interactive

I have a healthy interest in using computers for mathematics, in different ways.

Online mathematics

I've created interactive classifications:

• fanography.info
  a tool to visually study the geography of Fano 3-folds
• grassmannian.info
  a periodic table of (generalised) Grassmannians
• hyperkaehler.info
  the geography of irreducible holomorphic symplectic (or hyperkähler) varieties
• mgnbar.info
  the geometry of $\overline{\mathrm{M}}_{g,n}$
• superficie.info
  interactive geography of (minimal) complex algebraic smooth surfaces
• cubics.fanography.info
  cubic fourfolds and their associated K3 surfaces

I also maintain the (infrastructure for) the following websites:

• stacks.math.columbia.edu
  an open source textbook and reference work on algebraic geometry
• ncag.info
  a website for things related to noncommutative algebraic geometry
• kerodon.net
  an online resource for homotopy-coherent mathematics

Computational mathematics

I've implemented the following (hopefully useful) tools:

• Hodge diamond cutter
  a tool to work with Hodge diamonds, comes with many varieties and constructions built into it
• QuiverTools
  a toolset to work with quivers and moduli of quiver representations

• IntersectionTheory in OSCAR
  integration of Jieao Song's IntersectionTheory in OSCAR, joint with Wolfram Decker

• HomogeneousTools
  a toolset to work with sheaf cohomology on partial flag varieties and zero sections inside them
  • Lie.jl: a Julia reimplementation of (part of) LiE
  • PartialFlagVarieties.jl: a Julia implementation of computations for sheaf cohomology on partial flag varieties and zero sections, joint with Javier Fernández Píriz
  • Lie.js: a JavaScript port of Lie.jl and PartialFlagVarieties.jl

• twisted-hodge-hilbert, Sage library to compute twisted Hodge numbers of Hilbert schemes of points

• twisted-hodge-ci, Sage library to compute twisted Hodge numbers of complete intersections, joint with Piet Glas

Tools

• gerby-project, the system underlying the Stacks project, Kerodon, and other large online mathematical texts
• bibgetter, automatically resolve MathSciNet and arXiv identifiers in LaTeX for bibliography management

Various

I also have smaller (and older) projects:

• An atlas for $\mathop{\mathrm{Spec}}\mathbb{Z}[x]$, a collection of pictures for the geometric intuition behind $\mathop{\mathrm{Spec}}\mathbb{Z}[x]$

• comparison of topologies on $\mathrm{Sch}/S$, an incomplete comparison of all the Grothendieck topologies on the category of schemes I could find, together with their properties

  see github.com/pbelmans/topologies-comparison

• cohomology of twists of the structure sheaf and Hodge diamonds for complete intersections: in the case of a complete intersection it is not too difficult to compute the dimensions of the cohomology spaces of (twists of) the structure sheaf, thereby visualising Serre duality, and similarly one can compute the Hodge diamond

  see github.com/pbelmans/cohomology-tables

• 

  3d version of Conway's Game of life, an unfinished visualisation project (see also the blogpost); the code is open-source and you can have a go at implementing a GUI!