Research

My work concerns algebraic and geometric objects connected to mathematical physics, especially symplectic and Poisson singularities, their resolutions and deformations, Poisson (co)homology, and their connections with representation theory. I have also worked on deformations and quantisations of log symplectic structures.

Current interests

Topics I have worked on include symplectic resolutions and singularities; quiver and character varieties; Poisson geometry and Poisson (co)homology; deformations and quantisations of log symplectic structures; deformation theory; Hochschild and cyclic homology; algebraic differential operators; preprojective algebras; and noncommutative algebraic geometry.

Collaborators

• Gwyn Bellamy
• Pieter Belmans
• Thomas Bitoun
• Raf Bocklandt
• Alastair Craw
• Pavel Etingof
• Ching-Hwa Eu
• Wee Liang Gan
• Qëndrim Gashi
• Victor Ginzburg
• Sherry Gong
• Amihay Hanany
• Austin Hubbard
• Masoud Kamgarpour
• Daniel Kaplan
• Guhesh Kumaran
• Andrey Lazarev
• Deshuo Liu
• Ivan Losev
• Ruslan Maksimau
• Mykola Matviichuk
• Cris Negron
• Aldo Pacchiano
• Nicholas Proudfoot
• Brent Pym
• Qingchun Ren
• Steven Rayan
• Olivier Schiffmann
• Alexandre Soloviev
• David Speyer
• Ulrich Thiel
• Andrea Tirelli
• Boris Tsygan
• Zhengfang Wang
• Hartmut Weiss
• Michael Wemyss
• Julian Wykowski
• Yourong Zang

Publications

Complete list from my CV, in reverse chronological order. Links are to the journal or publisher page and to arXiv where available.

Preprints

1. New symplectic singularities from SU(2) gauge theories
   Posted preprint. with A. Hanany, G. Kumaran, and D. Liu. arXiv
2. Singularities in Calogero–Moser varieties
   Posted preprint. with G. Bellamy and R. Maksimau. arXiv
3. Nonprojective crepant resolutions of quiver varieties
   Posted preprint. with D. Kaplan. arXiv
4. Creating quantum projective spaces by deforming (q)-symmetric algebras
   Posted preprint. with M. Matviichuk and B. Pym. arXiv
5. The semi-invariant ring as the Cox ring of a GIT quotient
   Posted preprint. with G. Bellamy and A. Craw. arXiv
6. An investigation into Lie algebra representations obtained from regular holonomic D-modules
   Posted preprint; expository article. with J. Wykowski. arXiv
7. A local Torelli theorem for log symplectic manifolds
   Posted preprint. with M. Matviichuk and B. Pym. arXiv

