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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