| NSF Org: |
DMS Division Of Mathematical Sciences |
| Recipient: |
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| Initial Amendment Date: | July 28, 2018 |
| Latest Amendment Date: | June 16, 2021 |
| Award Number: | 1844206 |
| 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: | September 1, 2018 |
| End Date: | February 28, 2022 (Estimated) |
| Total Intended Award Amount: | $150,000.00 |
| Total Awarded Amount to Date: | $212,467.00 |
| Funds Obligated to Date: |
FY 2021 = $62,467.00 |
| History of Investigator: |
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| Recipient Sponsored Research Office: |
9500 GILMAN DR LA JOLLA CA US 92093-0021 (858)534-4896 |
| Sponsor Congressional District: |
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| Primary Place of Performance: |
9500 Gilman Drive La Jolla CA US 92093-0621 |
| 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 |
| Primary Program Source: |
01002122DB NSF RESEARCH & RELATED ACTIVIT |
| 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
Number theory is one of the oldest branches of mathematics, and yet it continues to have more and more applications within the sciences. In this project, the principal investigator (PI) will investigate the relationship between two of the main focuses of number theory, both of which are utilized in computer science as well as physics: (1) prime numbers and the divisibility of integers and (2) algebraic solutions to polynomial equations. The fundamental idea is to understand the extent to which global objects can be arithmetically determined by the collection of its local pieces. The strategies and techniques that will be utilized in this project originate in a broad range of other mathematical subjects, including analysis, geometry, algebra and in some cases, statistics. Some of the specific questions the PI is interested in are at a level accessible to undergraduate and high-school students, and throughout the course of the project, the PI plans to utilize this to continue in educational efforts supporting an increase in diversity within mathematics.
This project surrounds the widespread phenomenon of local-global principles throughout algebraic and analytic number theory, ranging from understanding obstructions of unique prime factorization in rings of integers to determining the asymptotic number of global fields with fixed invariants via the number of local extensions with fixed p-adic invariants to proving local-global compatibility results within the Langlands program. First, the PI will conduct research that furthers the statistical study of class groups that originated with the Cohen-Lenstra heuristics; amongst others, this will include proving asymptotics for class groups of families of orders. Second, the PI will study number field distributions and the local-global principles that can control their asymptotics, beginning with the case of octic quaternion number fields. The strategy for obtaining such results will rely on arithmetic invariant theory, sieve methods, and geometry-of-numbers techniques utilized frequently in the field of arithmetic statistics. On the automorphic side, the PI will investigate arithmetic and geometric properties of p-adic families of Galois representations arising from non-conjugate self-dual regular algebraic automorphic representations of the general linear group over CM fields. This will involve studying eigenvarieties and strengthening p-adic interpolation methods.
This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.
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.
Prof. Ila Varma (formerly at UC San Diego, now at University of Toronto) works primarily in a subfield of number theory commonly known as arithmetic statistics. A central problem in this area is to give asymptotic formulas for the count of number fields with fixed degree and Galois group and discriminant varying over some range. The general shape of such formulas is predicted by a conjecture of Malle.
One important feature that has emerged recently is that when one counts the same class of objects using different sort orders, the formulas can change in subtle ways. The PI has established an explicit instance of this for number of fields of degree 4 whose Galois group is dihedral of order 8. This produces a new example of an elusive "local-global principle" which predicts that in some favorable situations, the count of number fields is given by an infinite product indexed by prime numbers.
In addition, the PI has been highly active in promoting diverse participation in the mathematical profession, through support of graduate students and postdocs at UC San Diego; active participation in Women in Numbers collaborative research conferences; and the founding on an India-based offshoot of the PROMYS summer mathematics program for high school students.
Last Modified: 06/29/2022
Modified by: Kiran Kedlaya
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