Martijn Kool

Associate Professor

Mailing address
Department of Mathematics
Utrecht University
P.O. Box 80010
3508 TA Utrecht
The Netherlands

Visiting address
Hans Freudenthalgebouw
Room 502
Budapestlaan 6
3584 CD Utrecht
The Netherlands

E-mail
m.kool1[at]uu.nl

Research interests
Algebraic geometry. Enumerative geometry and moduli spaces of sheaves. Connections to combinatorics, modular forms, and theoretical physics.

Writings

  • with J.V. Rennemo, Proof of a magnificent conjecture, 59 pages, arXiv:2507.02852. link
  • with J. Oh, J.V. Rennemo, and R.P. Thomas, A Hopf index for isotropic sections of orthogonal bundles, 27 pages, arXiv:2502.13050. link
  • with Y. Bae and H. Park, Counting surfaces on Calabi-Yau 4-folds II: DT-PT_0 correspondence, 97 pages, arXiv:2402.06526. link
  • with J. Jelisiejew and R.F. Schmiermann, Behrend's function is not constant on Hilb^n(A^3), Geometry & Topology 29 (2025) 4469-4476. link
  • with D. van Bree, A. Gholampour, and Y. Jiang, A virtual PGL_r-SL_r correspondence for projective surfaces, Moduli (Foundation Compositio Mathematica) 2 e5 (2025) 1-41. link
  • with Y. Cao and S. Monavari, A Donaldson-Thomas crepant resolution conjecture on Calabi-Yau 4-folds, Transactions of the American Mathematical Society 376 (2023) 8225-8268. link
  • with Y. Bae and H. Park, Counting surfaces on Calabi-Yau 4-folds I: Foundations, to appear in Geometry & Topology, 150 pages, arXiv:2208.09474. link
  • with L. Göttsche and T. Laarakker, SU(r) Vafa-Witten invariants, Ramanujan's continued fractions, and cosmic strings, Michigan Mathematical Journal 75 (2025) 3-63. link
  • with L. Göttsche, Sheaves on surfaces and virtual invariants, Surveys in Differential Geometry 24 (2019) 67-116. link
  • with L. Göttsche, Virtual Segre and Verlinde numbers of projective surfaces, Journal of the London Mathematical Society 106 (2022) 2562-2608. link
  • with Y. Jiang, Twisted sheaves and SU(r)/Z_r Vafa-Witten theory, Mathematische Annalen 382 (2022) 719-743. link
  • with Y. Cao and S. Monavari, Stable pair invariants of local Calabi-Yau 4-folds, International Mathematics Research Notices 2022 (2022) 4753-4798. link
  • with Y. Cao and S. Monavari, K-theoretic DT/PT correspondence for toric Calabi-Yau 4-folds, Communications in Mathematical Physics 396 (2022) 225-264. link
  • with Y. Cao, Curve counting and DT/PT correspondence for Calabi-Yau 4-folds, Advances in Mathematics 375 (2020) 107371. link
  • with L. Göttsche and R.A. Williams, Verlinde formulae on complex surfaces: K-theoretic invariants, Forum of Mathematics, Sigma 9 (2021) 1-31. link
  • with L. Göttsche, Refined SU(3) Vafa-Witten invariants and modularity, Pure and Applied Mathematics Quarterly 14 (2018) 467-513. link
  • with Y. Cao, Counting zero-dimensional subschemes in higher dimensions, Journal of Geometry and Physics 136 (2019) 119-137. link
  • with L. Göttsche, A rank 2 Dijkgraaf-Moore-Verlinde-Verlinde formula, Communications in Number Theory and Physics 13 (2019) 165-201. link
  • with Y. Cao, Zero-dimensional Donaldson-Thomas invariants of Calabi-Yau 4-folds, Advances in Mathematics 338 (2018) 601-648. link
  • with A. Gholampour, Higher rank sheaves on threefolds and functional equations, Épijournal de Géométrie Algébrique 3 (2019) 1-29. link
  • with L. Göttsche, Virtual refinements of the Vafa-Witten formula, Communications in Mathematical Physics 376 (2020) 1-49. link
  • with J. Bryan, Donaldson-Thomas invariants of local elliptic surfaces via the topological vertex, Forum of Mathematics, Sigma 7 (2019) 1-45. link
  • with R. P. Thomas, Stable pairs with descendents on local surfaces I: the vertical component, Pure and Applied Mathematics Quarterly 13 (2017) 581-638. link
  • with J. Bryan and B. Young, Trace identities for the topological vertex, Selecta Mathematica, New Series 24 (2018) 1527-1548. link
  • with A. Gholampour, Rank 2 wall-crossing and the Serre correspondence, Selecta Mathematica, New Series 23 (2017) 1599-1617. link
  • with A. Gholampour and B. Young, Rank 2 sheaves on toric 3-folds: classical and virtual counts, International Mathematics Research Notices 2018 (2017) 2981-3069. link
  • with A. Gholampour, Stable reflexive sheaves and localization, Journal of Pure and Applied Algebra 221 (2017) 1934-1954. link
  • Stable pair invariants of surfaces and Seiberg-Witten invariants, The Quarterly Journal of Mathematics 67 (2016) 365-386. link
  • with A. Gholampour and Y. Jiang, Sheaves on weighted projective planes and modular forms, Advances in Theoretical and Mathematical Physics 21 (2017) 1455-1524. link
  • with R. P. Thomas, Reduced classes and curve counting on surfaces II: calculations, Algebraic Geometry (Foundation Compositio Mathematica) 1 (2014) 384-399. link
  • with R. P. Thomas, Reduced classes and curve counting on surfaces I: theory, Algebraic Geometry (Foundation Compositio Mathematica) 1 (2014) 334-383. link
  • with V. Shende and R. P. Thomas, A short proof of the Göttsche conjecture, Geometry & Topology 15 (2011) 397-406. link
  • Euler characteristics of moduli spaces of torsion free sheaves on toric surfaces, Geometriae Dedicata 176 (2015) 241-269. link
  • Fixed point loci of moduli spaces of sheaves on toric varieties, Advances in Mathematics 227 (2011) 1700-1755. link
  • Moduli spaces of sheaves on toric varieties, DPhil thesis, University of Oxford (2010), supervisor D. Joyce. link

