Symmetry and the Monster

1
Symmetry and the Monster

• One of the greatest quests in mathematics

2
A little early history

• Equations of degree 2meaning the highest power
  of x is x2 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 1500s.

3
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 1830s Évariste Galois

4
Galoiss 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.

5
x4 - 10x2  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 permutationsif a
  goes to b then b goes to a, if a goes to c then b
  goes to d.

6
The Galois Group

• Galois investigated when the solutions to a given
  equation can be expressed in terms of roots, and
  when they cant.
• The solutions can be deconstructed into roots
  precisely when the Galois group can be
  deconstructed into cyclic groups.

7
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.

8
Families of Simple Groups

• Galois discovered the first family of non-cyclic
  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.

9
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.

10
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 Chevalleys theme soon emerged, and
  by 1961 all finite groups of Lie type had been
  found.

11
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
  cross-sections, and Richard Brauer had shown
  that knowing one cross-section of a finite simple
  group gave a firm handle on the group itself.

12
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 groupscreated
  by Émile Mathieuare very exceptional. There is
  nothing else quite like them.

13
A Cat among the Pigeons

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

14
The Exceptions

• Some were found using the cross-section method
• Some were found by studying groups of
  permutations
• Some were found using geometry

15
The Hall-Janko group J2

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

16
The Leech Lattice

• John Leech used the largest Mathieu group M24 to
  create a remarkable lattice in 24 dimensions.
• John Conway studied Leechs lattice and turned up
  three new exceptions.
• Had he investigated it two years earlier, he
  would have found two morethe Leech Lattice
  contains half of the exceptional symmetry atoms.

17
Fischers 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.

18
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.

19
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.

20
McKays Observation

• 196,883  1  196,884, the smallest non-trivial
  coefficient of the j-function.
• McKay 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 j-function.

21
Oggs 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.

22
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

23
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.

24
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?v163  262537412640768743.99999999999925... is
  very close to being a whole number.