| NSF Org: |
DMS Division Of Mathematical Sciences |
| Recipient: |
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| Initial Amendment Date: | January 15, 2013 |
| Latest Amendment Date: | June 22, 2017 |
| Award Number: | 1254812 |
| Award Instrument: | Continuing Grant |
| Program Manager: |
James Matthew Douglass
mdouglas@nsf.gov (703)292-2467 DMS Division Of Mathematical Sciences MPS Directorate for Mathematical and Physical Sciences |
| Start Date: | June 1, 2013 |
| End Date: | May 31, 2019 (Estimated) |
| Total Intended Award Amount: | $414,004.00 |
| Total Awarded Amount to Date: | $414,004.00 |
| Funds Obligated to Date: |
FY 2014 = $90,312.00 FY 2015 = $71,683.00 FY 2016 = $83,887.00 FY 2017 = $95,466.00 |
| History of Investigator: |
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| Recipient Sponsored Research Office: |
W5510 FRANKS MELVILLE MEMORIAL LIBRARY STONY BROOK NY US 11794-0001 (631)632-9949 |
| Sponsor Congressional District: |
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| Primary Place of Performance: |
NY US 11794-3366 |
| 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, Division Co-Funding: CAREER |
| Primary Program Source: |
01001415DB NSF RESEARCH & RELATED ACTIVIT 01001516DB NSF RESEARCH & RELATED ACTIVIT 01001617DB NSF RESEARCH & RELATED ACTIVIT 01001718DB 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
This proposal is concerned with the study of the geometry of moduli spaces and the development of tools for studying them. In particular, the PI plans to apply and adapt some newly developed techniques in abstract Hodge theory to the geometric context of moduli spaces. The PI will focus on the investigation of special classes of varieties, such as Calabi-Yau threefolds and higher dimensional Hyperkahler manifolds. In a different, but related direction, the PI aims to construct geometric compactifications for certain classes of surfaces via the KSBA approach inspired by the minimal model program. The PI is actively involved with the training of undergraduate and graduate students, and in recent years he has organized several activities with a strong educational component. He will expand these activities. In particular, he will run a summer research activity for undergraduates. Also, as part of the thematic program on Calabi-Yau varieties to be held at the Fields institute (Fall 2013), the PI will organize an introductory school for graduate students and other training activities. Additional planed activities include developing new courses for undergraduates and students in the teacher education program, organizing a workshop on Hodge theory and moduli spaces, and writing a monograph.
The general area of the proposal is algebraic geometry with connections to complex geometry and arithmetic geometry. Algebraic Geometry studies the geometric properties of objects defined by algebraic (or equivalently polynomial) equations. Within Algebraic Geometry, the PI is interested in the study of moduli spaces. A moduli space is a geometric object that parameterizes the shapes of objects within a given topological class. By studying a moduli space, one obtains important information about geometric objects of a given kind, in particular about the existence or non-existence of objects with prescribed special properties. This study has numerous applications to algebraic geometry and other related fields including mathematical physics.
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.
The research area of this project is Algebraic Geometry, one of the oldest and most central subjects in Mathematics. Algebraic Geometry is concerned with the study of the geometric properties of objects defined by polynomial equations. Algebraic Geometry is related and strongly interacts with several other major subjects in Mathematics, such as Number Theory, Differential Geometry, and Topology. Algebraic Geometry has numerous real-life applications in various fields such as Cryptography, Mathematical Physics, and Mathematical Biology.
Concretely, this proposal studies the geometry of K-trivial algebraic varieties, which are one of the three main building blocks in Algebraic Geometry, and Mathematics more generally. Important classes of K-trivial varieties are the K3 surfaces, Calabi-Yau threefolds, and Hyperkaehler manifolds. In particular, the Calabi-Yau threefolds play a fundamental role in our understanding of the Universe; they can be used to model the Universe in String theory. The PI has made fundamental contributions to the study of K-trivial varieties, especially regarding the study of their moduli spaces. Among the highlights of Laza?s research supported by this grant, we note the development of a systematic program to study the birational geometry of the moduli space of K3 surface, a new construction of an exotic example of Hyperkaehler manifolds, and the development of tools for studying the period map and the degenerations of Calabi-Yau threefolds. These questions are at the core of the subject and progress on them should have far reaching implications. The research supported by the NSF grant led to numerous publications in top level mathematical journals. At the same time, all these publications are freely available on the online mathematical archive (arXiv).
In addition to a substantial research output, the grant has supported numerous training and development activities of the PI. Namely, approximately 10 students and postdocs of the PI were partially supported by this grant and directly involved in the research activities of the proposal. Among the many activities organized by the PI (partially supported by the CAREER grant), we mention a major international research conference that was held in Stony Brook, NY (April 2015), and several training schools targeted to students and postdocs. All these activities are an important contribution towards assuring a strong and vibrant research in this crucial research area in the future.
Last Modified: 11/23/2019
Modified by: Radu Laza
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