| Nicholas Proudfoot
Email: njp - at - uoregon.edu Office: 322 Fenton Hall Address: Department of Mathematics 1222 University of Oregon Eugene, OR 97403 CV |
I am part of a Research Training Group in Combinatorics, Geometry, Representation Theory, and Topology (we run the RATCOW seminar), and also part of a Focused Research Group on Matroids, Graphs, and Algebraic Geometry.
Advising
I received my Ph.D. from Berkeley in 2004, under the supervision of Allen Knutson. I've had seven students graduate so far; you can see them on my family tree.
Lecture Notes
Toric varieties (seven week course, fall 2010)
Symplectic geometry (ten week course, fall 2013)
The wonderful geometry of matroids (ten week course, fall 2019)
WARTHOG*
2010: Operator Algebras and Conformal Field Theory (with André Henriques)
2011: Cluster Algebras and Lusztig's Semicanonical Basis (with David Speyer)
2012: Categorical Representation Theory (with David Ben-Zvi)
2013: Quantum/Affine Schubert Calculus (with Allen Knutson)
2014: Kazhdan-Lusztig Theory and Soergel Bimodules (with Ben Elias)
2015: Positivity in Combinatorial Algebraic Geometry (with June Huh)
2016: Knot homologies, Hilbert schemes, and Cherednik algebras (with Jacob Rasmussen)
2017: Symplectic duality (the Abelian case) (with Ben Webster)
2018: Deligne-Lusztig theory (with Olivier Dudas)
2019: Foundations of Tropical Geometry (with Diane Maclagan and Jeff Giansiracusa)
2022: Infinite-dimensional methods in commutative algebra (with Andrew Snowden)
2023: Categorical Coulomb Branches (with Sabin Cautis and Harold Williams)
2024: Coherent-constructible equivalences in local Geometric Langlands and Representation Theory (with Pramod Achar, Gurbir Dhillon, and Simon Riche)
* Workshop on Algebra and Representation Theory Held on Oregonian Grounds
Ben Elias and I wrote an article in the Notices of the AMS about our experiences organizing these workshops.
Research
My work is somewhere in between algebraic geometry, combinatorics, representation theory, and algebraic topology. I work mostly with algebraic varieties that are built using the data of a hyperplane arrangement; examples include hypertoric varieties, Schubert varieties, and wonderful varieties. I then study the relationship between the algebraic invariants of these spaces and the combinatorial invariants of the input data. The flow of information goes in both directions: sometimes I use combinatorics to compute objects of intrinsic geometric interest (categories of sheaves, cohomology rings, etc.) and sometimes I use geometric machinery to prove purely combinatorial theorems.
Unfinished Business
I make a lot of conjectures. Here are brief descriptions of what I regard as the most interesting of my conjectures that remain open.
Equivariant log concavity of Orlik-Solomon algebras
Roots of Kazhdan-Lusztig polynomials and Z-polynomials of matroids
Classification of abstract hypertoric varieties
Intersection cohomology and quantum cohomology of conical symplectic resolutions
The quantum Hikita conjecture
Publications
Convergence of integrals on the moduli spaces of curves and cographical matroids
With Sasha Polishchuk; preprint. pdf
On the enumeration of series-parallel matroids
With Ben Young and Yuan Xu; Adv. Appl. Math. 163 (2025), 13 pp. pdf
Categorical valuative invariants of polyhedra and matroids
With Ben Elias, Dane Miyata, and Lorenzo Vecchi; to appear in Duke Math. J. pdf
What is the Dowling-Wilson conjecture?
With Tom Braden and Jacob Matherne; Not. Amer. Math. Soc. 71 (2024), 1197-1201. pdf
The combinatorics behind the leading Kazhdan-Lusztig coefficients of braid matroids
With Alice L.L. Gao, Arthur L.B. Yang, and Zhong-Xue Zhang; Electronic J. Combinatorics 31 (2024), Paper No. 3.21, 16 pp. pdf
K-rings of wonderful varieties and matroids
With Matt Larson, Shiyue Li, and Sam Payne; Adv. Math. 441 (2024), Paper No. 109554, 43 pp. pdf
Equivariant cohomology and conditional oriented matroids
With Galen Dorpalen-Barry and Jidong Wang; Int. Math. Res. Not. (2024), 9292-9322. pdf
Equivariant Kazhdan-Lusztig theory of paving matroids
With Trevor Karn, George Nasr, and Lorenzo Vecchi; Alg. Comb. 6 (2023), 677-688. pdf
Equivariant log concavity and representation stability
With Jacob Matherne, Dane Miyata, and Eric Ramos; Int. Math. Res. Not. (2023), 3885-3906. pdf
Stability phenomena for resonance arrangements
With Eric Ramos; Proc. Amer. Math. Soc. Ser. B 8 (2021) 219-223. pdf
A type B analogue of the category of finite sets with surjections
Electronic J. Combinatorics 29 (2022), Paper No. P3.34, 19 pp. pdf
Singular Hodge theory for combinatorial geometries
With Tom Braden, June Huh, Jacob Matherne, and Botong Wang; preprint. pdf
Equivariant incidence algebras and equivariant Kazhdan-Lusztig-Stanley theory
Alg. Comb. 4 (2021) 675-681. pdf
A semi-small decomposition of the Chow ring of a matroid
With Tom Braden, June Huh, Jacob Matherne, and Botong Wang; Adv. Math. 409 (2022), Paper No. 108646, 49 pp. pdf
The contraction category of graphs
With Eric Ramos; Representation Theory 26 (2022), 673-697. pdf
Functorial invariants of trees and their cones
With Eric Ramos; Selecta Math. 25 (2019), 28 pp. pdf
Equivariant Kazhdan-Lusztig polynomials of q-niform matroids
Alg. Comb. 2 (2019) 613-619. pdf
The quantum Hikita conjecture
With Joel Kamnitzer and Michael McBreen; Adv. Math. 390 (2021), Paper No. 107947, 53 pp. pdf
The algebraic geometry of Kazhdan-Lusztig-Stanley polynomials
EMS Surveys in Mathematical Sciences 5 (2018), 99-127. pdf See also the related Oberwolfach report.
