writing

These are the things I have written over the past few years, all preprints are on the arXiv.

Papers

Published

1. Embeddings of algebras in derived categories of surfaces, joint with Theo Raedschelders (arXiv:1501.04197),
   published as Proceedings of the American Mathematical Society, volume 145, number 7, 2017, pages 2757–2770
2. The point variety of quantum polynomial rings, joint with Kevin De Laet and Lieven Le Bruyn (arXiv:1509.07312),
   published as Journal of Algebra, volume 463, 2016, pages 10–22
3. Derived categories of noncommutative quadrics and Hilbert squares, joint with Theo Raedschelders (arXiv:1605.02795),
   published online ahead of print in International Mathematics Research Notices
4. Relative tensor triangular Chow groups for coherent algebras, joint with Sebastian Klein (arXiv:1607.03423),
   published as Journal of Algebra, volume 487, 2017, pages 386–428
5. Hochschild cohomology of noncommutative planes and quadrics (arXiv:1705.06098),
   published as Journal of Noncommutative Geometry, volume 13, 2019, pages 769–795
6. Constructing noncommutative surfaces with exceptional collections of length 4, joint with Dennis Presotto (arXiv:1705.06943),
   published as Journal of the London Mathematical Society, volume 98, 2018, pages 85–103
7. Examples violating Golyshev's canonical strip hypothesis, joint with Sergey Galkin and Swarnava Mukhopadhyay (arXiv:1806.07648, see also the related repositories canonical-strip-moduli-of-vector-bundles and ehrhart-polynomials-toric-fanos)
   published online ahead of print in Experimental Mathematics
8. Admissible subcategories in derived categories of moduli of vector bundles on curves, joint with Swarnava Mukhopadhyay (arXiv:1807.00216)
   published as Advances in Mathematics, volume 351, 2019, pages 653–675
9. Hilbert squares: derived categories and deformations, joint with Lie Fu and Theo Raedschelders (arXiv:1810.11873)
   published as Selecta Mathematica, volume 25, 2019, paper 37
10. Automorphisms of Hilbert schemes of points on surfaces, joint with Georg Oberdieck and Jørgen Vold Rennemo (arXiv:1907.07064)
    published online ahead of print in Transactions of the American Mathematical Society

Preprints

1. Comparison of two constructions of noncommutative surfaces with exceptional collections of length 4, joint with Dennis Presotto and Michel Van den Bergh (arXiv:1811.08810)

   We will retitle this to "The Hirzebruch isomorphism for exotic noncommutative surfaces".

2. Derived categories of (nested) Hilbert schemes, joint with Andreas Krug (arXiv:1909.04321)
3. Hochschild cohomology of generalised Grassmannians, joint with Maxim Smirnov (arXiv:1911.09414)
4. Moduli spaces of semiorthogonal decompositions in families, joint with Shinnosuke Okawa and Andrea Ricolfi (arXiv:2002.03303)
5. Derived categories of standard flips and cubic hypersurfaces, joint with Lie Fu and Theo Raedschelders (arXiv:2002.04940)
6. Indecomposability of derived categories in families, joint with Francesco Bastianelli, Shinnosuke Okawa and Andrea Ricolfi

Work in progress

1. Spanning classes on curves, joint with Swarnava Mukhopadhyay

Other

1. Semiorthogonal decompositions for moduli of sheaves on curves, Oberwolfach Reports 24/2018, Oberwolfach Workshop 1822, Interactions between Algebraic Geometry and Noncommutative Algebra, pages 1473–1476
2. appendix "Comparing definitions of Hochschild cohomology for algebraic stacks" to The Hochschild cohomology ring of a global quotient orbifold, by Cris Negron and Travis Schedler (arXiv:1809.08715)
   published online ahead of print in Advances in Mathematics
3. Artin–Schelter regular algebras versus smooth projective varieties (short note, to be extended at some point)

Reviews

I have written reviews for

•  Mathematical Reviews
•  zbMath

Notes

You might also be interested in my blog.

2018

• Hochschild (co)homology, and the Hochschild–Kostant–Rosenberg decomposition, lecture notes for Advanced Topics in Algebra

2017

• Hochschild–Kostant–Rosenberg decomposition for surfaces, dimensions of components in the Hochschild–Kostant–Rosenberg decomposition for surfaces relevant to deformation theory

2016

• Segre symbols, a short note defining Segre symbols as a method of classifying base loci of pencils of quadric hypersurfaces
• The restriction of a homotopy-injective complex to a Zariski open subset is not necessarily homotopy-injective, details for the construction of such a complex, based on an explanation by Leonid Positselski
• Cheat sheet for mutations, because remembering where left and right goes is difficult

2015

• Cheat sheet for the simplicial indexing category, in case you always forget the notation (like me)
• Cheat sheet for recollements, the definition of a recollement and how to remember it
• The torsor for a formally smooth morphism, rough notes related to my lecture on lifting along a smooth morphism for ANAGRAMS about Hodge-to-de Rham degeneration

2014

• Spectral sequences: examples in algebra and algebraic geometry, notes for my lecture for ANAGRAMS about spectral sequences
• Quivers of exceptional collections on smooth projective varieties, poster for the British Algebraic Geometry meeting
• Positselski's main lemma, a short note elaborating on a lemma of Positselski that recognises vector bundles in the derived category
• Stratification of triangulated categories, notes for the lecture series by Henning Krause at the summer school on derived categories in Nantes (with corrections by Henning Krause, thanks!)
• Depth, and measuring singularities, notes for my short lecture for ANAGRAMS in the series on dimension functions
• My lecture series on Grothendieck duality for ANAGRAMS, also available as a single set of notes:
  1. Riemann–Roch theorem and Serre duality, notes for the first lecture
  2. More on Riemann–Roch and Serre duality, with applications, notes for the second lecture
  3. Derived categories and Grothendieck duality, notes for the third lecture
  4. Sketches of some of the proofs, notes for the fourth lecture

2013

• The $(\infty,1)$-topos of derived stacks, notes for my lecture at the Winter school on derived algebraic geometry
• Understanding $\mathbf{D}^{\mathrm{b}}(kQ)$ using moduli spaces, slides for my lecture at Young researchers in representation theory
• The triangulated structure on the category of projectives, notes for my lecture at the Summer school on preprojective algebras
• The derived Torelli theorem, notes for my lectures at the summer seminar in algebraic geometry of Columbia University and my exam for Surfaces K3 et théorie des déformations
• Grothendieck topologies and étale cohomology, notes for my lecture for the course Fonctions zêta et L de variétés et de motifs
• The derived stack of perfect complexes, notes for my lecture in Autour de la Géométrie Algébrique Dérivée, subsumed in the notes of the seminar
• Differential graded categories, notes for my lecture in Autour de la Géométrie Algébrique Dérivée, subsumed in the notes of the seminar

Non-mathematical writing

• The fugue and the guitar (in Dutch)