symmetry and the monster one of the greatest quests in mathematics
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Symmetry and the Monster
One of the greatest quests in mathematics
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A little early history
Equations of degree 2—meaning 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.
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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
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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.
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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 permutations;if a goes to b then b goes to a, if a goes to c then b goes to d.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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The Exceptions
• Some were found using the cross-section method
• Some were found by studying groups of permutations
• Some were found using geometry
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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.
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The Leech Lattice
• John Leech used the largest Mathieu group M24 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.
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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.
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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.
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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 = 475971, the three largest primes dividing the size of the Monster.
• Later Robert Griess constructed the Monster by hand in 196,884 dimensions.
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McKay’s 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.
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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.
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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
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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.
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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 = 262537412640768743.99999999999925... is very close to being a whole number.