    • Some recorded lectures

  • Virtual structure on Hilbert schemes of affine 4-space, M-seminar, Kansas State University, 3-4-2025. link
  • Gauge origami and Oh-Thomas virtual cycles, Workshop Moduli Stacks and Enumerative Geometry, Cambridge, 19-6-2024. link
  • Algebraic geometry and Vafa-Witten theory, Physics Latam, ICTP Physics without Frontiers, 8-9-2023. link
  • Vafa-Witten invariants of projective surfaces - overview, Workshop New Four-dimensional Gauge Theories, MSRI Berkeley, 24-10-2022. link
  • Counting surfaces on Calabi-Yau fourfolds, Strings and Geometry, Utrecht, 3-5-2022. link
  • Counting surfaces on Calabi-Yau fourfolds, Connections between String Theory and Special Holonomy, Simons collaboration on Special Holonomy in Geometry, Analysis, and Physics, Oxford, 12-01-2022. link
  • DT/PT correspondence for Calabi-Yau 4-folds, Moduli and Invariants, Casa Matematica Oaxaca (CMO), 20-11-2019. link
  • Donaldson-Thomas invariants of local elliptic surfaces, Workshop on Geometric Correspondence of Gauge Theories, ICTP, Trieste, 12-9-2016. link
  • Curves on surfaces, Hilbert Schemes, Sheaves and Representations (VBAC 2013), SISSA, Trieste, 16-6--20-6-2013. link