| NSF Org: |
DMS Division Of Mathematical Sciences |
| Recipient: |
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| Initial Amendment Date: | March 26, 2024 |
| Latest Amendment Date: | March 26, 2024 |
| Award Number: | 2401152 |
| Award Instrument: | Standard Grant |
| Program Manager: |
Adriana Salerno
asalerno@nsf.gov (703)292-2271 DMS Division Of Mathematical Sciences MPS Directorate for Mathematical and Physical Sciences |
| Start Date: | April 1, 2024 |
| End Date: | November 30, 2024 (Estimated) |
| Total Intended Award Amount: | $15,000.00 |
| Total Awarded Amount to Date: | $15,000.00 |
| Funds Obligated to Date: |
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| History of Investigator: |
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| Recipient Sponsored Research Office: |
1125 W MAPLE ST STE 316 FAYETTEVILLE AR US 72701-3124 (479)575-3845 |
| Sponsor Congressional District: |
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| Primary Place of Performance: |
526 Old Main FAYETTEVILLE AR US 72701-1201 |
| 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: |
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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
This award supports US-based scientists to attend the conference "Modular Forms, L-functions, and Eigenvarieties". The event will take place in Paris, France from June 18, 2024, until June 21, 2024. Whole numbers are the atoms of our mathematical universe. Number theorists study why patterns arise among whole numbers. In the 1970's, Robert Langlands proposed connections between number theory and mathematical symmetry. His ideas revolutionized the field. Some of the most fruitful approaches to his ideas have come via calculus on geometric spaces. The conference funded here will expose cutting edge research on such approaches. The ideas disseminated at the conference will have a broad impact on the field. The presentations of leading figures will propel junior researchers toward new theories. The US-based participants will make a written summary of the conference. The summaries will encourage the next generation to adopt the newest perspectives. Writing them will also engender a spirit of collaboration within the research community. The summaries along with details of the events will be available on the website https://www.eventcreate.com/e/bellaiche/.
The detailed aim of the conference is exposing research on modular forms and L-functions in the context of eigenvarieties. An eigenvariety is a p-adic space that encodes congruence phenomena in number theory. Families of eigenforms, L-functions, and other arithmetic objects find their homes on eigenvarieties. The conference's primary goal is exposing the latest research on such families. The presentations will place new research and its applications all together in one place, under the umbrella of the p-adic Langlands program.
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.
This project funded US-based participants to attend the international reserach conference "Modular forms, L-functions, and eigenvarieties", held in Paris, France in June 2024.
The primary outcome of the conference was research discussions on modern number theory. The field itself is as old as human thought, with the foundational investigations all being related to how we can solve equations and what structure the solution sets have. The structure of those solution sets is one of the fundamental bits of knowledge that allowed mathematicians to develop public key cryptosystems in the past fifty years, and it allows us to scientifically study how secure those cryptosystems are.
On a theoretical level, one of the driving goals of the past few centuries has been the establishment of successively more sophisticated reciprocity laws. These laws explain how to relate solution sets for one equation to solution sets for another. To know how to solve one equation is to know how to solve the other.
The primary research discussions at the conference involved objects called modular forms and L-functions, each of which play a central role in modern approaches to reciprocity laws. The eigenvarieties are geometric parameter spaces for these objects. The research presented was about how geometric properties can be leveraged to solve equations and establish new laws.
The participants funded by the project are among the future leaders of this research field. By attending the conference, they were able to get in-person access to some of the field's current luminaries. The participation of the conference overall was extremely broad, with multiple continents and dozens of countries represented. So, the funded participants have now developed connections to the research field that will fuel their growth and support further developments of number theory research in the United States.
The project participants impacted the community more widely by producing lecture summaries. Although title and abstracts for research talks are always available, these project participants took care to produce reports on the discussions surrounding the conference's research presentations. Each report is 2-3 pages long and will remain available for free on the conference's website https://www.eventcreate.com/e/bellaiche/. The summaries will help people who were unable to come to Paris understand the newest advances in the field.
Last Modified: 12/31/2024
Modified by: John F Bergdall
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