Award Abstract # 1700814
Algebraic Geometry of Knot Homology

NSF Org: DMS
Division Of Mathematical Sciences
Recipient: UNIVERSITY OF CALIFORNIA, DAVIS
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: FY 2017 = $150,000.00
History of Investigator:
  • Evgeny Gorskiy (Principal Investigator)
    egorskiy@ucdavis.edu
Recipient Sponsored Research Office: University of California-Davis
1850 RESEARCH PARK DR STE 300
DAVIS
CA  US  95618-6153
(530)754-7700
Sponsor Congressional District: 04
Primary Place of Performance: University of California-Davis
One Shields Avenue
Davis
CA  US  95616-5270
Primary Place of Performance
Congressional District:
04
Unique Entity Identifier (UEI): TX2DAGQPENZ5
Parent UEI:
NSF Program(s): ALGEBRA,NUMBER THEORY,AND COM,
TOPOLOGY
Primary Program Source: 01001718DB NSF RESEARCH & RELATED ACTIVIT
Program Reference Code(s):
Program Element Code(s): 126400, 126700
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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(Showing: 1 - 10 of 15)
Bai, Yuzhe and Gorsky, Eugene and Kivinen, Oscar "Quadratic ideals and Rogers?Ramanujan recursions" The Ramanujan Journal , 2019 10.1007/s11139-018-0127-3 Citation Details
Borodzik, Maciej and Gorsky, Eugene "Immersed concordance of links and Heegaard Floer homology" Indiana University Mathematics Journal , v.67 , 2018 10.1512/iumj.2018.67.7375 Citation Details
Carlsson, Erik and Gorsky, Eugene and Mellit, Anton "The $${\mathbb {A}}_{q,t}$$ algebra and parabolic flag Hilbert schemes" Mathematische Annalen , v.376 , 2020 10.1007/s00208-019-01898-1 Citation Details
Gorsky, Eugene and Gukov, Sergei and Sto?i?, Marko "Quadruply-graded colored homology of knots" Fundamenta Mathematicae , v.243 , 2018 10.4064/fm30-11-2017 Citation Details
Gorsky, Eugene and Hawkes, Graham and Schilling, Anne and Rainbolt, Julianne "Generalized $q,t$-Catalan numbers" Algebraic Combinatorics , v.3 , 2020 https://doi.org/10.5802/alco.120 Citation Details
Gorsky, Eugene and Hogancamp, Matthew and Mellit, Anton and Nakagane, Keita "Serre duality for KhovanovRozansky homology" Selecta Mathematica , v.25 , 2019 https://doi.org/10.1007/s00029-019-0524-5 Citation Details
Gorsky, Eugene and Hogancamp, Matthew and Wedrich, Paul "Derived Traces of Soergel Categories" International Mathematics Research Notices , 2021 https://doi.org/10.1093/imrn/rnab019 Citation Details
Gorsky, Eugene and Hom, Jennifer "Cable links and L-space surgeries" Quantum Topology , v.8 , 2017 10.4171/QT/98 Citation Details
Gorsky, Eugene and Liu, Beibei and Moore, Allison "Surgery on links of linking number zero and the Heegaard Floer $d$-invariant" Quantum Topology , v.11 , 2020 https://doi.org/10.4171/QT/137 Citation Details
Gorsky, Eugene and Mazin, Mikhail and Vazirani, Monica "Recursions for rational q,t-Catalan numbers" Journal of Combinatorial Theory, Series A , v.173 , 2020 10.1016/j.jcta.2020.105237 Citation Details
Gorsky, Eugene and Negu, Andrei and Rasmussen, Jacob "Flag Hilbert schemes, colored projectors and Khovanov-Rozansky homology" Advances in Mathematics , v.378 , 2021 https://doi.org/10.1016/j.aim.2020.107542 Citation Details
(Showing: 1 - 10 of 15)

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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