The Z-polynomial of a matroid
With Yuan Xu and Ben Young; Electronic J. Combinatorics 25 (2018) Paper No. P1.26, 21 pp. pdf
Configuration spaces, FSop-modules, and Kazhdan-Lusztig polynomials of braid matroids
With Ben Young; New York Journal of Mathematics 23 (2017), 813-832. pdf
Kazhdan-Lusztig polynomials of matroids: a survey of results and conjectures
With Katie Gedeon and Ben Young; Séminare Lotharingien Combinatoire 78B (2017), Art. 80, 12. pdf
The equivariant Kazhdan-Lusztig polynomial of a matroid
With Katie Gedeon and Ben Young; Journal of Combinatorial Theory, Series A 150 (2017), 267-294. pdf
The Orlik-Terao algebra and the cohomology of configuration space
With Daniel Moseley and Ben Young; Experimental Mathematics 26 (2017), 373-380. pdf
Hypertoric varieties and zonotopal tilings
With Matt Arbo; Int. Math. Res. Not. 23 (2016), 7268-7301. pdf
Intersection cohomology of the symmetric reciprocal plane
With Max Wakefield and Ben Young; J. Alg. Comb. 43 (2016) 129-138. pdf
The Kazhdan-Lusztig polynomial of a matroid
With Ben Elias and Max Wakefield; Adv. Math. 299 (2016) 36-70. pdf
Intersection cohomology and quantum cohomology of conical symplectic resolutions
With Michael McBreen; Alg. Geom. 2 (2015) 623-641. pdf
Quantizations of conical symplectic resolutions II: category O and symplectic duality
With Tom Braden, Anthony Licata, and Ben Webster; Astérisque 384 (2016), 75-179. pdf
Poisson-de Rham homology of hypertoric varieties and nilpotent cones
With Travis Schedler; Selecta Math. 23 (2017), 179-202. pdf
Hypertoric Poisson homology in degree zero
Alg. Geom. 1 (2014) 261-270. pdf
Quantizations of conical symplectic resolutions I: local and global structure
With Tom Braden and Ben Webster; Astérisque 384 (2016), 1-73. pdf
Hypertoric category O
With Tom Braden, Anthony Licata, and Ben Webster; Adv. Math. 231 (2012) 1487-1545. pdf
Localization algebras and deformations of Koszul algebras
With Tom Braden, Anthony Licata, Chris Phan, and Ben Webster; Selecta Math. 17 (2011) 533-572. pdf
Resolving toric varieties with Nash blow-ups
With Atanas Atanasov, Christopher Lopez, Alexander Perry, and Michael Thaddeus; Experimental Math. 20 (2011) 288-303. pdf
Gale duality and Koszul duality
With Tom Braden, Anthony Licata, and Ben Webster; Adv. Math. 225 (2010) 2002-2049. pdf
The hypertoric intersection cohomology ring
With Tom Braden; Inv. Math. 177 (2009) 337-380. pdf
A survey of hypertoric geometry and topology
Toric Topology, Contemporary Mathematics 460, AMS, Providence, RI, 2006. pdf
Moduli spaces for Bondal quivers
With Aaron Bergman; Pacific J. Math. 237 (2008) 201-221. pdf
Moduli spaces for D-branes at the tip of a cone
With Aaron Bergman; Journal of High Energy Physics 03 (2006) 073. pdf
All the GIT quotients at once
Trans. Amer. Math. Soc. 363 (2011) 1687-1698. pdf
A nonhausdorff model for the complement of a complexified arrangement
Proc. Amer. Math. Soc. 135 (2007) 3989-3994. pdf
Geometric invariant theory and projective toric varieties
Snowbird Lectures in Algebraic Geometry, Contemporary Mathematics 388, AMS, Providence, RI, 2005. pdf
Intersection cohomology of hypertoric varieties
With Ben Webster; J. Alg. Geom. 16 (2007) 39-63. pdf
A broken circuit ring
With David Speyer; Beitrage zur Algebra und Geometrie 47 (2006) 161-166. pdf
Equivariant topology of real hyperplane arrangements
Announcement of results. pdf
Hyperplane arrangements and K-theory
Top. Appl. 153 (2006) 2866-2875. pdf
Hyperkahler analogues of Kahler quotients
U.C. Berkeley Ph.D. thesis, Spring 2004. pdf
Abelianization for hyperkahler quotients
With Tamas Hausel; Topology 44 (2005) 231-248. pdf
Hyperpolygon spaces and their cores
With Megumi Harada; Trans. Amer. Math. Soc. 357 (2005) 1445-1467. pdf
The equivariant Orlik-Solomon algebra
J. Alg. 305 (2006) 1186-1196. pdf
Properties of the residual circle action on a hypertoric variety
With Megumi Harada; Pacific J. Math. 214 (2004) 263-284. pdf
Brieskorn's construction of exotic spheres
Harvard University senior thesis, Spring 2000. pdf
Parallel connections and bundles of arrangements
With Michael Falk; Top. Appl. 118 (2002) 65-83. pdf
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In India, my friend and I had small roles in the soap opera Kasauti Zindagi Ki; you can see the clip here. |