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Tuesday, November 21, 2006

Reading List: Symmetry and the Monster

Ronan, Mark. Symmetry and the Monster. Oxford: Oxford University Press, 2006. ISBN 0-19-280722-6.
On the morning of May 30th, 1832, self-taught mathematical genius and revolutionary firebrand Évariste Galois died in a duel in Paris, the reasons for which are forgotten; he was twenty years old. The night before, he wrote a letter in which he urged that his uncompleted mathematical work be sent to the preeminent contemporary mathematicians Jacobi and Gauss; neither, however, ever saw it. The work in question laid the foundations for group theory, an active area of mathematical research a century and three quarters hence, and a cornerstone of the most fundamental theories of physics: Noether's Theorem demonstrates that conservation laws and physical symmetries are two aspects of the same thing.

Finite groups, which govern symmetries among a finite number of discrete items (as opposed to, say, the rotations of a sphere, which are continuously valued), can be arbitrarily complicated, but, as shown by Galois, can be decomposed into one or more simple groups whose only normal subgroups are the trivial subgroup of order one and the improper subgroup consisting of the entire group itself: these are the fundamental kinds of symmetries or, as this book refers to them, the “atoms of symmetry”, and there are only five categories (four of the five categories are themselves infinite). The fifth category are the sporadic groups, which do not fit into any of the other categories. The first was discovered by Émile Mathieu in 1861, and between then and 1873 he found four more. As group theory continued to develop, mathematicians kept finding more and more of these sporadic groups, and nobody knew whether there were only a finite number or infinitely many of them…until recently.

Most research papers in mathematics are short and concise. Some group theory papers are the exception, with two hundred pagers packed with dense notation not uncommon. The classification theorem of finite groups is the ultimate outlier; it has been likened to the Manhattan Project of pure mathematics. Consisting of hundreds of papers published over decades by a large collection of authors, it is estimated, if every component involved in the proof were collected together, to be on the order of fifteen thousand pages, many of which are so technical those not involved in the work itself have extreme difficulty understanding them. (In fact, a “Revision project” is currently underway with the goal of restating the proof in a form which future generations of mathematicians will be able to comprehend.) The last part of the classification theorem, itself more than a thousand pages in length, was not put into place until November 2004, so only then could one say with complete confidence that there were only 26 sporadic groups, all of which are known.

While these groups are “simple” in the sense of not being able to be decomposed, the symmetries most of them represent are of mind-boggling complexity. The order of a finite group is the number of elements it contains; for example, the group of permutations on five items has an order of 5! = 120. The simplest sporadic group has an order of 7920 and the biggest, well, it's a monster. In fact, that's what it's called, the “monster group”, and its order is (deep breath):

808,017,424,794,512,875,886,459,904,961,710,757,005,754,368,000,000,000 =
If it helps, you can think of the monster as the group of rotations in a space of 196,884 dimensions—much easier to visualise, isn't it? In any case, that's how Robert Griess first constructed the monster in 1982, in a 102 page paper done without a computer.

In one of those “take your breath away” connections between distant and apparently unrelated fields of mathematics, the divisors of the order of the monster are precisely the 15 supersingular primes, which are intimately related to the j-function of number theory. Other striking coincidences, or maybe deep connections, link the monster group to the Lorentzian geometry of general relativity, the multidimensional space of string theory, and the enigmatic properties of the number 163 in number theory. In 1983, Freeman Dyson mused, “I have a sneaking hope, a hope unsupported by any facts or any evidence, that sometime in the twenty-first century physicists will stumble upon the Monster group, built in some unsuspected way into the structure of the universe.” Hey, stranger things have happened.

This book, by a professional mathematician who is also a talented populariser of the subject, tells the story of this quest. During his career, he personally knew almost all of the people involved in the classification project, and leavens the technical details with biographical accounts and anecdotes of the protagonists. To avoid potentially confusing mathematical jargon, he uses his own nomenclature: “atom of symmetry” instead of finite simple group, “deconstruction” instead of decomposition, and so on. This sometimes creates its own confusion, since the extended quotes from mathematicians use the standard terminology; the reader should refer to the glossary at the end of the book to resolve any such puzzlement.

Posted at November 21, 2006 22:27