| NSF Org: |
OISE Office of International Science and Engineering |
| Recipient: |
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| Initial Amendment Date: | March 28, 2013 |
| Latest Amendment Date: | March 28, 2013 |
| Award Number: | 1318015 |
| Award Instrument: | Standard Grant |
| Program Manager: |
Akaysha Tang
OISE Office of International Science and Engineering O/D Office Of The Director |
| Start Date: | August 1, 2013 |
| End Date: | December 31, 2014 (Estimated) |
| Total Intended Award Amount: | $29,610.00 |
| Total Awarded Amount to Date: | $29,610.00 |
| 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: |
Beijing/Tianjin CH |
| 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, Catalyzing New Intl Collab |
| 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): | 47.079 |
ABSTRACT
The proposed project catalyzes a new collaboration between the Louisiana State University and three institutes in China, in the area of computational algebraic geometry.Research is planned in 3 distinct domains:
Problems in Arithmetic Algebraic Geometry. Specifically: (1) Moduli of genus 3 curves with special endomorphisms of their Jacobians. (2) Modularity questions for the Galois representations that arise from curves in part (1); (3) Special values of L-functions for Galois representations of the shape ρf ⊗ φ where f is a modular form and φ is a nonabelian Artin representation; (4) Dynamical systems arising from integer matrices.
Polynomial systems. Develop efficient algorithms for solving systems of polynomials via in- volutive bases such as Pommaret bases. Extend work already done for solving 0-dimensional systems to higher dimensions.
Generalized hypergeometric equations. Study the GKZ hypergeometric systems attached to interesting polytopes, for instance those coming from root systems. These give rise to interesting families of varieties, such as Calabi-Yau varieties.
All three areas are related to currently active research going on in algebraic geometry, and projects will involve both theoretical and computational tools.
Algebraic Geometry - is concerned with solutions to systems of polynomial equations. Polynomials are ubiquitous in pure and applied mathematics. They are fundamental objects occurring in practically every domain of science, and their study is central to many areas of current mathematical research. Part of the project is a study of spaces defined by polynomials. Another part is to study algorithms for the efficient solution to polynomial equations. This project also explores connections to other parts of mathematics, for instance number theory.
This proposal also supports one month visits each for a graduate student and a young post- doctoral researcher. The locus of this research is China, especially Beijing and Tianjin. The PI will be collaborating with Chinese mathematicians, and also teaching courses on related topics. This project allows two young U.S. mathematicians to participate in the exciting mathematical world developing in China. Establishing and extending such international collaborations can have a major positive impact not only on the narrow goals of a research project, but also on the long-term development of research and teaching in the US.
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.
Goals. This grant was to help support visits to China for the purpose of:
1. Continue existing research projects with Chinese mathematicians, and establish new ones.
2.]Teach Chinese mathematics students.
3. Mentor American mathematics students and introduce them to Chinese mathematicians.
Major progress was made on all three items. The PI made two trips to China in the duration of the grant: the first was for the period July 2013 - January 2014. The second was in summer 2014. The first visit was a Sabbatical leave from my position in the Mathematics Department of Louisiana State University (LSU). In both cases, the PI was at the Beijing International Center for Mathematics Research (BICMR) on the Peking University campus. The research conducted has led to several publications, and others in preparation. I taught courses at Peking University and the Chinese Academy of Sciences. I mentored two American graduate students and one American postdoc on a one month visit to the Chern Institute of Nankai University.
Sabbatical. I spent the Fall semester 2013 in China on a sabbatical leave from my regular position as Professor of Mathematicsat Louisiana State University. I was there from July 30, 2013 to January 7, 2014. During that time I was mostly in Beijing, but I made trips to Taiwan, and the cities of Lanzhou and Tianjin in China, for the purposes of research. This visit was partiallysupported by NSF grant F47000 OISE 1318015, ``Problems in Computational Algebraic Geometry'' and NSA grant H98230-11-1-0160, ``Computational Aspects of Certain Shimura Varieties''.
Outline of Activities.
1. During August I worked with Zhibin LIANG (Beijing International Center for Mathematical Research = BICMR), Dun LIANG (my Ph.D. student from LSU, who was also spending the Fall Semester in China), Yukiko SAKAI (postdoc currently at Kitasato University in Japan), Haohao WANG (Southeast Missouri State University), and Ryotaro OKAZAKI (Doshisha University, Japan) on the problem of finding explicit families of genus 3 algebraic curves X such that its Jacobian has special endomorphisms. This project was a success: we have two papers submitted for publication and my student Dun LIANG finished his Ph.D under my direction at LSU in December 2014. He is accepting a job at Tsinghua University in Beijing.
2. During that time I worked with Zhibin Liang, Yukiko Sakai and Xuehzi Zhao on a problem in discrete dynamical systems. A paper has been accepted for publication in the Advances in Mathematics
3. During September 2-8, I visited the NCTS (National Center for Mathematical Research) in Hsinchu, Taiwan. I gave a lecture on my work on genus 3 curves outlined above, and I gave a seminar talk on ``Counting $\ell$-adic sheaves (d'apr\`es Deligne and Flicker)''.
4. During the months September, October and November I gave two courses: (1) at Peking University: Modular Forms(taught with Zhibin LIANG); (2) at the Chinese Academy of Sciences: Toric Varieties.
5. October 15-20, visited Lanzhou University. Gave a series of 5 lectures: ``Elliptic curves and modular forms''.
6. December 6-30, visited the Chern Institute at Nankai University in Tianjin. Worked with Professor Lei FU on character sums attached to GKZ hypergeometric systems. During the period December 19-30, I was joined by 2 students Benjamin FILIPPENKO (MIT), Owen BARRETT (Yale), and a postdoc Joel GEIGER (MIT) for this project. These three had worked with me in Summer 2013 in the REU program at LSU. The visit of these three was supported in part by the NSF grant mentioned above. There was a seminar in the week December 23-27, where we all lectured ...
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