| NSF Org: |
DMS Division Of Mathematical Sciences |
| Recipient: |
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| Initial Amendment Date: | July 16, 2012 |
| Latest Amendment Date: | July 16, 2012 |
| Award Number: | 1201435 |
| Award Instrument: | Standard Grant |
| Program Manager: |
Andrew Pollington
adpollin@nsf.gov (703)292-4878 DMS Division Of Mathematical Sciences MPS Directorate for Mathematical and Physical Sciences |
| Start Date: | August 1, 2012 |
| End Date: | July 31, 2015 (Estimated) |
| Total Intended Award Amount: | $134,006.00 |
| Total Awarded Amount to Date: | $134,006.00 |
| Funds Obligated to Date: |
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| History of Investigator: |
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| Recipient Sponsored Research Office: |
202 HIMES HALL BATON ROUGE LA US 70803-0001 (225)578-2760 |
| Sponsor Congressional District: |
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| Primary Place of Performance: |
Lockett Hall Baton Rouge LA US 70803-2701 |
| Primary Place of
Performance Congressional District: |
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| Unique Entity Identifier (UEI): |
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| Parent UEI: |
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| NSF Program(s): |
ALGEBRA,NUMBER THEORY,AND COM, EPSCoR Co-Funding |
| Primary Program Source: |
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| Program Reference Code(s): |
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| Program Element Code(s): |
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| Award Agency Code: | 4900 |
| Fund Agency Code: | 4900 |
| Assistance Listing Number(s): | 47.049 |
ABSTRACT
The primary aim of this research program is to study the applications of modular and automorphic forms, and hypergeometric q-series. These applications include a wide selection of different areas of mathematics and mathematical physics, such as the theory of integer partitions, combinatorial probability and Markov processes, bootstrap percolation models, affine Lie superalgebras, and (quadratic) Hurwitz class numbers. The automorphic objects of interest include modular and Jacobi forms, as well as mock modular and Jacobi forms, which have seen a great deal of recent interest thanks to work of Borcherds, Bringmann, Bruinier, Funke, Ono, Zagier, and Zwegers. Mock modular forms are of particular interest due both to their connections with harmonic Maass forms, as well as their famous history as objects of mystery dating back to Ramanujan and Watson. As an example of the interplay between automorphic forms and other topics, the PI and Bringmann recently used combinatorial probability bounds for gap-avoiding sequences (that first arose in Holroyd, Liggett, and Romik's study of finite-size scaling in bootstrap percolation) in order to prove a cuspidal asymptotic expansion for a family of hypergeometric q-series considered by Andrews.
Due to its scope, this research program has the potential for wide-ranging applications. An underlying theme is the universality of the tools and techniques of modern number theory, whose best-known uses include cryptography and cellular communication. This research will also illustrate applications to high-energy physics (where wall-crossings and black holes are described by mock theta functions), and biological 'growth' processes (where the the large-scale behavior of cellular automata is determined by the asymptotic expansions of modular forms).
PUBLICATIONS PRODUCED AS A RESULT OF THIS RESEARCH
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PROJECT OUTCOMES REPORT
Disclaimer
This Project Outcomes Report for the General Public is displayed verbatim as submitted by the Principal Investigator (PI) for this award. Any opinions, findings, and conclusions or recommendations expressed in this Report are those of the PI and do not necessarily reflect the views of the National Science Foundation; NSF has not approved or endorsed its content.
During the course of research supported by this Award, the PI investigated applications of modular and automorphic forms, and hypergeometric q-series. The automorphic objects included classical theta functions, modular forms, and Jacobi forms, as well as classes of functions whose full structure are still being understood, notably including the Ramanujan's enigmatic mock theta functions and the corresponding harmonic weak Maass forms. In order to use the symmetries of these functions to answer questions about limiting behaviors, the PI also contributed to the further development of the Hardy-Ramanujan Circle Method and Wright's variant form. For example, in one application the PI proved a quantified and asymptotic version of Garvan's conjecture for inequalities among Dyson's partition statistics. In another application the PI answered a physical question first posed by Temperley in his study of saw-toothed crystal surfaces in thermal equilibrium, providing the precise governing constants. Other applications arose from the classical partition identities of Rogers-Ramanujan and Schur, as well as more recent extensions from Capparelli's calculations in the vertex operator program of Lepowsky-Wilson. The results here included a collection of new identities, generalizations, relations to modular and mock modular forms, and analytic properties for these classical partitions. The PI also began a new research program and collaborations on nonstandard analysis to additive number theory, which resulted in many structural results: if a set of integers (or elements in certain topological groups) is sufficiently "large", then it is forced to have regularity properties. One such result represented significant progress on a sumset conjecture of Erdos.
The PI also performed a number of synergistic activities. These included hosting and visiting collaborators, speaking at and attending conferences, workshops, and seminars, and mentoring and advising students at all levels.The PI is also very active in mathematics competitions at his sponsor institution, organizing a weekly Problem-Solving Seminar in order to train and recruit undergraduate student participation in events such as the Virginia Tech Regional Math Contest and the Putnam Mathematical Competition. Under the PI's guidance, student participation has significantly increased, including non-math majors and underrepresented groups. Additionally, the PI has encouraged several students over multiple years to subsequently enroll in graduate programs in mathematics.
Last Modified: 02/26/2016
Modified by: Karl Mahlburg
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