Why Beauty is Truth Page 19

  The library records at the university show him taking out more and more books on mathematical topics. In 1867, in the middle of the night, he was struck by a vision of his life’s work. His friend Ernst Motzfeldt was astonished to be woken from sleep by an excited Lie, who was shouting “I have found it, it is quite simple!”

  What he had found was a new way to think about geometry.

  Lie began to study the works of the great geometers, such as the German Julius Plücker and the Frenchman Jean-Victor Poncelet. From Plücker he got the idea of geometries whose underlying elements are not Euclid’s familiar points but other objects—lines, planes, circles. He published a paper outlining his big idea in 1869, at his own expense. Like Galois and Abel before him, he discovered that his ideas were too revolutionary for the old guard, and the regular journals did not wish to publish his researches. But Ernst refused to let his friend become discouraged, and kept him working on his geometry. Eventually, one of Lie’s papers was published in a prestigious journal and was favorably received. It gained Lie a scholarship. Now he had the money to travel, visit leading mathematicians, and discuss his ideas with them. He went to the hotbeds of Prussian and German mathematics, Göttingen and Berlin, and talked to the algebraists Leopold Kronecker and Ernst Kummer and the analyst Karl Weierstrass. He was impressed by Kummer’s way of doing mathematics, less so by Weierstrass’s.

  The most significant meeting, however, was in Berlin with Felix Klein—who, it happened, had been a student of Plücker, whom Lie greatly admired and wished to emulate. Lie and Klein had very similar mathematical backgrounds, but their tastes differed considerably. Klein, basically an algebraist with geometrical leanings, enjoyed working on special problems with inner beauty; Lie was an analyst who liked the broad sweep of general theories. Ironically, it was Lie’s general theories that gave mathematics some of its most important special structures, which were and still are extraordinarily beautiful, extraordinarily deep, and mostly algebraic. These structures might not have been discovered at all were it not for Lie’s push to generality. If you try to understand all possible mathematical objects of a certain kind, and succeed, you will inevitably find many that have unusual features.

  In 1870, Lie and Klein met again in Paris. And there, Jordan converted Lie to the cause of group theory. There was a growing realization that geometry and group theory were two sides of the same coin, but it took a long time for the idea to become fully formed. Lie and Klein did some joint work, trying to make the connection between groups and geometry more explicit. Eventually, Klein crystallized the thought in his “Erlangen Program” of 1872, according to which geometry and group theory are identical.

  In modern language, the idea sounds so simple that it should have been obvious all along. The group that corresponds to any given geometry is the symmetry group of that geometry. Conversely, the geometry corresponding to any group is whatever object it is the symmetry group of. That is, the geometry is defined by those things that are invariant under the group.

  For example, the symmetries of Euclidean geometry are those transformations of the plane that preserve lengths, angles, lines, and circles. This is the group of all rigid motions of the plane. Conversely, anything that is invariant under rigid motions naturally falls within the purview of Euclidean geometry. Non-Euclidean geometry simply employs different transformation groups.

  Why, then, bother converting geometry into group theory? Because it gives you two different ways to think about geometry, and two different ways to think about groups. Sometimes one way is easier to understand, sometimes the other. Two points of view are better than one.

  Relations between France and Prussia were deteriorating fast. Emperor Napoleon III thought he could shore up his declining popularity by starting a war with Prussia. Bismarck sent the French a stinging telegram, and the Franco-Prussian War was declared on 19 July 1870. Klein, a Prussian in Paris, deemed it prudent to head back to Berlin.

  Lie, however, was Norwegian and was greatly enjoying his visit, so he decided to stay in Paris. But he changed his mind when he realized that France was losing the war and the German army was advancing on Metz. Although he was a citizen of a neutral country, it was not safe to remain in a potential war zone.

