energy accelerators. The mathematician Robert Langlands was the first to conjecture a connection between these and other kinds of symmetry. Ian MacDonald took a big step toward making Langlands’ dream come true. The equation that I wrote down here is a small piece of MacDonald’s big step.

Computational complexity theory is the branch of theoretical computer science concerned with the fundamental limits on the efficiency of automatic computation. It focuses on problems that appear to require a very large number of computation steps for their solution. The inputs and outputs to a problem are sequences of symbols drawn from a finite alphabet; there is no limit on the length of the input, and the fundamental question about a problem is the rate of growth of the number of required computation steps as a function of the length of the input.

Some problems seem to require a very rapidly growing number of steps. One such problem is the independent set problem: given a graph, consisting of points called vertices and lines called edges connecting pairs of vertices, a set of vertices is called independent if no two vertices in the set are connected by a line. Given a graph and a positive integer n, the problem is to decide whether the graph contains an independent set of size n. Every known algorithm to solve the independent set problem encounters a combinatorial explosion, in which the number of required computation steps grows exponentially as a function of the size of the graph. On the other hand, the problem of deciding whether a given set of vertices is an independent set-in a given graph is solvable by inspection. There are many such dichotomies, in which it is hard to decide whether a given type of structure exists within an input object (the existence problem), but it is easy to decide whether a given structure is of the required type (the verification problem).

It is generally believed that existence problems are much harder to solve than the corresponding verification problems. For example, it seems hard to decide whether a jigsaw puzzle is solvable, but easy to verify that a given arrangement of the puzzle pieces is a solution. Similarly, it seems hard to solve Sudoku puzzles but easy to verify given solutions. Complexity theory provides precise definitions of “P”, the class of all existence problems that are easy to solve, and “NP”, the class of existence problems whose solutions are easy to verify. The general belief that verifying is easier than solving strongly suggests that the class NP properly includes the class P, but this claim has never been proven. The question of whether P = NP is the most central open question in theoretical computer science, and one of the most notorious open questions in all of mathematics.

In a 1972 paper entitled “Reducibility Among Combinatorial Problems” I demonstrated a technique that has made it possible to prove that thousands of problems, arising in mathematics, the sciences, engineering, commerce, and everyday life, are equivalent, in the sense that an efficient algorithm for any one of them would yield efficient algorithms for all the problems in NP, and thus establish that P = NP. Conversely, if P is unequal to NP, then none of these problems are easy to solve. These problems are called NP-complete. The moral of the story is that NP-completeness is a widespread phenomenon; most combinatorial problems arising in practice are NP-complete, and hence, in all likelihood, hard to solve.

The technique stems from a 1971 paper in which Stephen Cook of the University of Toronto showed that a particular problem in NP, the Satisfiability Problem (denoted Sat) of propositional logic is NP-complete. To do so, he showed that any problem in NP is efficiently reducible to Sat; i.e., for any problem A in NP, there is an efficient algorithm that converts any instance of A into an equivalent instance of Sat. It follows that, if Sat is easy to solve, then every problem in NP is easy to solve. Around the same time, Leonid Levin in the Soviet Union, who is now a professor at Boston University, proved a similar result.

In my 1972 paper I demonstrated the prevalence of NP-completeness by using a tree of efficient reductions to show that 21 canonical problems are NP-complete. The figure exhibits reductions among 13 of these problems. Each node of the tree is labeled with the name of problem in NP, and each edge indicates that the upper problem is efficiently reducible to the lower one; thus, if the lower problem is easy to solve, then so is the upper problem. Hence, if any problem in the tree were easy to solve, then Sat would be easy to solve and therefore, by Cook’s seminal result, every problem in NP would be easy to solve.

The prevailing, but not quite universal, opinion among complexity theorists is that P is unequal to NP, but no proof or disproof appears to be on the horizon. Perhaps some brilliant youngster, motivated by this essay, will find the elusive approach that will crack the P vs. NP problem.

Beauty in mathematics can be found at many levels: in the symmetry and elegance of mathematical curves, surfaces, and combinatorial structures; in the subtle logic of mathematical proofs; or, as in the case of NP-completeness, in the discovery that a large number of seemingly unrelated mathematical phenomena are all manifestations of a single underlying principle.

A conservation law says that the total amount of some quantity, like mass, momentum or energy contained in any domain, changes at the rate at which this quantity flows across or is created at the boundary of the domain. I find this idea beautiful because it is so basic. But once you specify details of the rate of creation you get a huge variety of phenomena. The laws governing the flow of fluids are conservation laws.

