Martijn Kool

Algebraic geometry. I am interested in moduli spaces of sheaves (or sheaf-like objects) on specific classes of varieties (e.g. toric varieties, local surfaces, elliptic fibrations, general type surfaces). Explicit descriptions of strata in these moduli spaces, or (virtual) cycles on them, can be used for calculating invariants such as topological Euler characteristics or Donaldson-Thomas, Pandharipande-Thomas, or Donaldson invariants. This often leads to unexpected links with other fields such as enumerative geometry, modular forms, combinatorics, or theoretical physics.

with Y. Cao, Curve counting and DT/PT correspondence for Calabi-Yau 4-folds, 27 pages, arXiv:1903.12171. link
with L. Göttsche and R.A. Williams, Verlinde formulae on complex surfaces I: K-theoretic invariants, 36 pages, arXiv:1903.03869. link
with L. Göttsche, Refined SU(3) Vafa-Witten invariants and modularity, 39 pages, arXiv:1808.03245. 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, to appear in Comm. Numb. Th. and Phys., 20 pages, arXiv:1801.01878. link
with Y. Cao, Zero-dimensional Donaldson-Thomas invariants of Calabi-Yau 4-folds, Adv. Math. 338 (2018) 601--648. arXiv:1712.07347. link
with A. Gholampour, Higher rank sheaves on threefolds and functional equations, 33 pages, arXiv:1706.05246. link
with L. Göttsche, Virtual refinements of the Vafa-Witten formula, 44 pages, arXiv:1703.07196. link
with J. Bryan, Donaldson-Thomas invariants of local elliptic surfaces via the topological vertex, Forum of Math. Sigma 7 (2019) 1-45. arXiv:1608.07369. 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. arXiv:1605.02576. link
with J. Bryan and B. Young, Trace identities for the topological vertex, Selecta Math. 24 (2018) 1527-1548. arXiv:1603.05271. link
with A. Gholampour, Rank 2 wall-crossing and the Serre correspondence, Selecta Math. 23 (2017) 1599-1617. arXiv:1602.03113. 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. arXiv:1509.03536. link
with A. Gholampour, Stable reflexive sheaves and localization, Jour. of Pure and Applied Alg. 221 (2017) 1934-1954. arXiv:1308.3688. link
Stable pair invariants of surfaces and Seiberg-Witten invariants, Quart. Jour. Math. 67 (2016) 365-386. arXiv:1303.5340. link
with A. Gholampour and Y. Jiang, Sheaves on weighted projective planes and modular forms, Adv. Theor. Math. Phys. 21 (2017) 1455-1524. arXiv:1209.3922. link
with R. P. Thomas, Reduced classes and curve counting on surfaces II: calculations, Alg. Geom. 3 (2014) 384-399. arXiv:1112.3070. link
with R. P. Thomas, Reduced classes and curve counting on surfaces I: theory, Alg. Geom. 3 (2014) 334-383. arXiv:1112.3069. link
with V. Shende and R. P. Thomas, A short proof of the Göttsche conjecture, Geom. Topol. 15 (2011) 397-406. arXiv:1010.3211. link
Euler characteristics of moduli spaces of torsion free sheaves on toric surfaces, Geom. Ded. 176 (2015) 241-269. arXiv:0906.3393. link
Fixed point loci of moduli spaces of sheaves on toric varieties, Adv. Math. 227 (2011) 1700-1755. arXiv:0810.0418. link

Moduli spaces of sheaves on toric varieties, DPhil thesis, University of Oxford (2010), supervised by D. Joyce. link
Anomalies of the (super)conformal algebra in string theory, MSc thesis in theoretical physics, University of Utrecht (2006), supervised by G. Arutyunov.
The McKay correspondence, derived categories and braid relations, MSc thesis in mathematics, University of Utrecht (2005), supervised by B. Szendrői.
Het lokaal-globaal principe voor kegelsneden en elliptische krommen, bachelor thesis in mathematics, University of Utrecht (2003), supervised by G. Cornelissen.

Mirror symmetry and Enumerative Geometry (Simons Collaboration on Homological Mirror Symmetry), UC Berkeley, 26-30 March 2018. video
Workshop on geometric correspondence of gauge theories, ICTP, 12-16 September 2016. video
KIAS winter school on virtual enumerative geometry, 6 until 9-1-2015. video, pdf
Curves on surfaces, Hilbert Schemes, Sheaves and Representations (VBAC 2013), SISSA, 16 until 20-6-2013. video
Four lectures on stable pairs, lecture series, International Workshop on Complex Geometry, Dep. of Math. and Inst. of the Math. Sc. CUHK, 18 until 19-4-2013. pdf
GQT graduate school and colloquium, Den Dolder, 28 Nov-2 Dec 2016. link, pdf

Staff Colloquium, co-organizer.
Geometry and Algebra, Geometry and Analysis Seminar.
Utrecht Geometry Centre (local).
Geometry and Quantum Theory Cluster (country-wide, contains job ads).

Organization: See organizational e-mail. Material covered and exercises:

Jan 8: Ch. 1-4 of Armstrong. Exc. 1.3, 1.4, 2.3, 2.7, 2.8, 3.2, 3.7*, 4.1, 4.6*, 4.8. Markers: Djin, Marije, Thomas.
Jan 9: Ch. 5, 6 of Armstrong. Exc. 5.4, 5.5, 5.6, 6.2, 6.6*, 6.9*. Markers: Geofrey, Pepijn, Friso.
Jan 10: parts of Ch. 7, 8 of Armstrong. Exc. 7.2, 7.11, 7.12*, 8.1, 8.6, 8.4*. Markers: Karel, Max, Thijs.
Jan 11: Ch. 10, 11 of Armstrong. Exc. 10.7, 10.12*, 11.1*, 11.3, 11.4. Markers: Nour, Hannah, Marijn.
Jan 12: Ch. 12, 15, 16 of Armstrong (we skipped 9, 13, 14). Exc. 12.1, 12.6, 12.9*, 15.2, 15.6*, 15.7. Markers: Alidad, Nick, and Louis.