Nicholas Proudfoot

Email: njp - at -
Office: 322 Fenton Hall
Address: Department of Mathematics
1222 University of Oregon
Eugene, OR 97403


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.

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 and Slides

Toric varieties (seven week course, fall 2010)
Geometric representation theory and categorification (three lecture minicourse, summer 2011)
Symplectic geometry (ten week course, fall 2013)
Kazhdan-Lusztig polynomials of matroids (20 minute AMS talk, spring 2018)
Category O, symplectic duality, and the Hikita conjecture (three lecture minicourse, summer 2018)
The wonderful geometry of matroids (ten week course, fall 2019)
Positivity theorems for hyperplane arrangements via intersection theory (UT Geometry Seminar, fall 2020)


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)

* 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.


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

  Categorical valuative invariants of polyhedra and matroids
      With Ben Elias, Dane Miyata, and Lorenzo Vecchi; preprint. pdf
  What is the Dowling-Wilson conjecture?
      With Tom Braden and Jacob Matherne; preprint. 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; preprint. 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; to appear in Int. Math. Res. Not. 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), #P3.34. 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) #P1.26. 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

In India, my friend and I had small roles in the soap opera Kasauti Zindagi Ki; you can see the clip here.