Conservation laws are a key to understanding shock waves. I got involved with shock waves in 1945 while in the army, when I was shipped to Los Alamos to work on the atomic bomb project, instead of the Pacific to participate in the invasion of Japan, which the atomic bomb made unnecessary. Since atomic bombs cannot be built by trial and error, it was of paramount importance to be able to calculate the flow that takes place when a bomb is detonated. Von Neumann recognized that such a calculation cannot be done without computers, and this gave him the original impetus to champion computers. Of course, he realized the paramount importance of computers for tasks other than designing atomic weapons.

Von Neumann had the beautiful and original idea to treat shocks in numerical calculations not as boundaries but as part of the flow. It is a powerful simplifying way of looking at flows with shocks. Many people are unaware that von Neumann was not only one of the leading theoretical mathematicians of the 20th century, but also one of the leading applied mathematicians.

A large part of what a mathematician does may be described as exploring “things” which they discover through reasoning, which become as real to them as the house they live in but which have no material existence at all. Some easy examples are the higher-dimensional analogs of the Platonic solids. One of the spires on Gaudi’s Sagrada Familia is expected to have an icosahedron on it – the twenty-sided regular polyhedron. This is a real tangible thing. But mathematicians all feel space can have more than three dimensions and in the 19th century Schläfli discovered some marvelous four-dimensional versions of Plato’s list that we can only imagine. My own passion for a long time focused on the things written as Mg in my formula, known as the “moduli space of curves of genus g”. Even by a mathematician’s standards for reality, when I was a student, these spaces seemed shrouded in fog, in a limbo between full-fledged mathematical existence and fantasy. I wanted to change that.

At that time, Alexander Grothendieck burst on the scene. A truly unique person, he had the skill of passing to higher levels of abstraction than anyone before and using these abstractions to shine a brilliant light on relatively concrete things we didn’t yet understand. It turned out that one of his deep results could be applied to my still dimly perceived space. I didn’t know then what to make of this. The power of the result only became apparent some twenty years later working with Joe Harris, when we knew enough to be able to treat Mg as a real object (technically, a quasi-projective algebraic variety).

So, what is the formula about and why is it beautiful? It states that two things (“line bundles”) are essentially the same (“isomorphic”). The one on the left is the key object that determines the geometry of any space like Mg. Going back to Gauss, we have known that in crudest terms there are three kinds of spaces: flat ones like a plane, positively curved ones like the surface of the earth and negatively curved ones like the surface of a saddle (where the sum of the angles of a triangle is less than 180°). The object on the left controls where our space sits in this tripartite categorization. The object on the right is what Raoul Bott called a “tautological” structure: a basic decoration produced by the very definition of the space Mg. Their isomorphism turned out to be the key that showed these spaces belong to the negative category if g is large enough.

The most startling thing about the formula is the number thirteen. You can look in many math journals and find that the only numbers greater than two are the page numbers. This thirteen has a pedigree in a long tradition of counting things, e.g., that a cubic surface has exactly 27 straight lines on it (including complex ones!) but, frankly, it still feels to me like a strange joke the creator is playing on us.

The discovery of the neutron in 1932 introduced the idea of nuclei composed of neutrons and protons. When we look deeper, we see that each neutron or proton is composed of three quarks, roughly speaking one of each “color.” It is the color force that binds quarks together to form the neutrons and protons. The color variable takes three different values, nicknamed “red,” “green,” and “blue.” “Colored” objects are confined. They cannot escape to be detected singly. The theory is perfectly symmetrical under the transformation group SU3(color) that transforms the three colors into one another.

The expression presented here embodies the Lagrangian of quantum chromodynamics (“QCD”), the mathematical representation encoding the dynamics of the strong interaction, one of the fundamental physical forces along with gravitation and the weak, and electromagnetic forces. It is “beautiful” because it contains some truth. There is also a beauty in its succinctness, but that terseness sweeps a bit under the rug. We have here three terms, where the first two Lgl,Lq, encode the effects (fields) due to gluons and quarks respectively, and Laddl, contains the “additional” terms and includes, among other things, the fields that ultimately predicted the recently discovered Higgs boson.

I recall that in arriving, along with some colleagues, at this formulation, it came not as a burst of intuition, but rather as an accretion of steady work, and this expression summarizes not just a truth about the world, but a lot of hard work over a long period of time, each term “plucked” from a body of discoveries over a number of years. As time went by, I and other people had insights about what would be included in this description. I might add that we were thinking about the strong interaction in a way that was a bit different from many in the community. At any stage we might have stopped, leaving more for the “additional” term, but this formulation felt good. It was self-contained and satisfied the symmetry conditions imposed by the group SU3. This kept us from venturing into territory that was at that time not yet fully explored. So, even though it is true, it is also in a sense not final, there are always more details to add – there are various scalar fields, not just the Higgs, that we know are there, but we don’t yet know what to do with them – so there is still more to be discovered and there is a beauty in that too.

