Welcome to my homepage! I am a Bonn Junior Fellow at the Hausdorff Center for Mathematics. In the Wintersemester 2022 I teach a class on K3 surfaces. Talks for the Enumerative Geometry seminar can be found here.
Phone: +49 228 73-62271
Office: Endenicher Allee 60, 1.028
Endenicher Allee 60
Algebraic Geometry, in particular K3 surfaces, holomorphic-symplectic varieties, Hilbert schemes of points, Gromov-Witten and Donaldson-Thomas theory. Modular and Jacobi forms.
- Gromov-Witten invariants of the Hilbert scheme of points of a K3 surface
Geometry & Topology, 22 (2018), no. 1, 323-437.
- Curve counting on K3 x E, the Igusa cusp form chi_10, and descendent integration (with Rahul Pandharipande)
K3 surfaces and their moduli, Birkhauser Prog. in Math. 315, 245-278, 2014.
- Curve counting on abelian surfaces and threefolds
with Jim Bryan, Rahul Pandharipande, and Qizheng Yin
Algebraic Geometry, 5 (2018), no. 4, 398-463.
A SAGE program for calculating the number of polarized isogenies of abelian varieties.
- Gromov-Witten theory of K3 x P1 and quasi-Jacobi forms
International Mathematics Research Notices, Vol. 2019, no. 16, 4966-5011.
- On reduced stable pair invariants
Mathematische Zeitschrift, 289 (2018), no. 1-2, 323-353.
- Curve counting on elliptic Calabi-Yau threefolds via derived categories (with Junliang Shen)
Journal of the European Mathematical Society, 22 (2020), no. 3, 967-1002.
- Holomorphic anomaly equations and the Igusa cusp form conjecture (with Aaron Pixton)
Inventiones mathematicae, 213 (2018), no. 2, 507-587.
- Reduced Donaldson-Thomas invariants and the ring of dual numbers (with Junliang Shen)
Proceedings of the London Mathematical Society, (3) 118 (2019), no. 1, 191-220.
- Gromov-Witten theory of elliptic fibrations: Jacobi forms and holomorphic anomaly equations
with Aaron Pixton
Geometry & Topology, 23 (2019), no. 3, 1415-1489.
- Rational curves in holomorphic symplectic varieties and Gromov-Witten invariants
with Junliang Shen and Qizheng Yin
Advances in Mathematics, Volume 357 (2019), 106829, 8 pp.
- Automorphisms of Hilbert schemes of points on surfaces
with Pieter Belmans and Jorgen Vold Rennemo
Transactions of the American Mathematical Society, 373 (2020), no. 9, 6139–6156.
- Elliptic curves in Hyperkähler varieties (with Denis Nesterov)
International Mathematics Research Notices, 2021, no. 4, 2962–2990.
- CHL Calabi-Yau threefolds: Curve counting, Mathieu moonshine and Siegel modular forms
with Jim Bryan, and containing a joint Appendix with Sheldon Katz
Communications in Number Theory and Physics, 14 (2020), no. 4, 785–862.
- A Lie algebra action on the Chow ring of the Hilbert scheme of points of a K3 surface
Commentarii Mathematici Helvetici 96 (2021), no. 1, 65--77.
- Motivic decompositions for the Hilbert scheme of points of a K3 surface
with Andrei Negut and Qizheng Yin
Journal für die Reine und Angewandte Mathematik (Crelle's journal), to appear, 2019.
- Gromov-Witten theory of K3 surfaces and a Kaneko-Zagier equation for Jacobi forms
with Jan-Willem van Ittersum and Aaron Pixton
Selecta Mathematica, to appear, 2020.
SAGE code used in computations.
- Donaldson-Thomas invariants of abelian threefolds and Bridgeland stability conditions
with Dulip Piyaratne and Yukinobu Toda.
Journal of Algebraic Geometry, to appear.
The code used for the calculations in the Appendix.
