text-only page produced automatically by Usablenet Assistive Skip all navigation and go to page content Skip top navigation and go to directorate navigation Skip top navigation and go to page navigation
National Science Foundation
design element
Search Awards
Recent Awards
Presidential and Honorary Awards
About Awards
Grant Policy Manual
Grant General Conditions
Cooperative Agreement Conditions
Special Conditions
Federal Demonstration Partnership
Policy Office Website

Award Abstract #1352398

CAREER: Knot invariants, moduli spaces of sheaves and representation theory

Division Of Mathematical Sciences
divider line
Initial Amendment Date: January 17, 2014
divider line
Latest Amendment Date: July 29, 2015
divider line
Award Number: 1352398
divider line
Award Instrument: Continuing grant
divider line
Program Manager: Matthew Douglass
DMS Division Of Mathematical Sciences
MPS Direct For Mathematical & Physical Scien
divider line
Start Date: September 1, 2014
divider line
End Date: August 31, 2019 (Estimated)
divider line
Awarded Amount to Date: $165,114.00
divider line
Investigator(s): Alexei Oblomkov oblomkov@math.umass.edu (Principal Investigator)
divider line
Sponsor: University of Massachusetts Amherst
Research Administration Building
AMHERST, MA 01003-9242 (413)545-0698
divider line
Division Co-Funding: CAREER
divider line
Program Reference Code(s): 1045
divider line
Program Element Code(s): 1264, 1267, 8048


The subject of this project is the geometry of configuration spaces of collections of points inside varieties of small dimension, and more generally, the moduli spaces of sheaves on these varieties. The main objective is to reveal new and further explore previously known links between the moduli spaces and objects in other fields of mathematics, in particular Representation Theory and Lower Dimensional Topology. The PI will work toward a proof of the mathematical conjecture relating the topological invariants of the Hilbert scheme of points on plane singular curves and the HOMFLY knot homology of the links of the singularities of the curve (Hilb/HOMFLY formula). The conjecture also reveals unexpected symmetries of the homology of torus knots: conjecturally, they form an irreducible representation of the rational Cherednik algebra of type A. The PI will explore the generalized Hilb/HOMFLY conjecture that relates the representation theory of the symplectic reflection algebras and the rational Cherednik algebras of types other than A. Finally, the PI describes the cohomology ring of the compactified Jacobians of quasi-homogeneous singularities. The PI (jointly with Zhiwei Yum) conjectures a relation between the cohomology ring of the compactified Jacobian of the curve and the structure ring of the moduli space of the rational maps to the curve: a local variation of the Gromov-Witten/Donaldson-Thomas relation. The educational component of the project offers a new model for the UMass REU program.

Knot invariants and topological invariants allow us to analyze the global structure of complicated shapes by collecting local information about the shape. Complicated shapes occur naturally in biology (e.g. proteins, DNA), theoretical physics (strings), and other areas of natural science. Thus developing new invariants and computational methods for understanding of the global structure of complex shapes is an important mathematical problem with many potential applications. The PI strives to understand the hidden symmetries of already discovered invariants, develop new invariants, and find unexpected applications of these invariants to other areas of mathematics. The PI will also involve undergraduate students in cutting edge research through a summer research program integrating mentorship by faculty and graduate students. The PI aims to attract more students from underrepresented groups to mathematical research by reserving specific spaces in the summer research program for students from two local women's colleges. The PI will prepare graduate student mentors during the year by teaching related graduate classes and a reading seminar. This new summer research program structure will increase diversity and strengthen vertical integration in academia and improve the communication and flow of ideas between different generations of present and future researchers.


Please report errors in award information by writing to: awardsearch@nsf.gov.



Print this page
Back to Top of page
Research.gov  |  USA.gov  |  National Science Board  |  Recovery Act  |  Budget and Performance  |  Annual Financial Report
Web Policies and Important Links  |  Privacy  |  FOIA  |  NO FEAR Act  |  Inspector General  |  Webmaster Contact  |  Site Map
National Science Foundation Logo
The National Science Foundation, 4201 Wilson Boulevard, Arlington, Virginia 22230, USA
Tel: (703) 292-5111, FIRS: (800) 877-8339 | TDD: (800) 281-8749
  Text Only Version