Long before Newton, the concept already was used by the Greeks for finding the square root of a positive number. Since Newton, the iteration has been used more generally to give an approximation to a solution of the equation f (x) = 0. Early in my mathematical career I was intrigued by the question, why does this method work so fast and so well and what are the limitations.

In the special case that f is a polynomial, the Fundamental Theorem of Algebra asserts that the equation f (x) = 0 has a solution. The solution x could be a real or a complex number. Gauss, early in the 19th century, gave proof of this existence result based on an algorithm which can be realized as a succession of applications of Newton’s method (his proof had a gap). My 1981 paper “The Fundamental Theorem of Algebra and Complexity Theory” was based on Newton’s Method and related to the Gauss idea.

Complexity (computational) theory is perhaps the central theme of theoretical computer science; in this theory a problem is called tractable in case there is an algorithm solving it using a polynomial bound (relative to the input size) on the number of bit-sized operations. On the one hand I was inspired by this theme of computer science. On the other hand, I found the formalism of this subject useless for analyzing the complexity of Newton’s method.

In the above paper I counted the arithmetic operations, not the bit operations to measure the complexity. Moreover the concept of “condition number” of numerical analysis played a role in my complex treatment of the Fundamental Theorem of Algebra. I showed tractability from this point of view.

The problem of finding a zero of a polynomial has a natural generalization to a system of polynomial equations. Toward dealing with this extension, I was joined by Mike Shub in writing a series of joint papers we called “Complexity of Bezout’s Theorem.” Our goal was to make this problem tractable by finding an algorithm that yields an approximation in polynomial time. We failed in this effort and today it remains an important open problem. However, Peter Burgisser and Felipe Cucker have come very close to a solution in a recent paper in the Annals of Mathematics journal. They used developments of Carlos Beltran and Luis Pardo along the way, and certainly Newton’s method played a central role in their work.

Mike Shub and I were joined by Lenore Blum and together we generalized the Turing Machine of computer science to give foundational support for the zero-finding studies. The associated real number algorithms of this 3-person project became imbedded in setting of polynomial time, NP-completeness, tractability, and all this made good sense. Eventually Felipe Cucker joined us to write the book, “Complexity and Real Computation.” A reference is Volume 3 (650 pages) of my collected works.

John Keats has written, “Beauty is truth, truth beauty ….” He has also written “A thing of beauty is a joy forever.” I wish to add, beauty is simple and profound. I hope that my few words will convince you that Newton’s Method is a concept of great beauty.

This equation presents the original version of what has become the standard theory of two fundamental forces of nature, the electromagnetic force and the weak nuclear force. The latter force, though less familiar than electromagnetism, is responsible for an important kind of radioactivity, known as beta decay, and for the first step in the chain of nuclear reactions that gives heat to the sun and stars. This equation was Eq. (4) in my first paper on this subject, published in 1967. This was for some years the most widely cited paper ever published in elementary particle physics and may still be.

The electroweak theory is a field theory. Its fundamental ingredients are fields, including the electric and magnetic fields. The quantity denoted by £ on the left side of the equation is a combination of fields and their rates of change, known as the Lagrangian density of this theory. The Lagrangian density is something like an energy density, and it provides a convenient way of summarizing all the equations governing the fields of the theory, following rules that have been used by physicists since the 1930s.

Most of the symbols on the right-hand side of the equation denote the various fields of the theory. The weak and electromagnetic forces are transmitted by the fields A and B • the electric and magnetic fields are combinations of A and B . The neutrino and the left-handed part of the electron field (that is, the field that describes electrons that are spinning around their direction of motion like the fingers of the left-hand curling around the thumb) are united in the symbol L; the right-handed part of the electron field is denoted /?. The quantities ^and g’are numerical constants, related to the charge of the electron, whose values have to be taken from experiment.

The third and fourth lines of the equation describe the mechanism by which the symmetry of the theory between neutrinos and left-handed electrons, and between the weak and electromagnetic forces, is broken. The symbol p denotes a quartet of fields, whose interaction with the other fields gives mass to the electron, leaving the neutrino massless, and gives mass to the three particles that transmit the weak forces, leaving the photon (the particle of light) massless. The quantities Ge, M\ additional numerical constants, related to the mass of the electron and the strength of the weak forces. One of the quartet of fields included in (p corresponds to a new particle, which was not discovered experimentally until 2012.

This equation may not look beautiful. Its beauty lies in its rigidity — once its ingredients are specified, its structure is well fixed by conditions of mathematical consistency. Leave out one line, or just change a minus sign to a plus sign and the whole thing would become inconsistent.

For brevity, this equation left out the muon, a heavier electron-like particle, and a corresponding type of neutrino. It was obvious that they should be included the same way as the electron and its neutrino.

In 1971 the theory was expanded to include quarks, the elementary particles that make up protons and neutrons. Since then, the theory has been repeatedly confirmed by experiments.