Preface
An international conference “Ramanujan 125” was held November 5–7, 2012, in
Gainesville, Florida. The conference, which was organized by Krishnaswami Alladi
and Frank Garvan of the University of Florida, and Ae Ja Yee of The Pennsylvania
State University, attracted 70 active research participants from around the world.
The conference was supported by grants from the National Science Foundation and
the National Security Agency and by funds from The Pennsylvania State University
through the NSF grant of Ae Ja Yee. We are most grateful for this support that
was crucial to the success of the conference.
The conference featured ten plenary talks of one hour each by leaders in the
world of Ramanujan’s mathematics, and 40 shorter presentations including sev-
eral by graduate students. These lectures discussed significant progress in various
branches of mathematics in the quarter century since Ramanujan’s centennial
progress directly related to Ramanujan’s work or topics whose origins can be traced
to Ramanujan’s discoveries. This Contemporary Mathematics volume is the refer-
eed proceedings of the conference and contains research and expository papers based
on talks delivered at the conference. All papers have been arranged in alphabetical
order of the first author’s last name.
In his last letter to Hardy in January 1920, Ramanujan communicated his
discovery of the mock theta functions, which mimic the theta functions in the sense
that their coefficients can be estimated to the same degree of precision as in the
case of objects expressible in terms of theta functions. The mock theta functions
are now considered to be among Ramanujan’s deepest contributions. Ramanujan
had obtained asymptotic evaluations of these mock theta functions, and in his letter
had observed that if certain well-behaved analytic expressions were subtracted from
the mock theta functions, then the resulting error would be bounded. He also
indicated bounds in certain instances. For many years the exact links between
mock theta functions and modular forms were unknown, and this was one of many
such tantalizing mysteries.
In the last decade, Ken Ono, Kathrin Bringmann, and their collaborators have
connected mock theta functions to harmonic Maass forms, thereby providing the
key to unlock this mystery by developing the ideas in a fundamental 2003 PhD
thesis of Sander Zwegers that was written under the direction of Don Zagier in
Bonn.
On the opening day of this conference, Ono announced for the first time his
recent work with Amanda Folsom and Robert Rhoades, in which they obtain a
precise expression for the bounded error Ramanujan indicated. We are pleased
that the three authors have contributed a paper on this topic to this Contemporary
Mathematics volume.
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