Award Abstract # 2053243
FRG: Collaborative Research: Matroids, Graphs, and Algebraic Geometry

NSF Org: DMS
Division Of Mathematical Sciences
Recipient: UNIVERSITY OF OREGON
Initial Amendment Date: June 28, 2021
Latest Amendment Date: June 28, 2021
Award Number: 2053243
Award Instrument: Standard Grant
Program Manager: Stefaan De Winter
sgdewint@nsf.gov
 (703)292-2599
DMS
 Division Of Mathematical Sciences
MPS
 Directorate for Mathematical and Physical Sciences
Start Date: July 1, 2021
End Date: June 30, 2025 (Estimated)
Total Intended Award Amount: $295,664.00
Total Awarded Amount to Date: $295,664.00
Funds Obligated to Date: FY 2021 = $295,664.00
History of Investigator:
  • Nicholas Proudfoot (Principal Investigator)
    njp@uoregon.edu
Recipient Sponsored Research Office: University of Oregon Eugene
1776 E 13TH AVE
EUGENE
OR  US  97403-1905
(541)346-5131
Sponsor Congressional District: 04
Primary Place of Performance: University of Oregon Eugene
5219 University of Oregon
Eugene
OR  US  97403-5219
Primary Place of Performance
Congressional District:
04
Unique Entity Identifier (UEI): Z3FGN9MF92U2
Parent UEI: Z3FGN9MF92U2
NSF Program(s): ALGEBRA,NUMBER THEORY,AND COM,
Combinatorics
Primary Program Source: 01002122DB NSF RESEARCH & RELATED ACTIVIT
Program Reference Code(s): 1616
Program Element Code(s): 126400, 797000
Award Agency Code: 4900
Fund Agency Code: 4900
Assistance Listing Number(s): 47.049

ABSTRACT

Recent advances in matroid and graph theory fuse the methods of combinatorics with concepts from algebraic geometry to resolve longstanding conjectures and provide deep insights into widespread phenomena such as unimodality and log concavity of integer sequences. The influences between combinatorics and algebraic geometry flow fruitfully in both directions; combinatorial constructions such as graph complexes have recently led to resolutions of long-standing conjectures in the geometry of moduli spaces of curves. The PIs will join forces and forge timely new collaborations to address the most pressing open problems at the interface between matroids, graphs, and algebraic geometry. The project includes the participation of graduate students and postdocs.

This focused research group will build on recent breakthroughs to accomplish the following goals: 1. Study matroidal generalizations of Kontsevich?s graph complex and pursue applications to the top weight cohomology of moduli spaces of abelian varieties; 2. Investigate K-theoretic analogs of the Chow ring of a matroid, with a view toward a matroidal analog of the Hecke algebra and applications to matroidal Kazhdan-Lusztig theory; 3. Prove a categorification of the Hodge-Riemann bilinear relations in the presence of a finite group action, and pursue equivariant log concavity for the characteristic polynomial of a matroid with automorphisms; 4. Use methods inspired by the hard Lefschetz theorem to attack both the Welsh conjecture on the number of isomorphism classes of matroids of given size and rank and the Harary edge reconstruction conjecture for graphs.

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.

PUBLICATIONS PRODUCED AS A RESULT OF THIS RESEARCH

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Braden, Tom and Matherne, Jacob P and Proudfoot, Nicholas "What is... the DowlingWilson Conjecture?" Notices of the American Mathematical Society , v.71 , 2024 https://doi.org/10.1090/noti3019 Citation Details
Gao, Alice LL and Proudfoot, Nicholas James and Yang, Arthur LB and Zhang, Zhong-Xue "The Combinatorics Behind the Leading Kazhdan-Lusztig Coefficients of Braid Matroids" The Electronic Journal of Combinatorics , v.31 , 2024 https://doi.org/10.37236/12778 Citation Details
Karn, Trevor and Nasr, George D. and Proudfoot, Nicholas and Vecchi, Lorenzo "Equivariant KazhdanLusztig theory of paving matroids" Algebraic Combinatorics , v.6 , 2023 https://doi.org/10.5802/alco.281 Citation Details
Larson, Matt and Li, Shiyue and Payne, Sam and Proudfoot, Nicholas "K-rings of wonderful varieties and matroids" Advances in Mathematics , v.441 , 2024 https://doi.org/10.1016/j.aim.2024.109554 Citation Details
Proudfoot, Nicholas and Xu, Yuan and Young, Benjamin "On the enumeration of series-parallel matroids" Advances in Applied Mathematics , v.163 , 2025 https://doi.org/10.1016/j.aam.2024.102801 Citation Details

PROJECT OUTCOMES REPORT

Disclaimer

This Project Outcomes Report for the General Public is displayed verbatim as submitted by the Principal Investigator (PI) for this award. Any opinions, findings, and conclusions or recommendations expressed in this Report are those of the PI and do not necessarily reflect the views of the National Science Foundation; NSF has not approved or endorsed its content.

This award supported a collaboration between research teams at Brown, Oregon, Princeton, and Texas.  The focus was on algebraic invariants of matroids, especially those that can be lifted from numerical invariants to categorical ones that carry actions of the symmetry groups of the matroids in question.  A major source of such invariants is algebraic geometry, as one can consider cohomomological invariants of algebraic varieties associated with the realization of a matroid.

One major accomplishment was a paper on K-rings of wonderful varieties and matroids, featuring authors from all four groups, which discovered new connections between K-theory and cohomology in the matroidal setting.  A surprising result in this paper is that the integral cohomology and K-theory of the Deligne-Mumford-Knudsen moduli space of stable rational curves with marked points are isomorphic, and that this isomorphism is equivariant with respect to permutations that fix one point, but not with respect to arbitrary permutations.

Another significant result was the development of the theory of intersection cohomology of matroids, which was a collaboration involving the Oregon and Princeton groups.  This has led to proofs of the Dowling-Wilson conjecture and non-negativity of Kazhdan-Lusztig polynomials, with further applications being developed in the course of ongoing research.

A third project led to a series of papers that were aimed at understanding valuative invariants of matroids in an equivariant context.  This work allows us to calculate equivariant invariants such as the Orlik-Solomon algebra, the Chow and augmented Chow rings, and intersection cohomology.

Finally, recent work at Oregon has used the theory of zonotopal algebras to prove a conjecture relating two different geometrically defined representations of the automorphism group of a graph, namely the intersection cohomology of the associated hypertoric variety and the cohomology of a configuration space parameterizing certain maps from the vertex set into the Lie group SU(2).


Last Modified: 07/17/2025
Modified by: Nicholas J Proudfoot

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