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. I study moduli spaces of sheaves and virtual sheaf counting invariants (Donaldson-Thomas, Pandharipande-Thomas, Vafa-Witten, …) in concrete geometric settings, mostly in dimensions 2, 3, 4. By specializing general theory to a specific setting, one is led to: (1) solutions of open problems (which sometimes existed before the virtual invariants were defined), (2) connections to other fields (for instance, combinatorics, modular forms, and string theory), (3) formulating/testing new conjectures (which can generate new research directions).

Writings

  • with L. Göttsche and T. Laarakker, SU(r) Vafa-Witten invariants, Ramanujan's continued fractions, and cosmic strings, 60 pages, arXiv:2108.13413. link
  • with L. Göttsche, Sheaves on surfaces and virtual invariants, Surveys Diff. Geom. 24 (2019) 67-116. link
  • with L. Göttsche, Virtual Segre and Verlinde numbers of projective surfaces, to appear in Jour. LMS, 38 pages, arXiv:2007.11631. link
  • with Y. Jiang, Twisted sheaves and SU(r)/Z_r Vafa-Witten theory, Math. Ann. 382 (2022) 719-743. link
  • with Y. Cao and S. Monavari, Stable pair invariants of local Calabi-Yau 4-folds, Internat. Math. Res. Not. 2022 (2022) 4753-4798. link
  • with Y. Cao and S. Monavari, K-theoretic DT/PT correspondence for toric Calabi-Yau 4-folds, to appear in Comm. Math. Phys., 45 pages, arXiv:1906.07856. link
  • with Y. Cao, Curve counting and DT/PT correspondence for Calabi-Yau 4-folds, Adv. Math. 375 (2020). link
  • with L. Göttsche and R.A. Williams, Verlinde formulae on complex surfaces: K-theoretic invariants, Forum of Math. Sigma 9 (2021) 1-31. link
  • with L. Göttsche, Refined SU(3) Vafa-Witten invariants and modularity, Pure and Appl. Math. Quart. 14 (2018) 467-513. link
  • with Y. Cao, Counting zero-dimensional subschemes in higher dimensions, Jour. Geom. Phys. 136 (2019) 119-137. link
  • with L. Göttsche, A rank 2 Dijkgraaf-Moore-Verlinde-Verlinde formula, Comm. Numb. Theor. and Phys. 13 (2019) 165-201. link
  • with Y. Cao, Zero-dimensional Donaldson-Thomas invariants of Calabi-Yau 4-folds, Adv. Math. 338 (2018) 601-648. link
  • with A. Gholampour, Higher rank sheaves on threefolds and functional equations, Épiga 3 (2019) 1-29. link
  • with L. Göttsche, Virtual refinements of the Vafa-Witten formula, Comm. Math. Phys. 376 (2020) 1-49. link
  • with J. Bryan, Donaldson-Thomas invariants of local elliptic surfaces via the topological vertex, Forum of Math. Sigma 7 (2019) 1-45. link
  • with R. P. Thomas, Stable pairs with descendents on local surfaces I: the vertical component, Pure and Appl. Math. Quart. 13 (2017) 581-638. link
  • with J. Bryan and B. Young, Trace identities for the topological vertex, Selecta Math. 24 (2018) 1527-1548. link
  • with A. Gholampour, Rank 2 wall-crossing and the Serre correspondence, Selecta Math. 23 (2017) 1599-1617. link
  • with A. Gholampour and B. Young, Rank 2 sheaves on toric 3-folds: classical and virtual counts, Internat. Math. Res. Not. 2018 (2017) 2981-3069. link
  • with A. Gholampour, Stable reflexive sheaves and localization, Jour. Pure and Applied Alg. 221 (2017) 1934-1954. link
  • Stable pair invariants of surfaces and Seiberg-Witten invariants, Quart. Jour. Math. 67 (2016) 365-386. link
  • with A. Gholampour and Y. Jiang, Sheaves on weighted projective planes and modular forms, Adv. Theor. Math. Phys. 21 (2017) 1455-1524. link
  • with R. P. Thomas, Reduced classes and curve counting on surfaces II: calculations, Alg. Geom. 1 (2014) 384-399. link
  • with R. P. Thomas, Reduced classes and curve counting on surfaces I: theory, Alg. Geom. 1 (2014) 334-383. link
  • with V. Shende and R. P. Thomas, A short proof of the Göttsche conjecture, Geom. Topol. 15 (2011) 397-406. link
  • Euler characteristics of moduli spaces of torsion free sheaves on toric surfaces, Geom. Ded. 176 (2015) 241-269. link
  • Fixed point loci of moduli spaces of sheaves on toric varieties, Adv. Math. 227 (2011) 1700-1755. link
  • Moduli spaces of sheaves on toric varieties, DPhil thesis, University of Oxford (2010), supervisor D. Joyce. link
  • Recent talks

  • 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