NSF Org:	DMS Division Of Mathematical Sciences
Recipient:	REGENTS OF THE UNIVERSITY OF CALIFORNIA AT RIVERSIDE
Initial Amendment Date:	February 2, 2024
Latest Amendment Date:	February 2, 2024
Award Number:	2337451
Award Instrument:	Continuing Grant
Program Manager:	Swatee Naik snaik@nsf.gov  (703)292-4876 DMS  Division Of Mathematical Sciences MPS  Directorate for Mathematical and Physical Sciences
Start Date:	July 1, 2024
End Date:	June 30, 2029 (Estimated)
Total Intended Award Amount:	$469,971.00
Total Awarded Amount to Date:	$79,647.00
Funds Obligated to Date:	FY 2024 = $79,647.00
History of Investigator:	Brian Collier (Principal Investigator) brian.collier@ucr.edu
Recipient Sponsored Research Office:	University of California-Riverside 200 UNIVERSTY OFC BUILDING RIVERSIDE CA  US  92521-0001 (951)827-5535
Sponsor Congressional District:	39
Primary Place of Performance:	University of California-Riverside 200 UNIVERSTY OFC BUILDING RIVERSIDE CA  US  92521-0001
Primary Place of Performance Congressional District:	39
Unique Entity Identifier (UEI):	MR5QC5FCAVH5
Parent UEI:
NSF Program(s):	ALGEBRA,NUMBER THEORY,AND COM, GEOMETRIC ANALYSIS
Primary Program Source:	01002425DB NSF RESEARCH & RELATED ACTIVIT 01002526DB NSF RESEARCH & RELATED ACTIVIT 01002627DB NSF RESEARCH & RELATED ACTIVIT 01002728DB NSF RESEARCH & RELATED ACTIVIT 01002829DB NSF RESEARCH & RELATED ACTIVIT
Program Reference Code(s):	1045
Program Element Code(s):	126400, 126500
Award Agency Code:	4900
Fund Agency Code:	4900
Assistance Listing Number(s):	47.049

ABSTRACT

This project focuses on the mathematical study of curved surfaces by connecting algebraic objects to them and thereby generalizing the scope of their application. One of the main notions used is that of a surface group representation, a concept which connects surfaces to generalizations of classical geometries such as Euclidean and hyperbolic geometry. The study of surfaces has surprising applications throughout many fields of mathematics and physics. Consequently, the project lies at the intersection of multiple disciplines. In addition to cutting edge mathematical research, the project will promote the progress of science and mathematics through different workshops aimed at graduate students as well as community outreach events. The educational component will also focus on creating an engaging and inclusive place for mathematical interactions for students and early career researchers.

In the past decades, both the theories of Higgs bundles and Anosov dynamics have led to significant advancements in our understanding of the geometry of surface groups. Recent breakthroughs linking these approaches are indirect and mostly involve higher rank generalizations of hyperbolic geometry known as higher rank Teichmuller spaces. The broad aim of this project is to go beyond higher rank Teichmuller spaces by using Higgs bundles to identify subvarieties of surface group representations which generalize the Fuchsian locus in quasi-Fuchsian space. The cornerstone for the approach is the role of Slodowy slices for Higgs bundles. Specifically, the PI aims to establish Anosov properties of surface group representations associated to Slodowy slices in the Higgs bundle moduli space. This approach will significantly extend applications of Higgs bundles to both Anosov representations and (G,X) geometries. It will complete the component count for moduli of surface group representations.

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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