Mark Ronan's website
Mathematicians involved
in the Classification
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Here are some of the
mathematicians involved in my book Symmetry and the Monster. Late 18th
century to mid 20th century Joseph Louis Lagrange
(1736–1813) Born Guiseppe
Lodovico Langrangia in northern Italy, he became professor in Berlin for more
than 20 years, before taking up a position in Paris. He was one of the great
mathematicians, working on many different aspects of mathematics: the three
body problem; differential equations; number theory; probability; mechanics;
and the stability of the solar system. In particular he published an
influential paper (Reflections on the Algebraic Solution of Equations) in 1770.
This paper inspired the work of many others, including Galois. For
biographical information see the St-Andrews website,
and Wikipedia. Galois died in
1832 at the age of twenty. He was fatally wounded in a duel, but the night
before the duel he wrote a long letter explaining his mathematical ideas.
Among other things he studied the question of when an algebraic equation has
solutions that can be expressed in terms of radicals (meaning square roots,
cube roots, and so on). His method involved treating the solutions as objects
that could be permuted among one another. The group of allowable permutations
— the Galois group of the equation — reveals immediately whether
the solutions can be expressed in terms of radicals, without knowing a single
solution. Galois' ideas were published in 1846, and have been extremely
influential, leading to what is now known as Galois theory. For biographical
information see the St-Andrews website,
and Wikipedia. Augustin-Louis
Cauchy (1789–1857) Cauchy used
clear and rigorous methods in studying calculus, and wrote several
influential books on the topic. He also had wide-ranging interests and played
a role in the early history of group theory. He proved a theorem showing that
if the size of a group is divisible by a prime number p, then it has a subgroup of size p. For biographical information see the St-Andrews website,
and Wikipedia. Jordan's 1870
Treatise on permutations and algebraic equations clarified and expanded the
new subject of group theory, particularly in connection with Galois's work. For
biographical information see the St-Andrews website,
and Wikipedia. Lie was born in
Oslo in 1842 (though at that time the city was called Christiania). He was a
larger than life character who developed new methods for studying solutions
to differential equations (equations involving rates of change). In this
context he introduced groups in which each operation could be gradually
modified — they are now known as Lie groups. Lie took up a chair in
Germany in 1886, but returned to a chair in Norway a few months before his
death in February 1899. For biographical information see the St-Andrews website,
and Wikipedia. Killing discovered Lie algebras
independently of Lie's work. He then went on to classify them, and from this
classification the table of most finite 'symmetry atoms' was created. For
biographical information see the St-Andrews website,
and Wikipedia. In his PhD
thesis, Cartan revised Killing's proofs of the classification of Lie
algebras. He then went on to make significant contributions to differential
equations and geometry. More details, click here. For biographical
information see the St-Andrews website,
and Wikipedia. Burnside wrote
the first book on group theory in English, published in 1897, and developed
the subject from the modern abstract point of view. In 1904 he proved that
the size of any finite simple group that is non-cyclic must be divisible by
at least three different prime numbers. For biographical information see the
St-Andrews website,
and Wikipedia. Leonard Eugene Dickson
(1874–1954) In 1901 he
published a book showing how to obtain finite versions for most families of
Lie groups. This was the start of the 'periodic table' of finite simple
groups. For biographical information see the St-Andrews website,
and Wikipedia. Brauer founded
the 'cross-section' (i.e. involution centralizer) approach to classifying the
finite simple groups. He also did leading work on the character theory of
finite groups. For biographical information see the St-Andrews website,
the National Academy of Sciences website,
and Wikipedia. Chevalley worked
on group theory and ring theory and in 1955 published a paper showing how to
obtain finite versions of Lie groups in all families. For biographical
information see the St-Andrews website,
and Wikipedia. Like Chevalley,
Tits was also pursuing finite versions of Lie groups in all families, but in
a geometric way rather than using Chevalley's algebraic approach. It led him
to create the theory of buildings (which are 'multi-crystals', not buildings
in the usual sense), which he went on to develop in other important ways. In
2008, Tits was awarded the Abel Prize, jointly with John Thompson. For
biographical information, see the St-Andrews website,
and Wikipedia. Feit was an
expert on the character theory of finite groups, and collaborated with John
Thompson to prove the celebrated theorem (the Feit-Thompson theorem) showing
that a finite simple group that is not cyclic must have even size. For
biographical information, see the St-Andrews website,
Wikipedia, and the website
from the mathematics department at Yale, where he worked. Thompson's early
work led to his collaboration with Walter Feit on the great Feit-Thompson
theorem (above). He went on to deal with the cross-section method of
classifying finite simple groups, and was involved in studying the Monster
and the new simple groups inside it, one of which is named after him. In
2008, Thompson was awarded the Abel Prize, jointly with Jacques Tits. For
biographical information see the St-Andrews website,
and Wikipedia. Gorenstein was
the first person to put forward a plan for classifying all the finite simple
groups, and he was closely involved with steering this project forward. When
it appeared complete, he started the project, in collaboration with Lyons and
Solomon, of revising and rewriting it so that it would stand the scrutiny of
future generations. For biographical information see the St-Andrews website,
and Wikipedia. The
Classification and Discovery of the Sporadic Groups A great many
mathematicians were involved in the Classification project, but only a few
are mentioned in the book, and the same is true here. No disrespect is
intended to those who are missing—only people whose work appears in the
book are mentioned here, and the book is not a complete history of the
Classification. For that one needs to read the books by Gorenstein, and by
Gorenstein, Lyons and Solomon. Here the main topic is the discovery of the
sporadic groups. A French
mathematical physicist who, as a student, studied permutation groups that are
multiply transitive. His results yielded five simple groups that are not of
'Lie type'. These are the Mathieu groups M11, M12, M22, M23
and M24. For biographical
information see the St-Andrews website,
and Wikipedia. Witt created the
Witt design on 24 symbols. It gives a simple way of understanding the Mathieu
groups, and proves their existence. For biographical information see the
St-Andrews website,
and Wikipedia. Leech discovered
the Leech Lattice in 24 dimensions by using the Witt design, and started
studying its symmetry group. For biographical information see the St-Andrews website. Conway studied
the symmetries of the Leech Lattice from which he produced three new finite
simple groups, along with others that had already been found by other
methods. He later worked on the Monster and its moonshine connections. For
biographical information see the St-Andrews website,
and Wikipedia. In 1966, Janko
published the first new exception since Mathieu's groups a century earlier.
