Symmetry and the Monster One of the greatest

Symmetry and the Monster One of the greatest quests in mathematics

A little early history Equations of degree 2—meaning the highest power of x is x 2: solved by the Babylonians in about 1800 BC Equations of degree 3: solved using a graphical method by Omar Khayyám in about 1100 AD Equations of degrees 3 and 4: solved by Italian mathematicians in the first half of the 1500 s.

The quintic equation • Equations of degree 5 were a problem. No one could come up with a formula. • 1799 Paolo Ruffini • 1824 Niels Hendrik Abel • Early 1830 s Évariste Galois

Galois’s Ideas • If the equation is irreducible any solution is equivalent to any other. • The solutions can be permuted among one another. • Not all permutations are possible, but those that are form the Galois group of the equation.

x 4 - 10 x 2 + 1 = 0 • There are four solutions, a, b, c, d • The negative of a solution is a solution, so we can set: a + b = 0, and c + d = 0. • This restricts the possible permutations; if a goes to b then b goes to a, if a goes to c then b goes to d.

The Galois Group • Galois investigated when the solutions to a given equation can be expressed in terms of roots, and when they can’t. • The solutions can be deconstructed into roots precisely when the Galois group can be deconstructed into cyclic groups.

Atoms of Symmetry • A group that cannot be deconstructed into simpler groups is called simple. • For each prime number p the group of rotations of a regular p-gon is simple; it is a cyclic group. • The structure of a non-cyclic simple group can be very complex.

Families of Simple Groups • Galois discovered the first family of noncyclic finite simple groups. • Other families were discovered in the later nineteenth century. • All these families were later seen as ‘groups of Lie type’, stemming from work of Sophus Lie.

Sophus Lie • Lie wanted to do for differential equations what Galois had done for algebraic equations. • He created the concept of continuous groups, now called Lie groups. • ‘Simple’ Lie groups were classified into seven families, A to G, by Wilhelm Killing.

Finite groups of Lie type • Finite versions of Lie groups are called groups of Lie type. • Most of them were created by Leonard Dickson in 1901. • In 1955 Claude Chevalley found a uniform method yielding all families A to G. • Variations on Chevalley’s theme soon emerged, and by 1961 all finite groups of Lie type had been found.

The Feit-Thompson Theorem • In 1963, Walter Feit and John Thompson proved the following big theorem: • A non-cyclic finite simple group must contain an element of order 2. • Elements of order 2 give rise to ‘crosssections’, and Richard Brauer had shown that knowing one cross-section of a finite simple group gave a firm handle on the group itself.

The Classification • By 1965 it looked as if a finite simple group must be a group of Lie type, or one of five exceptions discovered in the mid-nineteenth century. • These five exceptions, the Mathieu groups—created by Émile Mathieu—are very exceptional. There is nothing else quite like them.

A Cat among the Pigeons • In 1966, Zvonimir Janko in Australia produced a sixth exception. • He discovered it via one of its crosssections. • This led Janko and others to search for more exceptions, and within ten years another twenty turned up.

The Exceptions • Some were found using the crosssection method • Some were found by studying groups of permutations • Some were found using geometry

The Hall-Janko group J 2 • Janko found it using the cross-section method. • Marshall Hall found it using permutation groups. • Jacques Tits constructed it using geometry.

The Leech Lattice • John Leech used the largest Mathieu group M 24 to create a remarkable lattice in 24 dimensions. • John Conway studied Leech’s lattice and turned up three new exceptions. • Had he investigated it two years earlier, he would have found two more—the Leech Lattice contains half of the exceptional symmetry atoms.

Fischer’s Monsters • Bernd Fischer in Germany discovered three intriguing and very large permutation groups, modelled on the three largest Mathieu groups. • He then found a fourth one of a different type, and even larger, called the Baby Monster. • Using this as a cross-section, he turned up something even bigger, called the Monster.

Computer Constructions • When the exceptional groups were ‘discovered’, it was not always clear that they existed. • Proving existence could be tricky, and computers were sometimes used. • For example the Baby Monster was constructed on a computer. • BUT the Monster was too large for computer methods.

Constructing the Monster • Fischer, Livingstone and Thorne constructed the character table of the Monster, a 194 -by 194 array of numbers. • This showed the Monster could not live in fewer than 196, 883 dimensions. • 196, 883 = 47 59 71, the three largest primes dividing the size of the Monster. • Later Robert Griess constructed the Monster by hand in 196, 884 dimensions.

Mc. Kay’s Observation • 196, 883 + 1 = 196, 884, the smallest non -trivial coefficient of the j-function. • Mc. Kay wrote to Thompson who had further data on the Monster available. • Thompson confirmed that other dimensions for the Monster seemed to be related to coefficients of the jfunction.

Ogg’s Observation • Shortly after evidence for the Monster was announced, Andrew Ogg attended a lecture in Paris. • Jacques Tits wrote down the size of the Monster, as a product of prime numbers. • Ogg noticed these were precisely the primes that appeared in connection with his own work on the j-function.

Moonshine • The mysterious connections between the Monster and the j-function were dubbed Moonshine. • John Conway and Simon Norton investigated them in detail, proved they were real, and made conjectures about a deeper connection. • Their paper was called Monstrous Moonshine

Vertex Algebras and String Theory • The Moonshine connections involved the Monster acting in finite dimensional spaces. • Frenkel, Leopwski and Meurman combined these in an infinite dimensional space. • Their space had a vertex algebra structure, which brought in the mathematics of string theory.

Conway-Norton Conjectures • The conjectures by Conway and Norton were later proved by Richard Borcherds, who received a Fields Medal for his work, but as he points out, there are still mysteries to resolve • For example the space of ‘j-functions’ associated with the Monster has dimension 163. Is this just a coincidence? • e √ 163 = 262537412640768743. 99999925. . . is very close to being a whole number.