Published and accepted work

8. Birational geometry of quiver varieties and other GIT quotients
   Compos. Math. 162 (2026), 630–668. with G. Bellamy and A. Craw. journal arXiv
9. Holonomic Poisson geometry of Hilbert schemes
   Geom. Topol. 29 (2025), 2047–2103. with M. Matviichuk and B. Pym. journal arXiv
10. Crepant resolutions of stratified varieties via gluing
    Int. Math. Res. Not. IMRN (2024), no. 17, 12161–12200. with D. Kaplan. journal arXiv
11. All 81 crepant resolutions of a finite quotient singularity are hyperpolygon spaces
    J. Algebraic Geom. 33 (2024), 757–793. with G. Bellamy, A. Craw, S. Rayan, and H. Weiss. journal arXiv
12. The 2-Calabi–Yau property for multiplicative preprojective algebras
    Algebra Number Theory 17 (2023), 831–883. with D. Kaplan. journal arXiv
13. Symplectic resolutions of character varieties
    Geom. Topol. 27 (2023), 51–86. with G. Bellamy. journal arXiv
14. Symplectic resolutions of quiver varieties
    Selecta Math. 27 (2021), article 36. with G. Bellamy. journal arXiv
15. The Hochschild cohomology ring of a global quotient orbifold
    Adv. Math. 364 (2020), 106978. with C. Negron; contains appendices by P. Belmans and by P. Etingof with the authors. journal arXiv
16. Symplectic resolutions for multiplicative quiver varieties and character varieties for punctured surfaces
    In Representation Theory and Algebraic Geometry, Birkhäuser (2022), 393–459. with A. Tirelli. journal arXiv
17. On symplectic resolutions and factoriality of Hamiltonian reductions
    Math. Ann. 375 (2019), 165–176. with G. Bellamy. journal arXiv
18. Hyperplane arrangements associated to symplectic quotient singularities
    Banach Center Publ. 116 (2018), 25–45. with G. Bellamy and U. Thiel. journal arXiv
19. On D-modules related to the b-function and Hamiltonian flow
    Compos. Math. 154 (2018), no. 11, 2426–2440. with T. Bitoun. journal arXiv
20. Holonomic Poisson manifolds and deformations of elliptic algebras
    In Geometry and Physics: A Festschrift in honour of Nigel Hitchin, Oxford University Press (2018), 681–704. with B. Pym. journal arXiv
21. Poisson traces, D-modules, and symplectic resolutions
    Lett. Math. Phys. 108 (2018), 633–678; survey article. with P. Etingof. journal arXiv
22. Filtrations on Springer fiber cohomology and Kostka polynomials
    Lett. Math. Phys. 108 (2018), 679–698. with G. Bellamy. journal arXiv
23. On the (non)existence of symplectic resolutions for imprimitive symplectic reflection groups
    Math. Res. Lett. 23 (2016), no. 6, 1537–1564. with G. Bellamy. journal arXiv
24. Coinvariants of Lie algebras of vector fields on algebraic varieties
    Asian J. Math. 20 (2016), no. 5, 795–868. with P. Etingof. journal arXiv
25. Equivariant slices for symplectic cones
    Int. Math. Res. Not. (2017), no. 12, 3801–3847. journal arXiv
26. Poisson-de Rham homology of hypertoric varieties and nilpotent cones
    Selecta Math. 23 (2017), 179–202. with N. Proudfoot. journal arXiv
27. A new construction of cyclic homology
    Proc. Lond. Math. Soc. 112 (2016), no. 3, 549–587. with V. Ginzburg. journal arXiv
28. Noncommutative algebraic geometry
    Book in the MSRI lecture series, Cambridge University Press (2016). with G. Bellamy, D. Rogalski, J. T. Stafford, and M. Wemyss. journal
29. Zeroth Hochschild homology of preprojective algebras over the integers
    Adv. Math. 299 (2016), 451–542. journal arXiv
30. Geometrization of principal series representations of reductive groups
    Ann. Inst. Fourier 65 (2015), no. 5, 2273–2330. with M. Kamgarpour. journal arXiv
31. Invariants of Hamiltonian flow on locally complete intersections
    Geom. Funct. Anal. 24 (2014), no. 6, 1885–1912. with P. Etingof. journal arXiv
32. Poisson traces for symmetric powers of symplectic varieties
    Int. Math. Res. Not. (2014), 3396–3438. with P. Etingof. journal arXiv
33. Type D Weyl groups
    Appendix to the preceding paper (2014). journal arXiv
34. Ramified Satake Isomorphisms for strongly parabolic characters
    Documenta Math. 18 (2013), 1275–1300. with M. Kamgarpour. journal arXiv
35. A new linear quotient of C⁴ admitting a symplectic resolution
    Math. Z. 273 (2013), no. 3–4, 753–769. with G. Bellamy. journal arXiv
36. Normality and quadraticity for special ample line bundles on toric varieties arising from root systems
    Glasgow Math. J. 55 (2013), no. A, 113–134. with Q. Gashi. journal arXiv
37. Computational approaches to Poisson traces associated to finite subgroups of Sp(2n,C)
    Exp. Math. 21 (2012), no. 2, 141–170. with P. Etingof, S. Gong, A. Pacchiano, and Q. Ren. journal arXiv
38. Curved infinity-algebras and their characteristic classes
    J. Topol. 5 (2012), no. 3, 503–528. with A. Lazarev. journal arXiv
39. On the asymptotic Sₙ-structure of invariant differential operators on symplectic manifolds
    J. Algebra 356 (2012), 39–89. with Q. Ren. journal arXiv
40. Looping of the numbers game and the alcoved hypercube
    J. Combin. Theory Ser. A 119 (2012), no. 3, 713–730. with Q. Gashi and D. Speyer. journal arXiv
41. Free products, cyclic homology, and the Gauss-Manin connection
    Adv. Math. 231 (2012), no. 3–4, 2352–2389. with V. Ginzburg; appendix by B. Tsygan. journal arXiv
42. Connes-Kreimer quantizations and PBW theorems for pre-Lie algebras
    Séminaire et Congrès 26, Société Mathématique de France (2012/2013), 223–251. journal arXiv
43. Zeroth Poisson homology of symmetric powers of isolated quasihomogeneous surface singularities
    J. Reine Angew. Math. 667 (2012), 67–88. with P. Etingof. journal arXiv
44. On dominance and minuscule Weyl group elements
    J. Algebraic Combin. 33 (2011), no. 3, 383–399. with Q. Gashi. journal arXiv
45. Traces on finite W-algebras
    Transform. Groups 15 (2010), no. 4, 843–850. with P. Etingof. journal arXiv
46. A graph-theoretical translation of a lemma of Alexakis
    Appendix to S. Alexakis, “The decomposition of global conformal invariants IV: A proposition on local Riemannian invariants”, Adv. Math. 225 (2010), 515–597. journal arXiv
47. Differential operators and BV structures in noncommutative geometry
    Selecta Math. (N.S.) 16 (2010), no. 4, 673–730. with V. Ginzburg. journal arXiv
48. Poisson traces and D-modules on Poisson varieties
    Geom. Funct. Anal. 20 (2010), no. 4, 958–987. with P. Etingof; appendix by I. Losev. journal arXiv
49. Superpotentials and higher-order derivations
    J. Pure Appl. Algebra 214 (2010), no. 9, 1501–1522. with R. Bocklandt and M. Wemyss. journal arXiv
50. Poisson algebras and Yang-Baxter equations
    In Advances in Quantum Computation, Contemp. Math. 482, AMS (2009), 91–106. journalarXiv
51. Calabi-Yau Frobenius Algebras
    J. Algebra 321 (2009), no. 3, 774–815. with C.-H. Eu. journal arXiv
52. The necklace Lie coalgebra and renormalization algebras
    J. Noncommut. Geom. 2 (2008), no. 2, 195–214. with W. L. Gan. journal arXiv
53. Moyal quantization and stable homology of necklace Lie algebras
    Mosc. Math. J. 6 (2006), no. 3, 431–459. with V. Ginzburg. journal arXiv
54. A Hopf algebra quantizing a necklace Lie algebra canonically associated to a quiver
    Int. Math. Res. Not. (2005), no. 12, 725–760. journal arXiv
55. Trigonometric solutions of the associative Yang-Baxter equation
    Math. Res. Lett. 10 (2003), 301–321. journal arXiv
56. Proof of the GGS conjecture
    Math. Res. Lett. 7 (2000), no. 5–6, 801–826. journal arXiv
57. Explicit quantization of dynamical r-matrices for finite-dimensional semisimple Lie algebras
    J. Amer. Math. Soc. 13 (2000), no. 3, 595–609. with P. Etingof and O. Schiffmann. journal arXiv
58. Set-theoretical solutions of the quantum Yang-Baxter equation
    Duke Math. J. 100 (1999), no. 2, 169–209. with P. Etingof and A. Soloviev. journal arXiv

