| NSF Org: |
DMS Division Of Mathematical Sciences |
| Recipient: |
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| Initial Amendment Date: | May 5, 2015 |
| Latest Amendment Date: | June 11, 2017 |
| Award Number: | 1501370 |
| Award Instrument: | Continuing Grant |
| Program Manager: |
janet striuli
DMS Division Of Mathematical Sciences MPS Directorate for Mathematical and Physical Sciences |
| Start Date: | June 1, 2015 |
| End Date: | May 31, 2019 (Estimated) |
| Total Intended Award Amount: | $347,630.00 |
| Total Awarded Amount to Date: | $347,630.00 |
| Funds Obligated to Date: |
FY 2016 = $115,847.00 FY 2017 = $115,861.00 |
| History of Investigator: |
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| Recipient Sponsored Research Office: |
400 HARVEY MITCHELL PKY S STE 300 COLLEGE STATION TX US 77845-4375 (979)862-6777 |
| Sponsor Congressional District: |
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| Primary Place of Performance: |
MS 3368 College Station TX US 77843-3368 |
| 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, Combinatorics |
| Primary Program Source: |
01001617DB NSF RESEARCH & RELATED ACTIVIT 01001718DB NSF RESEARCH & RELATED ACTIVIT |
| Program Reference Code(s): | |
| Program Element Code(s): |
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| Award Agency Code: | 4900 |
| Fund Agency Code: | 4900 |
| Assistance Listing Number(s): | 47.049 |
ABSTRACT
Algebraic geometry, which is the mathematical study of solutions to systems of polynomial equations, is a core mathematical discipline noted for its theoretical depth and its interactions with other areas of mathematics. Algebraic geometry is also a tool for applications, since physical objects can be described by polynomial equations, and relations between concepts in science and engineering may be modeled by polynomials. Whatever their source, once polynomials enter the picture, the theoretical base, trove of classical examples, and modern computational tools of algebraic geometry may be brought to bear on the problem at hand. This research project aims to strengthen the role of algebraic geometry as a tool for applications, in two ways. First, a major focus will be on developing the interface between algebraic geometry and applications. Second is work on combinatorial aspects of algebraic geometry, for this develops tools and ideas to better understand objects in algebraic geometry with special structure, and these highly structured objects are those that appear most commonly in the interactions between algebraic geometry and other fields, both within mathematics and in the applied sciences. This project will also involve training of students and postdocs, the investigator's outreach activities at the local level through a mathematics circle, and his continued interaction with mathematics in Nigeria.
Oftentimes, objects from algebraic geometry that arise in other parts of mathematics or science have strong combinatorial structures (e.g., toric varieties or Grassmannians) or else the application demands real solutions. Consequently, the research in areas of combinatorial and real algebraic geometry in this project will serve both to advance our basic understanding of these topics and to help build a foundation for applications. This project involves research in three topics within algebraic geometry: real toric varieties, tropical geometry, and Schubert calculus. In each of these areas the investigator will work with collaborators and students on projects ranging from developing foundations to understanding key examples to work inspired by problems from applications. Of particular note are the goals of developing a rich and robust theory of irrational toric varieties, understanding the geometry and topology of complements of tropical objects, establishing positivity in type C Schubert calculus via a useful theory of shifted dual equivalence, and understanding Galois groups in the Schubert calculus.
This award is jointly funded by the Algebra and Number Theory and Combinatorics programs.
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.
The intellectual merits of this project addressed theoretical and practical questions involving geometric objects that arise in applications of mathematics involving polynomial equations. A notable outcome was the development of a theory of real toric varieties from irrational fans, with graduate student Ata Pir. This completed the an understanding of these objects that often arise in applications, such as statistics and geometric modeling.
A theory of splines defined for curvilinear domains was developed. Splines are functions that are widely used to approximate complicated functions, in many applications.
Another notable oucome was the development of algorithms and software to solve geometric problems from the Schubert Calculus, a class of problems noted
for their ubiquity and detailed combinatorial structure. One such structure concerns the Galois groups of Schubert problems, and another outcome was a partial classification of Schubert Galois groups, which was a result of studying all 35000+ problems on a single Grassmannians. This also led to an understanding of some combinatorial structures that appear to control the Galois group.
Among the broader impacts of this proposal were the training of six Ph.D. students, Robert Williams, Ata Pir, Li Ying, Taylor Brysiewicz, Elise Walker, and Thomas Yahl (the first three have graduated). This helped Sottile organize a thematic semester on Nonlinear Algebra at the Institute for Computational and Experimental Research in Mathematics in Providence in Fall 2018, and play an role in the Fall 2016 program on Combinatorial Algebraic Geometry at the Fields Institute for Research in the Mathematical Sciences in Toronto. It also supported Sottile in the planning and execution of a research school in June 2017 in Nigeria, helping a graduate student to attend. It also directly supported the Texas A&M math circle that Sottile ran. This is a weekly mathematics outreach program for middle school students.
Last Modified: 07/18/2019
Modified by: Frank J Sottile
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