
NSF Org: |
DMS Division Of Mathematical Sciences |
Recipient: |
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Initial Amendment Date: | August 21, 2017 |
Latest Amendment Date: | August 21, 2017 |
Award Number: | 1700814 |
Award Instrument: | Standard 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: | September 1, 2017 |
End Date: | August 31, 2020 (Estimated) |
Total Intended Award Amount: | $150,000.00 |
Total Awarded Amount to Date: | $150,000.00 |
Funds Obligated to Date: |
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History of Investigator: |
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Recipient Sponsored Research Office: |
1850 RESEARCH PARK DR STE 300 DAVIS CA US 95618-6153 (530)754-7700 |
Sponsor Congressional District: |
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Primary Place of Performance: |
One Shields Avenue Davis CA US 95616-5270 |
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, TOPOLOGY |
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
A knot is a closed curve in three-dimensional space. More generally, a link is a collection of several such curves. Two links are equivalent if one can be transformed to another by a continuous deformation and/or stretching (without tearing). A link invariant is a quantity that does not change under the deformations of a link. Such invariants can be used to distinguish links and to study geometric properties of knots, and many powerful link invariants have been developed in recent decades. In another direction, the mathematical subject of algebraic geometry studies spaces described by polynomial equations and has applications in a wide range of fields, including statistics, control theory, and computer science. This project is focused on surprising interactions between the invariants of links and algebraic geometry that appear in two different mathematical frameworks. It is anticipated that the research will provide new perspectives on the relationships between low-dimensional topology, algebraic geometry, representation theory, and combinatorics.
In more detail, the first framework is classical: given a complex plane curve singularity, its intersection with a small sphere is a link. Similarly, for a complex surface singularity its link is a three-manifold. The investigator will the study the interactions between the algebraic properties of these curves and surfaces, and the Heegaard-Floer homology of their links. The second framework is less direct. The investigator and collaborators have conjectured that the Khovanov-Rozansky homology of links can be computed by studying the algebraic and geometric properties of the Hilbert scheme of points (the configuration space of points in four-dimensional space). The Hilbert scheme is well studied in algebraic geometry, representation theory and combinatorics, so this interpretation would yield explicit computations of Khovanov-Rozansky homology that were previously out of reach. In this project the investigator will develop the connection between knots and the Hilbert scheme, with the goals of computing these invariants for various classes of links and proving the conjecture.
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 PI studied interactions between the invariants of links (such as Heegaard Floer and Khovanov-Rozansky homology) and algebraic geometry.
Heegaard Floer homology was explored in a series of papers by PI, Lidman, Liu and Moore (and in several other papers by PI's graduate student Liu alone). One of the results of this work is that that Heegaard Floer homology detects Whitehead and Borromean links.
Khovanov-Rozansky homology and related categorical invariants were explored in the series of papers of PI, Hogancamp, Mellit, Nakagane and Wedrich. Their connections to combinatorics were explored in papers of PI with Hawkes, Mazin, Schilling, Rainbolt and Vazirani, while their connections to Hilbert schemes were studied in a series of papers of PI, Carlsson, Mellit, Simental and Vazirani (and in several papers by PI's graduate student Kivinen alone). One of the results of this work is the explicit computation of Khovanov-Rozansky homology for the powers of the full twist braid.
In addition to all this, the work of PI with Negut and Rasmussen spurred a lot of activity in link homology community. Several of the conjectures proposed in that paper were proved by the PI and his collaborators, and by other groups of researchers.
The PI has mentored two graduate students (both graduated in 2019, now postdocs in University of Toronto and Max Planck Institute), two undergraduate students (both started graduate school) and one postdoc. The PI has also co-organized five workshops closely related to the topic of this project.
Last Modified: 12/11/2020
Modified by: Evgeny Gorskiy
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