
NSF Org: |
DMS Division Of Mathematical Sciences |
Recipient: |
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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: |
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History of Investigator: |
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Recipient Sponsored Research Office: |
1776 E 13TH AVE EUGENE OR US 97403-1905 (541)346-5131 |
Sponsor Congressional District: |
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Primary Place of Performance: |
5219 University of Oregon Eugene OR US 97403-5219 |
Primary Place of
Performance Congressional District: |
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Unique Entity Identifier (UEI): |
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Parent UEI: |
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NSF Program(s): |
ALGEBRA,NUMBER THEORY,AND COM, Combinatorics |
Primary Program Source: |
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Program Reference Code(s): |
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Program Element Code(s): |
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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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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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