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Research Spending & Results

Award Detail

Awardee:UNIVERSITY OF KENTUCKY
Doing Business As Name:University of Kentucky Research Foundation
PD/PI:
  • Kathleen Ponto
  • (859) 257-7729
  • kate.ponto@uky.edu
Award Date:05/10/2021
Estimated Total Award Amount: $ 158,033
Funds Obligated to Date: $ 158,033
  • FY 2021=$158,033
Start Date:07/01/2021
End Date:06/30/2024
Transaction Type:Grant
Agency:NSF
Awarding Agency Code:4900
Funding Agency Code:4900
CFDA Number:47.049
Primary Program Source:040100 NSF RESEARCH & RELATED ACTIVIT
Award Title or Description:FRG: Collaborative Research: Trace Methods and Applications for Cut-and-Paste K-Theory
Federal Award ID Number:2052905
DUNS ID:939017877
Parent DUNS ID:007400724
Program:TOPOLOGY
Program Officer:
  • Krishnan Shankar
  • (703) 292-4703
  • kshankar@nsf.gov

Awardee Location

Street:109 Kinkead Hall
City:Lexington
State:KY
ZIP:40526-0001
County:Lexington
Country:US
Awardee Cong. District:06

Primary Place of Performance

Organization Name:University of Kentucky Research Foundation
Street:500 S Limestone 109 Kinkead Hall
City:Lexington
State:KY
ZIP:40526-0001
County:Lexington
Country:US
Cong. District:06

Abstract at Time of Award

This research brings together ideas, techniques, and insights from two long-standing programs in mathematics: scissors congruence and algebraic K-theory. Scissors congruence originated in Hilbert's 3rd Problem, which asks when two polyhedra in three-dimensional space are "scissors congruent," meaning one can be obtained from the other by cutting it into smaller polyhedra and reassembling in a different way. This question, together with its solution by Dehn, initiated an extensive program of research. Over the past 120 years these ideas have grown and now connect to almost every branch of geometry. Ground-breaking recent work provides a fundamental link between this program and algebraic K-theory, which is itself a deep and rapidly developing area of research. Algebraic K-theory intertwines three major fields of mathematics: topology, algebraic geometry, and number theory. Developing the connection between scissors congruence and algebraic K-theory will significantly advance research in both. This work also provides the platform for striking new research avenues that will bring to bear the tools and techniques of modern algebraic K-theory research on a wide range of geometric questions. This project additionally includes a number of efforts to support students and new researchers in the field, expanding and broadening access to these innovative ideas. This broad new program of research develops the foundations of combinatorial, or "cut-and-paste," algebraic K-theory, applies these new tools to resolve outstanding geometric questions, and expands the scope of combinatorial K-theory to new applications. It brings modern techniques in algebraic K-theory to the emerging K-theoretic approach to cut-and-paste invariants, and applies this approach to a variety of problems in algebraic topology, differential topology, and algebraic geometry. Algebraic K-theory has seen a stunning revolution in the last thirty years due to the invention of trace methods, but these tools have not yet been developed for combinatorial K-theory, a deficiency that this project hopes to remedy. This requires developing the foundations of this new theory and exploiting connections to equivariant homotopy theory. New computational and analytic tools for combinatorial K-theory will lead to progress on a wide variety of geometric problems, including applications to manifolds and invertible TQFTs, varieties and motivic measures, and fixed point theory. Many questions in these fields have natural interpretations in terms of cut-and-paste invariants. 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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