FRG: COLLABORATIVE RESEARCH: TRACE METHODS AND APPLICATIONS FOR CUT-AND-PASTE K-THEORY -THIS RESEARCH BRING…

Assistance award details

Assistance award identifier

ASST_NON_2528362_049

FAIN

2528362

Recipient

REGENTS OF THE UNIVERSITY OF MINNESOTA

Awarding agency

National Science Foundation

Funding agency

National Science Foundation

Assistance listing (CFDA)

47.049

Total obligated

$31,705

Period

Mar 1, 2025 – Jun 30, 2025

Fiscal year

2025

Place of performance state

MN

Source file

FY2025_All_Assistance_Full_20261006.zip

Description

FRG: COLLABORATIVE RESEARCH: TRACE METHODS AND APPLICATIONS FOR CUT-AND-PASTE K-THEORY -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 I