- Equivariant categories of symplectic surfaces and fixed loci of Bridgeland moduli spaces
with Thorsten Beckmann
Algebraic Geometry, to appear, 2020.
- Gromov-Witten theory and Noether-Lefschetz theory for holomorphic-symplectic varieties
Forum Sigma, to appear, 2021. SageCode for Fano and DV case.
- Hilbert schemes of K3 surfaces, generalized Kummer, and cobordism classes of hyper-Kähler manifolds
with Jieao Song and Claire Voisin
PAMQ, to appear.
- Stable pairs and Gopakumar-Vafa type invariants on holomorphic symplectic 4-foldswith Yalong Cao and Yukinobu Toda, to appear in Advances in Mathematics, 2022
- Universality of descendent integrals on moduli spaces of stable sheaves on K3 surfaces
SIGMA 18 (2022), 076, 15 pages
Submitted papers and preprints:
- On equivariant derived categories
with Thorsten Beckmann, 2020.
- Multiple cover formulas for K3 geometries, wallcrossing, and Quot schemes, 2021
- Marked relative invariants and GW/PT correspondences, 2021
- Gopakumar-Vafa type invariants of holomorphic symplectic 4-folds
with Yalong Cao and Yukinobu Toda, 2022
- Holomorphic anomaly equations for the Hilbert scheme of points of a K3 surface, 2022
- Lagrangian Planes hyperkahler varieties of K3[n]-type, 2022
- PhD Thesis
- A Serre derivative for even weight Jacobi forms. Unpublished Note., 2012.
- Notes on the monodromy of K3[n]-Hyperkahler (after Markman)
- Oberwolfach report: Curves on the Hilbert scheme of a K3 surface
- Oberwolfach report: Noether-Lefschetz theory of hyper-Kähler varieties via Gromov-Witten invariants
- On the Chern character numbers of the Hilbert scheme of points on a K3 surface
- Slides for the talk Kaneko-Zagier equations for Jacobi forms and curve counting on CHL manifolds at the Quantum Gravity & Modularity Workshop in Dublin, May 2021.
- Notes on the LLV decomposition (joint with Jieao Song)
- Lectures on holomorphic anomaly equations (for Swampland workshop)
- Denis Nesterov
- Maximilian Schimpf
Supervised theses: Here
- Gromov-Witten theory of hyperkähler varieties (Summer 2021)
- Algebraic Stacks (Fall 2020)
- Topics in Enumerative Geometry (Fall 2019)
- Algebraic Geometry II (Spring 2019)
- Algebraic Geometry I (Fall 2018)
Past student seminars:
- Hilbert schemes of points and infinite dimensional Lie algebras (Sommersemester 2022)
- Seminar on Donaldson-Thomas theory (Fall 2021)
- Hilbert schemes and applications, Sommer 2020
- Quantum Groups & Quantum Cohomology of Symplectic Varieties (Fall 2019, with C. Stroppel)
- Beyond GIT (Fall 2018)
- Motivic PT Seminar (Fall 2014)
Research seminar organization:
- Seminar Algebraic Geometry (2018-)
- Algebraic Geometry, Physics, Gromov-Witten theory Seminar (Fall 2020+)
- Moduli and Representation Theory seminar (MSRI, Spring 2018)
- Workshop Curves and K3 surfaces, May 9-12, 2022.
- Hausdorff School Perverse Sheaves in Enumerative Geometry, February 10-14, 2020.
- Bethe Forum Number Theoretic Methods in Quantum Physics, July 15 - 19, 2019.
- Google Scholar
- Complex Geometry Group
- Co-authors: Thorsten Beckmann, Pieter Belmans, Jim Bryan, Yalong Cao, Jan-Willem van Ittersum, Sheldon Katz, Andrei Negut, Denis Nesterov, Rahul Pandharipande, Aaron Pixton, Dulip Piyaratne, Jørgen Vold Rennemo, Junliang Shen, Jieao Song, Yukinobu Toda, Claire Voisin, Qizheng Yin.
- Poonen's suggestions on mathematical writing