| NSF Org: |
DMS Division Of Mathematical Sciences |
| Recipient: |
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| Initial Amendment Date: | January 28, 2021 |
| Latest Amendment Date: | January 28, 2021 |
| Award Number: | 2109144 |
| Award Instrument: | Standard Grant |
| Program Manager: |
Christopher Stark
DMS Division Of Mathematical Sciences MPS Directorate for Mathematical and Physical Sciences |
| Start Date: | January 1, 2021 |
| End Date: | July 31, 2022 (Estimated) |
| Total Intended Award Amount: | $193,421.00 |
| Total Awarded Amount to Date: | $66,423.00 |
| Funds Obligated to Date: |
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| History of Investigator: |
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| Recipient Sponsored Research Office: |
3 RUTGERS PLZ NEW BRUNSWICK NJ US 08901-8559 (848)932-0150 |
| Sponsor Congressional District: |
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| Primary Place of Performance: |
33 Knightsbridge Road Piscataway NJ US 08854-3925 |
| 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): | GEOMETRIC ANALYSIS |
| Primary Program Source: |
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| 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
Geometric shapes can be either smooth or singular. Smooth ones have been studied effectively by the calculus, while singular geometric shapes are much more difficult to study in general although they appear with abundance in the world around us. One method to understand singular shapes is by measuring distances between their points. To do that, the best way is to use the so-called Einstein structures, which originate from general relativity. In this project, the investigator plans to study Einstein structures on a class of geometric shapes called algebraic varieties, which are central objects in many branches of mathematics. This study will allow us to measure distances and reveal certain mysterious structures of algebraic varieties. This project requires combinations of many techniques and will bring experts from different fields to interact. Its outcome will have potential applications in the development of several theories, including canonical metrics in differential geometry, stability theory in algebraic geometry, and string theory in mathematical physics.
The investigator will study the Yau-Tian-Donaldson conjecture about the equivalence of K-stability and the existence of Kahler-Einstein metrics on singular Fano varieties. This requires new strategies to overcome difficulties due to the presence of singularities. The investigator has introduced a new process, the minimization of normalized volumes, for detecting local geometries of algebraic singularities. On the algebraic side, the investigator will continue his research on the K-stability of Fano varieties by studying minimization of normalized volumes and applying deep techniques of the minimal model program from algebraic geometry. This could lead to new criteria for the K-stability of singular varieties. On the analytic side, the investigator will apply various newly-developed techniques, including a variational approach via pluripotential theory, a priori estimates for singular complex Monge-Ampere equations, Cheeger-Colding-Tian's regularity theory from metric geometry, algebraic structures on Gromov-Hausdorff limits, and asymptotical analysis of singular metrics. The combination of these techniques will be effective in solving Kahler-Einstein equations on singular varieties.
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 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.
Chi Li has completed the project titled Kahler-Einstein Metrics on Fano Varieties supported by NSF. During four years of the project, he has carried out extensive research on finding new metric structures on projective varieties. In particular, he together with collaborators are able to construct Kahler-Einstein structures on uniformly stable Fano varieties with singularities. This is the first time that the existence of such singular metric structures is discovered. The PI then studied infinitesimal structure of these singularities and also long time behavior of flows of Kahler metrics. He showed that the behaviors are uniquely determined by the underlying algebraic structure of varieties. The PI has further studied the more general constant scalar curvature Kahler metrics and obtained a sufficient algebraic condition for the existence of such metric structure. This is currently the most frontier result towards a central conjecture called Yau-Tian-Donaldson conjecture in complex geometry.
To obtain these results, the PI needs to establish new connections between analytic geometry and algebraic geometry, and develop new techniques in various directions in analysis and algebra, including non-Archimedean geometry and pluripotential theory.
The PI's NSF project has led to a lot of research activities in the field including thesis projects of several Ph.D. students and joint works with a postdoc. The results described above have been widely cited among researchers studying Kahler geometry and have stimulated significant progress in related subjects.
The PI was invited to give a 45-min talk about the above results in International Congress of Mathematicians (ICM) 2022.
The PI co-organized a section on Complex Geometry in an AMS sectional meeting in March 2022 at Purdue University. He also organized several seminars at Purdue University and Rutgers University. He is a co-organizer of the annual Rutgers Geometric Analysis Conference. His expertise in complex geometry has attracted and brought many mathematicians in the field to visit and communicate.
Last Modified: 11/29/2022
Modified by: Chi Li
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