NSF Org:	DMS Division Of Mathematical Sciences
Recipient:	MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Initial Amendment Date:	February 13, 2019
Latest Amendment Date:	July 6, 2023
Award Number:	1845034
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:	July 1, 2019
End Date:	June 30, 2025 (Estimated)
Total Intended Award Amount:	$400,000.00
Total Awarded Amount to Date:	$400,000.00
Funds Obligated to Date:	FY 2019 = $116,706.00 FY 2020 = $34,639.00 FY 2021 = $47,884.00 FY 2022 = $99,524.00 FY 2023 = $101,247.00
History of Investigator:	Andrei Negut (Principal Investigator) anegut@mit.edu
Recipient Sponsored Research Office:	Massachusetts Institute of Technology 77 MASSACHUSETTS AVE CAMBRIDGE MA  US  02139-4301 (617)253-1000
Sponsor Congressional District:	07
Primary Place of Performance:	Massachusetts Institute of Technology 77 Massachusetts Ave, Rm 2-269 Cambridge MA  US  02139-4307
Primary Place of Performance Congressional District:	07
Unique Entity Identifier (UEI):	E2NYLCDML6V1
Parent UEI:	E2NYLCDML6V1
NSF Program(s):	ALGEBRA,NUMBER THEORY,AND COM
Primary Program Source:	01001920DB NSF RESEARCH & RELATED ACTIVIT 01002021DB NSF RESEARCH & RELATED ACTIVIT 01002122DB NSF RESEARCH & RELATED ACTIVIT 01002223DB NSF RESEARCH & RELATED ACTIVIT 01002324DB NSF RESEARCH & RELATED ACTIVIT
Program Reference Code(s):	1045
Program Element Code(s):	126400
Award Agency Code:	4900
Fund Agency Code:	4900
Assistance Listing Number(s):	47.049

ABSTRACT

This project will enhance the understanding of enumerative invariants, which are important numbers that arise both in the mathematical study of various geometries, as well as fundamental physical quantities in quantum physical theories, through the study of the underlying symmetries of the geometry itself. Enumerative geometry is connected with many other parts of mathematics and physics, with concrete applications to fields from knot theory to combinatorics. As such, the field is of interest to graduate students from a variety of backgrounds, and to this end the PI will organize yearly summer schools on topics connected with the project.

The spaces whose geometry to be studied fall into one of two major classes. First, quiver varieties that are built from the combinatorial datum of a graph. Symmetries of quiver varieties are connected to Kac-Moody Lie algebras, Yangians and quantum groups. Second, moduli spaces of sheaves on surfaces and threefolds, especially in the important example of Hilbert schemes, parametrize global objects, and their study involves different techniques. The intersection of the two classes of spaces consists of local geometries such as the plane or toric surfaces, wherein many physical quantities can be explicitly computed. In this context, the enumerative invariants of interest are obtained by integrating cohomology classes over the space, taking the Euler characteristic of line bundles, or at a higher level even studying the derived categories of the spaces themselves. This project will study all three approaches.

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.

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