  Lie decided to go on a hiking trip, heading for Italy. He did not get far; the French authorities caught him at Fontainebleau, about 25 miles southeast of Paris, carrying a number of documents covered in incomprehensible symbols. Since these were evidently in code, Lie was obviously spying for the Germans, and he was placed under arrest. It took the intervention of a leading French mathematician, Gaston Darboux, to convince the authorities that the writings were mathematics. Lie was let out of prison, the French army surrendered, the Germans began a blockade of Paris, and Lie once more headed for Italy—this time successfully. From there he returned to Norway. Along the way he dropped in on Klein, who had remained safe in Berlin.

  Lie received his doctorate in 1872. The Norwegian academic world was so impressed by his work that the University of Christiania created a position especially for him in the same year. With his former teacher Ludwig Sylow, he took on the task of editing Abel’s collected works. In 1874 he married Anna Birch, eventually fathering three children.

  By now, Lie had focused on a particular topic that he felt was ripe for development. There are many kinds of equations in mathematics, but two types are especially important. The first is algebraic equations, of the kind studied so effectively by Abel and Galois. The second is differential equations, introduced by Newton in his work on the laws of nature. Such equations involve concepts from calculus, and instead of dealing directly with some physical quantity, they describe how that quantity changes as time passes. More precisely, they specify the rate of change of the quantity. For example, Newton’s most important law of motion states that the acceleration experienced by a body is proportional to the total force acting on it. Acceleration is the rate of change of velocity. Instead of telling us directly what the body’s velocity is, the law tells us the rate of change of velocity. Similarly, another equation that Newton developed, to explain how the temperature of an object changes when it cools, states that the rate of change of temperature is proportional to the difference between the temperature of the object and the temperature of its surroundings.

  Most of the important equations in physics—those that concern the flow of a fluid, the action of gravity, the motion of the planets, the transfer of heat, the movement of waves, the action of magnetism, and the propagation of light and sound—are differential equations. As Newton first realized, nature’s patterns generally become simpler and easier to spot if we look at the rates of change of the quantities that we want to observe, not at those quantities themselves.

  Lie posed himself a momentous question. Is there a theory of differential equations analogous to Galois’s theory of algebraic ones? Is there a way to decide when a differential equation can be solved by specified methods?

  The key, once more, was symmetry. Lie now realized that some of his results in geometry could be reinterpreted in terms of differential equations. Given one solution of a particular differential equation, Lie could apply a transformation (from a particular group) and prove that the result was also a solution. From one solution you could get many, all connected by the group. In other words, the group consisted of symmetries of the differential equation.

  It was a broad hint that something beautiful awaited discovery. Consider what Galois’s application of symmetries had done for algebraic equations. Now imagine doing the same thing for the far more important class of differential equations!

  The groups studied by Galois are all finite. That is, the number of transformations in the group is a whole number. The group of all permutations on the five roots of a quintic, for example, has 120 elements. Many sensible groups are infinite, however, including symmetry groups of differential equations.

  One common infinite group is the symmetry group of a circle, which contains transformations t
hat rotate the circle through any angle whatsoever. Since there are infinitely many possible angles, the rotation group of the circle is infinite. The symbol for this group is SO(2). Here “O” stands for “orthogonal,” meaning that the transformations are rigid motions of the plane, and “S” means “special”—rotations do not flip the plane over.

  Circles also have infinitely many axes of reflectional symmetry. If you reflect a circle in any diameter, you get the same circle. Adding in the reflections leads to a bigger group, O(2).

  The groups SO(2) and O(2) are infinite, but it is a tame type of infinity. The different rotations can all be determined by specifying a single number—the relevant angle. When two rotations are composed, you just add the corresponding angles. Lie called this kind of behavior “continuous,” and in his terminology, SO(2) was therefore a continuous group. And because only one number is needed to specify an angle, SO(2) is one-dimensional. The same goes for O(2), because all we need is a way to distinguish reflections from rotations, and this is a matter of a plus or minus sign in the algebra.