See also my Imperial publications page and arXiv author search.

Selected invited conference talks

This is intentionally selective rather than a complete record. Conference titles link to event pages where available.

• 2026. SCOTIE — Symplectic Singularities, Representation Theory and Integrability, University of Glasgow, 26–29 May.
• 2026. Gone Fishing 2026, Centre de Recherches Mathématiques, Montréal, 23–26 April.
• 2026. Deformations and Birational Geometry of Algebraic Varieties, in celebration of Yoshinori Namikawa's 60th birthday, RIMS, Kyoto, 19–23 January. Talk: “Building blocks of symplectic singularities and resolutions.”
• 2025. Around Singularities in Poisson Geometry, BIRS-IASM, Hangzhou, 3–8 August.
• 2024. Poisson Geometry and Artin-Schelter Regular Algebras, BIRS-IASM, Hangzhou, 13–18 October (online talk).
• 2024. MFO-RIMS Tandem Workshop: Algebraic Geometry and Noncommutative Projective Varieties, Oberwolfach, 22–27 September. Talk: “Calabi–Yau deformations of q-symmetric algebras” (remote/asynchronous).
• 2024. Hochschild (Co)Homology and Applications, Oberwolfach, 14–19 April. Talk: “Koszul, Calabi–Yau deformations of q-symmetric algebras,” given remotely from SLMath in Berkeley.
• 2023. Cluster Algebra, Representation Theory and Algebraic Geometry, University of Hong Kong, 4–9 September. Talk: “Birational geometry of GIT quotients and symplectic resolutions.”
• 2022. Noncommutative Geometry and Noncommutative Invariant Theory, BIRS, Banff, 25–30 September.
• 2022. Interactions between Algebraic Geometry and Noncommutative Algebra, Oberwolfach, 1–7 May (also invited to earlier editions in 2018 and 2014).
• 2020. Higher Structures and Field Theory, Erwin Schrödinger Institute, Vienna (online programme).
• 2020. Workshop on Lie Theory and Integrable Systems in Symplectic and Poisson Geometry, Fields Institute, Toronto (online). Talk: “Deformations of toric Poisson structures.”
• 2020. Quiver Varieties, Character Varieties and DT-invariants, Institut Henri Poincaré, Paris, 24–28 February.
• 2018. Poisson Geometry of Moduli Spaces, Associators and Quantum Field Theory, Simons Center for Geometry and Physics.
• 2016. Poisson 2016, ETH Zürich, 4–8 July. Talk: “Poisson-de Rham homology and special polynomials.”
• 2012. Representation Theory and Symplectic Algebraic Geometry, CIRM, Marseille, 9–13 July.

Selected organisation and research visits

• Co-organiser, Qolloquium: Quivers, Representations, Resolutions, PIMS, Canada (2020).
• Co-organiser, Symplectic Representation Theory, CIRM, Marseille (2019).
• Co-organiser, Interactions between Representation Theory and Algebraic Geometry, University of Chicago (2017).
• Organiser, Why be noncommutative?, Imperial College London (2016).
• Organiser and lecturer, MSRI summer graduate workshop on noncommutative algebraic geometry (2012).
• Visiting member, Max Planck Institute for Mathematics, Bonn (2016, 2018, 2020); group leader, Hausdorff Institute Junior Trimester Program (2017); research member, Simons Center (2018) and MSRI (2013).