It is now known as J1. He went on to
discover three more: J2, J3 and J4.
All Janko's sporadic groups were discovered by the cross-section (involution
centralizer) method. For more information see Wikipedia. Suzuki made
important contributions to the classification project in the early days and
proved a version of the Feit-Thompson theorem in an important special case.
In the early 1960s he also discovered a new family of finite simple groups
that subsequently turned out to be groups of Lie type. Later in the 1960s he
discovered a sporadic group that bears his name. For biographical
information, see Wikipedia,
and the website
from the University of Illinois where he worked. Fischer
discovered several sporadic groups, three of which are known by his name.
These are the Fischer groups Fi22, Fi23 and Fi24 (the last one is not simple but contains a large simple subgroup).
Fischer also discovered the Baby Monster, from which emerged the Monster.
This in turn produced two new sporadic groups, which are named after those
who did most of the work on them: Thompson in one case, and Harada and Norton
in the other. For further information see Wikipedia. Donald
Livingstone (1924–2001) Livingstone and
Fischer together created the character table of the Monster, with Michael
Thorne writing the computer programs that they needed for the calculations. Hall constructed
Janko's group J2 as a group of
permutations on 100 symbols. For biographical information see the St-Andrews website,
and Wikipedia. Higman worked on the construction of several
sporadic groups that had been discovered by the cross-section (involution
centralizer) method. For biographical information see the St-Andrews website, or Wikipedia. Higman, in collaboration with Charles Sims,
adapted Hall's construction of J2 to produce another group of permutations on 100
symbols that was a new finite simple group. This is the Higman-Sims group.
For more information see Wikipedia. In addition to
being a co-discoverer of the Higman-Sims group, Sims used permutation
techniques to construct several other sporadic groups. In collaboration with
Jeffrey Leon he constructed the Baby Monster. For biographical information
see Wikipedia. Griess predicted the Monster independently of
Fischer, using Fischer's Baby Monster. He later constructed the Monster as
the group of symmetries for an algebra in 196,884 dimensions. For further
information see his homepage, or Wikipedia. McLaughlin created a new sporadic group (the
McLaughlin group) as a group of permutations. Rudvalis
predicted the existence of a new sporadic group (the Rudvalis group) as a
group of permutations. It was later constructed by Conway and David Wales.
For more information see Wikipedia. Held discovered
the sporadic group that bears his name. He used the cross-section (involution
centralizer) method, and the group was later constructed by Graham Higman and
John McKay. O'Nan discovered
the sporadic group that bears his name. He used the cross-section (involution
centralizer) method, and the group was later constructed by Charles Sims. Lyons discovered
the sporadic group that bears his name. He used the cross-section (involution
centralizer) method, and the group was later constructed by Charles Sims.
Then with Ronald Solomon and Daniel Gorenstein he undertook the Revision of
the Classification, a project that continues to this day. Harada studied
one of the two previously undiscovered simple groups that emerged as
subgroups of the Monster. It is named after him and Simon Norton. Norton
calculated a large amount of information on the Harada-Norton group, and on
the Monster itself. He also collaborated with Conway on the strange moonshine
connections with the j‑function
in number theory. For more information see Wikipedia. McKay made the
first observation of a numerical coincidence between the Monster and the j‑function in number theory. He also made
other intriguing observations, some of which have since been elucidated. For
more information see Wikipedia. With James
Lepowsky and Arne Meurman, he created the Moonshine module, connecting the
Monster and the j‑function. For more
information, see Wikipedia. With James
Lepowsky and Arne Meurman, he created the Moonshine module, connecting the
Monster and the j‑function. For
more information see Wikipedia. With James
Lepowsky and Arne Meurman, he created the Moonshine module, connecting the
Monster and the j‑function. For
more information see Wikipedia. Borcherds
created a Monster Lie algebra that led him to a proof of the Conway-Norton
conjectures for the Moonshine module, an achievement for which he was awarded
the Fields Medal
in 1998. For biographical information see Wikipedia. A large
amount of information is also available on his homepage. Aschbacher was
the greatest contributor to the Classification program, apart from Thompson.
He and Stephen Smith eventually filled in the missing part of the program,
the quasi-thin case. For some biographical information, see Wikipedia. In joint work,
Smith and Aschbacher filled in a gap in the Classification by finally nailing
the quasi-thin case. For biographical information see his homepage. Solomon,
together with Richard Lyons and Daniel Gorenstein undertook the Revision of
the Classification, a project that continues to this day. For biographical
information see his homepage. |
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