  The group SO(2) is the simplest example of a Lie group, which has two types of structure at the same time: it is a group and also a manifold—a multidimensional space. For SO(2), the manifold is a circle, and the group operation combines two points on the circle by adding the corresponding angles.

  The circle has infinitely many rotational symmetries (left) and infinitely many reflectional symmetries (right).

  Lie discovered a beautiful feature of Lie groups: the group structure can be “linearized.” That is, the underlying curved manifold can be replaced by a flat Euclidean space. This space is the tangent space to the manifold. Here’s how it looks for SO(2):

  From Lie group to Lie algebra: the tangent space to a circle.

  The group structure, when linearized in this fashion, gives the tangent space an algebraic structure of its own, which is a kind of “infinitesimal” version of the group structure, describing how transformations very close to the identity behave. It is called the Lie algebra of that group. It has the same dimension as the group, but its geometry is much simpler, being flat.

  There is a price to pay for this simplicity, of course: the Lie algebra captures most important properties of the corresponding group, but some fine detail gets lost. And those properties that are captured undergo subtle changes. Nonetheless, you can learn a lot about a Lie group by passing to its Lie algebra, and most questions are more easily answered in the Lie algebra setting.

  It turns out—and this was one of Lie’s great insights—that the natural algebraic operation on the Lie algebra is not the product AB, but the difference AB – BA, which is called the commutator. For groups like SO(2), where AB = BA, the commutator is zero. But in a group like SO(3), the rotation group in three dimensions, AB – BA is nonzero unless the axes of rotation of A and B are either the same or at right angles. So the geometry of the group shows up in the behavior of commutators.

  Lie’s dream of a “Galois theory” of differential equations was eventually realized with the creation of a theory of “differential fields” in the early 1900s. But the theory of Lie groups turned out to be far more important, and more widely applicable, than Lie expected. Instead of being a tool to determine whether a differential equation can be solved in specific ways, the theory of Lie groups and Lie algebras has pervaded almost every branch of mathematics. “Lie theory” escaped its creator and became greater than he ever imagined.

  In hindsight, the reason is symmetry. Symmetry is deeply involved in every area of mathematics, and it underlies most of the basic ideas of mathematical physics. Symmetries express underlying regularities of the world, and those are what drive physics. Continuous symmetries such as rotations are closely related to the nature of space, time, and matter; they imply various conservation laws, such as the law of conservation of energy, which states that a closed system can neither gain nor lose energy. This connection was worked out by Emmy Noether, a student of Hilbert.

  The next step, of course, is to understand the possible Lie groups, just as Galois and his successors sorted out many properties of finite groups. Here a second mathematician joined in the hunt.

  Anna Catharina was worried about her son.

  Her doctor had told her that young Wilhelm was “quite weakly and besides very awkward” and “always excited, but a completely impractical bookworm.” Wilhelm’s health improved as he grew, but his bookworm tendencies did not. Just before his 39th birthday, he would publish a piece of mathematical research that has been described, with justification, as “the greatest mathematical paper of all time.” Such designations are of course subjective, but Wilhelm’s paper would certainly be high on anyone’s list.

  Wilhelm Karl Joseph Killing was the son of Josef Killing and Anna Catharina Kortenbach. He had one brother, Karl, and one sister, Hedwig. Josef was a legal clerk, and Anna was a pharmacist’s daughter. They were married in Burbach, on the eastern side of central Germany, and soon afterward moved to Medebach when Josef became the mayor there. Then he was made mayor of Winterberg, and after that mayor of Rüthen.

  The family was quite well off and could afford a private tutor to prepare Wilhelm for the gymnasium, which in his case was in Brilon, 50 miles west of Dortmund. At school he liked classics—Latin, Hebrew, Greek. A teacher named Harnischmacher introduced him to mathematics; Wilhelm turned out to be very good at geometry, and resolved to become a mathematician. He attended what is now the Westphalian Wilhelm University of Münster, but was then merely a Royal Academy. The academy did not teach advanced mathematics, so Killing taught himself. He read Plücker’s geometrical work and tried to derive some new theorems of his own. He also read Gauss’s Disquisitiones Arithmeticae.

