| NSF Org: |
DMS Division Of Mathematical Sciences |
| Recipient: |
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| Initial Amendment Date: | July 8, 2019 |
| Latest Amendment Date: | September 9, 2020 |
| Award Number: | 1903892 |
| 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: | July 15, 2019 |
| End Date: | June 30, 2023 (Estimated) |
| Total Intended Award Amount: | $550,000.00 |
| Total Awarded Amount to Date: | $550,000.00 |
| Funds Obligated to Date: |
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| History of Investigator: |
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| Recipient Sponsored Research Office: |
845 N PARK AVE RM 538 TUCSON AZ US 85721 (520)626-6000 |
| Sponsor Congressional District: |
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| Primary Place of Performance: |
617 N. Santa Rita Ave Tucson AZ US 85721-0089 |
| 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): |
INFRASTRUCTURE PROGRAM, ALGEBRA,NUMBER THEORY,AND COM |
| 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 University of Arizona hosts "Arizona Winter School" (AWS), an annual week-long conference and workshop in which mathematics graduate students and undergraduates learn from and work under the guidance of leading experts on questions at the forefront of research in number theory and arithmetic geometry. This award supports the 2020, 2021, and 2022 meetings in the AWS series. The 2020 Arizona Winter School will be held from March 9-13 on the topic of "Non-Abelian Chabauty." Future topics will be based on important mathematical developments and the availability of key participants. AWS advances mathematics by catalyzing new research and generating a wealth of pedagogical materials, including detailed lecture notes, research project descriptions, problem session outlines, and high-quality video recordings of all the lectures. These resources from past AWS conferences, along with information about upcoming Winter School topics and application materials, are available through the AWS website: http://swc.math.arizona.edu/.
AWS 2020 will focus on the method of Chabauty and Coleman, a cornerstone of the arithmetic of curves and a key computational and theoretical tool. Non-abelian techniques subsequently gave a motivic proof of finiteness of solutions to the unit equation and laid the foundations of a program to push p-adic analytic techniques beyond the limitations of abelian integrals. The last decade has witnessed rapid progress (and applications), despite the technical depth of the subject. To effectively disseminate such a technical topic, the lecture series will be closely coordinated, and include classical Chabauty, computational aspects and applications, heights, arithmetic intersection theory, quadratic Chabauty, and a series of lectures about the conceptual and conjectural framework. Potential future topics for the AWS series include "Shimura Varieties," which would prepare students to be users of this ubiquitous theory; "Unlikely Intersections," which would survey recent advances on finiteness theorems for geometrically interesting intersections (e.g. the conjectures of Manin-Mumford and the Andre-Oort conjecture); "Complexity of Arithmetic Geometry Algorithms," which would focus on the theoretical analysis of algorithms in number theory and algebraic geometry; and "Automorphic Forms Beyond GL_2," which would introduce students to automorphic forms on groups beyond GL_2, emphasizing their applications to concrete number-theoretic problems.
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.
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 Arizona Winter School (AWS) is a five day workshop for graduate students in the mathematical field of arithmetic geometry, held annually in March at the University of Arizona in Tucson, Arizona. Since the first Winter School in 1998, the AWS has developed a significant reputation around the world as a highly successful, model school in graduate mathematics education. The NSF award was the primary source of support for the 2020-2022 meetings and provided supplemental support for AWS 2023.
Arithmetic geometry lies at the interface between the fields of number theory and algebraic geometry. At its core, it studies arithmetic questions, such as the finding of integral solutions to Diophantine equations, from a modern geometric perspective. In practice, it is a field of vast scope that utilizes and informs theory from much of modern mathematics.
Each Winter School is held on a topic of special interest to researchers in the field of arithmetic geometry, with lecture series given by four distinguished speakers on a particular topic related to their research. The topics of the schools supported by the award were:
AWS 2020: Nonabelian Chabauty
AWS 2021: Virtual School in Number Theory
AWS 2022: Automorphic Forms Beyond GL2
(AWS 2023: Unlikely Intersections)
The speakers prepare lecture notes prior to the school, which are posted online. Videos of the lectures are also posted online shortly after the talks. In evening working sessions, the speakers and a project assistant supervise smaller groups of Ph.D. students on research projects created prior to the school, and the students present their results on the last day of the school. Other groups of students work in one of two problem sessions led by a postdoctoral expert, or in one of four study groups that each follow one of the lectures led by an expert postdoctoral guide.
By numerous accounts, the totality of the events at the school make for an intensely rewarding experience for the Ph.D. students, aiding significantly in their development as research mathematicians. It is typical for many students to continue collaborations begun at the school and to publish the results of their work in research journals.
The 2020, 2022, and 2023 in-person meetings averaged 238 participants per year and included an average of 186 graduate students per year. These students hail from around the world, from institutions large and small, to participate in the event. Due to the pandemic, an in-person meeting was not possible in 2021, so an entirely on-line, 12-week Virtual School in Number Theory was organized. This instructional program was built around four 6-week courses, and was aimed at undergraduate students from underrepresented minority groups with the goal of increasing participation by these students in future in-person Arizona Winter Schools. Starting in 2023, the AWS has included a virtual, 6-week preliminary AWS (PAWS) each Fall, modeled on the 2021 school, and aiming to increase participation by underrepresented groups in the following AWS. By all indications, the reach, reputation, and impact of the AWS have continued to grow throughout the course of the award.
The AWS is organized by a group of five mathematicians linked to the Southwest U.S. and has an advisory board of distinguished mathematicians who help in the selection of the topics and advise on the selection of speakers for forthcoming schools. The organizers work to ensure the participation of a diverse group at all levels of the school and in the evening working groups, in particular by members of underrepresented groups. At the end of each meeting, the organizers provide a survey to all participants on the various aspects of the school and its organization. The organizers use this as a tool in making continued improvements to the structure and organization of the school.
Last Modified: 08/14/2023
Modified by: Bryden Cais
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