| NSF Org: |
DMS Division Of Mathematical Sciences |
| Recipient: |
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| Initial Amendment Date: | September 20, 2021 |
| Latest Amendment Date: | September 20, 2021 |
| Award Number: | 2153115 |
| Award Instrument: | Continuing 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: | June 1, 2021 |
| End Date: | May 31, 2024 (Estimated) |
| Total Intended Award Amount: | $330,000.00 |
| Total Awarded Amount to Date: | $133,960.00 |
| Funds Obligated to Date: |
FY 2020 = $65,674.00 FY 2021 = $65,078.00 |
| History of Investigator: |
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| Recipient Sponsored Research Office: |
1 NASSAU HALL PRINCETON NJ US 08544-2001 (609)258-3090 |
| Sponsor Congressional District: |
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| Primary Place of Performance: |
Fine Hall Princeton NJ US 08544-1000 |
| 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: |
01002021DB NSF RESEARCH & RELATED ACTIVIT 01002122DB 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
The Principal Investigator (PI) will study varieties. Varieties are defined as the set of solutions of systems of polynomial equations. They are fairly easy to compute and Nash proved every space can be well approximated by varieties. The main aim of the project is to understand how varieties vary if we change the coefficients of the defining polynomial equations, especially for the ones which are positively curved. Such varieties are called Fano varieties. In particular the research will try to understand situations when a family of Fano varieties degenerates to one with singularities.
The PI intends to prove that among all Fano varieties, the K-polystable ones can be parametrised by a universal space, called moduli space. As part of the this project, the PI aims to show the moduli space is Hausdorff and compact. The PI aims to understand which Fano varieties are K-semistable by understanding concrete examples as well as some general phenomena. The PI aims to understand the degeneration of Calabi-Yau manifolds, through the interplay between birational and non-archimedean geometry.
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.
In both the physical world and theoretic studies, many objects can be described by polynomial equations. In many problems people are interested in how things change when one varies the coefficients. In algebraic geometry, this type of study is called moduli theory. While when the objects given by the polynomial equations are negatively curved, the moduli theory had been developped for decades, the case when the objects are positively curved had been mysterious for a long time.
The aim of the project was to remedy the situation. It turns on this can be achieved by connecting it with a topic originated in physics, namely the Kahler-Einstein problem. The corresponding algebraic notion appeared is called K-stability. In a series of paper, the PI, joint with others, develops a satisfactory theory of K-stability in higher dimensional geometry. The most important output then is to provide a theoretical construction of moduli spaces of positive curved objects. Naturally, it also leads to solutions to many other deep questions in both the fields of Kahler-Einstein problem and moduli theory. A remaining open-ended question, which is an active field now, is to write down explicit examples in the framework of this theory.
Besides many research papers, the PI has written many survey articles as well as a book to make the topic more accessible. A large number of young researchers are now doing research in this newly established field.
Last Modified: 07/01/2024
Modified by: Chenyang Xu
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