  After two years at the Royal Academy he moved to Berlin, where the quality of mathematical teaching was much superior, and came under the influence of Weierstrass, Kummer, and Hermann von Helmholtz, a mathematical physicist who clarified the link between conservation of energy and symmetry. Killing wrote a PhD thesis on the geometry of surfaces, based on some ideas of Weierstrass, and took a job as a teacher of mathematics and physics, with a sideline in Greek and Latin.

  In 1875, he married a music lecturer’s daughter, Anna Commer. Their first two children, both sons, died in infancy; the next two, daughters named Maria and Anka, thrived. Later, Killing fathered two more sons.

  By 1878, he had gone back to his old school, but now as a teacher. He had a heavy workload, about 36 contact hours per week, but somehow he found the time to continue his mathematical research—the greats always do. He published a series of important papers in top journals.

  In 1882, Weierstrass secured Killing a professorship at the Lyceum Hosianum in Braunsberg, where he spent the next ten years. Braunsberg had no strong mathematical tradition and offered no colleagues with whom to discuss research, but Killing seems not to have needed such stimulation. For it was there that he made one of the most important discoveries in the whole of mathematics. It left him rather disappointed.

  What he had hoped to achieve was hugely ambitious: a description of all possible Lie groups. The Lyceum did not buy the journals in which Lie published, and Killing had very little idea of Lie’s work, but he independently discovered the role of Lie algebras in 1884. So Killing knew that each Lie group was associated with a Lie algebra, and he quickly recognized that Lie algebras would probably be more tractable than Lie groups, so his problem reduced to the classification of all possible Lie algebras.

  This problem turns out to be desperately hard—we now know that it probably has no sensible answer, in the sense that no simple construction can produce all Lie algebras by a uniform and transparent procedure. So Killing was forced to settle for something far less ambitious: to describe the basic building blocks from which all Lie algebras can be assembled. This is a bit like wanting to describe all possible architectural styles but having to settle for a list of all possible shapes and sizes of brick.

  These basic building bloc
ks are known as simple Lie algebras. They are distinguished by a very similar property to Galois’s idea of a simple group, one with no normal subgroups except trivial ones. In fact, a simple Lie group has a simple Lie algebra, and the converse is very nearly true as well. Amazingly, Killing succeeded in listing all possible simple Lie algebras—mathematicians call such a theorem a “classification.”

  In Killing’s eyes, that classification was a very limited version of something far more general, and he was frustrated by several restrictive assumptions he had been forced to make in order to get anywhere. He was particularly irked by the need to assume simplicity, which forced him to switch to Lie algebras over the complex numbers rather than the reals. The former are better behaved but less directly related to the geometrical problems that fascinated Killing. Because of these self-imposed limitations, he did not consider his work worth publishing.

  He did manage to make contact with Lie; not very fruitfully, as it turned out. First he wrote to Klein, who put him in touch with Lie’s assistant Friedrich Engel, then working in Christiania. Killing and Engel hit it off immediately, and Engel became a staunch supporter of Killing’s work, helped him get over some tricky points, and encouraged him to push the ideas further. Without Engel, Killing might have given up.

  At first, Killing thought he knew the complete list of simple Lie algebras, and that these were the Lie algebras so(n) and su(n) associated with two infinite families of Lie groups: the special orthogonal groups SO(n), consisting of all rotations in n-space, and their analogues SU(n) in complex n-space, the special unitary groups. The historian Thomas Hawkins imagined “the amazement with which Engel read Killing’s letter with its bold conjectures. Here was an obscure professor at a Lyceum dedicated to the training of clergymen in the far-away reaches of East Prussia, discoursing with authority and conjecturing profound theorems on Lie’s theory of transformation groups.”