My research centres on questions from algebraic geometry, noncommutative algebra and geometric representation theory, with applications in aspects of mathematical physics relating to quiver gauge theories in string theory. My interests include the study of representations of quivers, tilting theory and derived categories, geometric invariant theory, the McKay correspondence, symplectic singularities, as well as Mori's programme in birational geometry. A lot of my work has focused on the study of moduli spaces, especially (equivariant) Hilbert schemes of points, and quiver moduli spaces.
Papers
The Cautis-Logvinenko conjecture (with Hsueh-Yung Lin and Ryo Yamagishi).
Submitted. [arXiv]
Hilbert schemes of points on canonical surfaces (with Ryo Yamagishi).
Submitted. [arXiv]
Hilbert schemes of points on crepant partial resolutions (with Ruth Wye).
Submitted. [arXiv]
The semi-invariant ring as the Cox ring of a GIT quotient (with Gwyn Bellamy and Travis Schedler).
Submitted. [arXiv]
The Le Bruyn-Procesi theorem following Lusztig (with Ryo Yamagishi).
Submitted. [arXiv]
Gale duality and the linearisation map for noncommutative crepant resolutions.
To appear Algebra & Number Theory (2026). [arXiv]
Birational geometry of quiver varieties and other GIT quotients (with Gwyn Bellamy and Travis Schedler).
Compositio Mathematica 162 (2026), no 3, 630 - 668. [arXiv]
Orbifold Quot schemes via the Le Bruyn-Procesi theorem,
Cheltsov, I., & Martinez-Garcia, J. (Eds.). (2025). ZAG Handbook of Algebraic Geometry (1st ed.). Chapman and Hall/CRC. [arXiv]
All 81 crepant resolutions of a finite quotient singularity are hyperpolygon spaces (with Gwyn Bellamy, Steve Rayan, Travis Schedler and Hartmut Weiss),
Journal of Algebraic Geometry, 33 (2024), 757-793. [arXiv, video]
An introduction to Hilbert schemes of points on ADE singularities,
Advanced Studies in Pure Mathematics 88 (2023), McKay correspondence, mutation and related topics, 119-157. [arXiv]
Quot schemes for Kleinian orbifolds (with Soren Gammelgaard, Adam Gyenge and Balazs Szendroi)
SIGMA 17 (2021), 099, 21 pages. [journal]
Punctual Hilbert schemes of Kleinian singularities as quiver varieties (with Soren Gammelgaard, Adam Gyenge and Balazs Szendroi)
Algebraic Geometry 8 (2021), no. 6, 680-704. [arXiv, video]
Combinatorial Reid's recipe for consistent dimer models (with Liana Heuberger and Jesus Tapia Amador),
Épijournal de Géométrie Algébrique 5 (2021), Article no. 4. [journal]
Birational geometry of symplectic quotient singularities (with Gwyn Bellamy),
Inventiones Mathematicae 222 (2020), no. 2, 399-468. [arXiv]
Reconstructing toric quiver flag varieties from a tilting bundle (with James Green),
European Journal of Mathematics 4 (2018), no. 1, 185-199. [arXiv]
Multigraded linear series and recollement (with Yukari Ito and Joseph Karmazyn),
Mathematische Zeitschrift 289 (2018), no. 1, 535-565. [arXiv]
Nef divisors for moduli spaces of complexes with compact support (with Arend Bayer and Ziyu Zhang),
Selecta Mathematica (New Series) 23 (2017), no. 2, 1507-1561. [arXiv]
Derived Reid's recipe for abelian subgroups of SL(3,C) (with Sabin Cautis and Timothy Logvinenko),
Journal fur die reine und angewandte Mathematik 727 (2017), 1-48. [arXiv]
Geometric Reid's recipe for dimer models (with Raf Bocklandt and Alexander Quintero Velez),
Mathematische Annalen 361 (2015), 689-723 [arXiv] [the Correction is Mathematische Annalen 380, 911–913 (2021)]
Cohomology of wheels on toric varieties (with Alexander Quintero Velez),
Hokkaido Mathematical Journal 44 (2015), 47-79. [arXiv]
Mori Dream Spaces as fine moduli of quiver representations (with Dorothy Winn),
Journal of Pure and Applied Algebra 217 (2013), 172-189. [arXiv]
Cellular resolutions of noncommutative toric algebras from superpotentials (with Alexander Quintero Velez),
Advances in Mathematics 229 (2012), no. 3, 1516-1554. [arXiv]
Quiver flag varieties and multigraded linear series,
Duke Mathematical Journal 156 (2011), no. 3, 469-500. [arXiv]
The Special McKay correspondence as an equivalence of derived categories,
Quarterly Journal of Mathematics 62 (2011), 573-591. [arXiv]
Projective toric varieties as fine moduli spaces of quiver representations (with Gregory G. Smith),
American Journal of Mathematics 130 (2008), no. 6, 1509-1534. [arXiv]
Moduli of McKay quiver representations II: Grobner basis techniques (with Diane Maclagan and Rekha R. Thomas),
Journal of Algebra 316 (2007), no. 2, 514-535. [arXiv]
Fiber fans and toric quotients (with Diane Maclagan),
Discrete and Computational Geometry 37 (2007), no. 2, 251-266. [arXiv]
Moduli of McKay quiver representations I: the coherent component (with Diane Maclagan and Rekha R. Thomas),
Proceedings of the London Mathematical Society 95 (2007), no. 1, 179-198. [arXiv]
An explicit construction of the McKay correspondence for A-Hilb(C^3),
Journal of Algebra 285 (2005), no. 2, 682-705. [arXiv]
Flops of G-Hilb and equivalences of derived categories by variation of GIT quotient (with Akira Ishii),
Duke Mathematical Journal 124 (2004), no. 2, 259-307. [arXiv]
An introduction to motivic integration,
In Strings and Geometry, 203-225, Clay Mathematical Proceedings 3, AMS, Providence, RI, 2004. [arXiv]
How to calculate A-Hilb(C^3) (with Miles Reid),
In Geometry of toric varieties, 129--154, Seminaires et Congres 6, SMF, Paris, 2002. [arXiv]
The McKay correspondence and representations of the McKay quiver,
University of Warwick Ph.D thesis, xviii + 134 pp, June 2001. [link]
Published reports and lecture notes
Noncommutative toric geometry,
Oberwolfach Reports 9 (2012), no. 2, 1254-1257. [link]
Toric varieties are fine moduli spaces,
Oberwolfach Reports 3 (2006), no. 2, 266-268. [link]
(for W. Fulton) Enumerative Geometry,
Quantum Field Theory, Supersymmetry, and Enumerative Geometry (IAS/PCMI, V. 11), AMS (2006). [link]
(for C.H. Clemens) Introduction to Hodge Theory,
Appunti dei Corsi Tenuti da Docenti della Scuola. Scuola Normale Superiore, Pisa, 1998. 72 pp. [link]
Unpublished work
Quiver representations in toric geometry,
Produced following summer schools in 2007. [arXiv]
The Kodaira vanishing theorem via Hodge theory,
These notes accompanied a series of lectures at the University of Warwick in 1998. [link]