University of Florida, Mathematics Department
FIFTH
RAMANUJAN^{*}
COLLOQUIUM
by
Professor
John G. Thompson, FRS
^{**}
Emeritus: University of Cambridge and University of Florida
on
The Divisor Matrix, Dirichlet Series and SL(2,Z)
Date and Time: 4:00  5:00pm, Monday, March 21, 2011
Room: The Atrium (LIT 339)
Refreshments: At 3:30pm in the Atrium (LIT 339)



Abstract:
The lecture will be a report on joint work with Professor Sin which
began with a study of
the divisor matrix D = (d(i,j)) , where d(i,j) = 1 if i
divides j and is 0 otherwise; and i , j range over the
natural numbers. This matrix is in the ring of
columnfinite matrices and
there is a homomorphism of the modular group into the group of
invertible elements of this ring which maps T to D and maps
S to a square root of minus the identity element,
where T is (1 1; 0 1), and S is (0 1;0 1).
It is this representation of the modular group
which has been the focus of our study. It leads naturally to
an action of the modular group on the ring of ordinary
Dirichlet series, and to an explicit elliptic curve. I shall
discuss other properties of this representation as well.
NOTE: After the Ramanujan colloquium, Professor Thompson
will
give the following seminars on the same topic.
Number Theory Seminar, Tue, Mar 22 at 1:55pm in The Atrium
Algebra Seminar, Wed, Mar 23 at 11:45am in The Atrium

^{*}
ABOUT RAMANUJAN: Srinivasa Ramanujan (18871920), a selftaught genius
from South India, dazzled mathematicians at Cambridge University by
communicating bewildering formulae in a series of letters. G. H. Hardy
invited Ramanujan to work with him at Cambridge, convinced that
Ramanujan was a "Newton of the East"! Ramanujan's work has had a
profound and wide impact within and outside mathematics. He is considered
one of the greatest mathematicians in history.
^{**}
ABOUT THE SPEAKER:
John Griggs Thompson is noted for his work in the classification of
finite simple groups.
In 1971, Thompson was elected to the United States National Academy of Sciences. In 1982, he was awarded the Senior Berwick Prize of the London Mathematical Society, and in 1988, he received the honorary degree of Doctor of Science from the University of Oxford. Thompson received the Wolf Prize in 1992, and is the only person other than JeanPierre Serre to have won all three of the Fields Medal, the Wolf Prize, and the Abel Prize. Thompson was awarded the United States National Medal of Science in 2000. He is a Fellow of the Royal Society (United Kingdom), and a recipient of its Sylvester Medal. He is a member of the Norwegian Academy of Science and Letters.
He was Graduate Research Professor at the University of Florida from 1993
to 2010.
ABOUT THE SPONSOR: Evan Pugh Professor George Andrews of The Pennsylvania
State University is the world's premier authority in the theory of partitions
and work of the Indian mathematical genius Srinivasa Ramanujan combined.
He is a Member of the National Academy of Sciences.
He has close ties
with the UF Mathematics Department which has one of the strongest programs
on mathematics related to Ramanujan's work. He was a recipient of an
Honorary Doctorate from UF in December 2002. Since 2005, he is a Distinguished
Visiting Professor each year in the Spring term in the Mathematics Department.
He has just finished his term as President of the American Mathematical Society.
Ramanujan Colloquium *
University of Florida *
Mathematics *
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