| NSF Org: |
DMS Division Of Mathematical Sciences |
| Recipient: |
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| Initial Amendment Date: | September 11, 2019 |
| Latest Amendment Date: | September 11, 2019 |
| Award Number: | 1949812 |
| 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: | June 1, 2019 |
| End Date: | June 30, 2022 (Estimated) |
| Total Intended Award Amount: | $136,876.00 |
| Total Awarded Amount to Date: | $136,876.00 |
| Funds Obligated to Date: |
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| History of Investigator: |
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| Recipient Sponsored Research Office: |
615 W 131ST ST NEW YORK NY US 10027-7922 (212)854-6851 |
| Sponsor Congressional District: |
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| Primary Place of Performance: |
2960 Broadway New York NY US 10027-6902 |
| 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): | |
| Program Element Code(s): |
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| Award Agency Code: | 4900 |
| Fund Agency Code: | 4900 |
| Assistance Listing Number(s): | 47.049 |
ABSTRACT
Algebraic Geometry has connections to many areas in mathematics, including topology, differential geometry, number theory, representation theory, combinatorics and the theory of differential equations. Over last 20 year important connections with string theory were discovered as well. Algebraic Geometry is the study of algebraic varieties: geometric objects that can be described as the collections of points satisfying a set of polynomial equations. One of the aims of the field is to classify algebraic varieties. This can be done by first associating discrete invariants to algebraic varieties and then studying all algebraic varieties with a given set of invariants. A basic invariant used in algebraic geometry, as well as in differential geometry, is the first Chern class. Algebraic varieties can be divided into classes according to the positivity properties (or lack thereof) of this invariant. One of the most important of these classes is that of varieties with first Chern class equal to zero. These varieties have a crucial role also in physics and in differential geometry. With this project the PI aims to advance our knowledge of hyper-K\"ahler manifolds which are, together with complex tori and Calabi-Yau manifolds, one of the building blocks of varieties with trivial first Chern class.
More specifically, the PI plans to carry out the research in following directions: investigating the relation between hyper-Kahler manifolds and cubic fourfolds, improving the current knowledge of Lagrangian fibrations, using Lagrangian fibrations to expand our knowledge of the known examples, and carrying out a systematic study of symplectic resolutions. These lines of research build on past work of the PI as well as on recent progress in this field.
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.
Algebraic geoemtry is the study of algebraic varieties, geometric objects which are defined by system of polynomial equations. Algebraic geometry relates the algebraic properties of the polynomial equations defining an algebraic variety with the geometry of the variety itself.
Within algebraic geometry, the work of the PI focuses on a particular class of algebraic varieties called hyper-Kahler manifolds. These have attracted significant attention in the last decades, both because they are one of the building blocks for more general algebraic varieties and because of their ties with other areas of mathematics such as differential geometry, mathematical physics, and representation theory.
The PI has used a wide spectrum of techniques, from Hodge theory and topological methods, to birational geometry and the minimal model program, to moduli spaces and derived categories to make substantial progress in the study of hyper-Kahler manifolds. Most notably, the work of the PI has advanced our understanding of the topology of exceptional examples of hyper-Kahler manifolds and of the relationship between hyper-Kahler manifolds and Fano varieties (e.g. cubic fourfolds), a class of algebraic varieties which at a first glance appears very distant from hyper-Kahler manifolds but which infact turns out to have many points of contact with them.
The PI has been training graduate and undergraduate students and has been strongly involved in activities meant to encourage the representation of women in mathematics. The research supported by this award has resulted in several pubications and preprints, and has been disseminated in a large number of national and international talks.
Last Modified: 11/29/2022
Modified by: Giulia Sacca
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