Simulations
Small interactive pieces to accompany talks and courses. Each one runs entirely in the browser, so you can change the parameters and watch what happens. I wrote these simulations with the help of a large language model.
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How to draw a perfect maze at random
Six interactive slides for the talk “How do you draw a perfect maze at random?” at Hablando de Matemáticas. A perfect maze is a spanning tree, so drawing one uniformly at random is Wilson's algorithm: walk at random, erase the loops as they appear, and the corridors that remain are the tree while the walls are its dual.This one is in Spanish.
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The geometry of the uniform spanning tree in three dimensions
The slides from my course at the XI School in Probability and Stochastic Processes. Eleven interactive slides, from Bernoulli percolation to Wilson's algorithm run inside a three-dimensional box: use the arrows or the arrow keys to move between them, and press F for full screen. -
Bienaymé–Galton–Watson
A branching process grown one generation at a time, shown at once as a genealogy, as the sequence of generation sizes, and as a walk. A second panel puts an Erdős–Rényi graph beside it, so the local limit can be read off the picture. Choose the offspring law and its parameter and watch the process die out or survive.This one is in Spanish.
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Intrinsic volumes as averaged shadows
Kubota's formula says an intrinsic volume of a convex body is the average area of its shadow under a uniformly random rotation. Here the body is rotated, the shadow is cast, and the running average is compared against the exact value. -
Intrinsic volumes in higher dimensions
The same experiment run in dimensions two through twelve, projecting onto a uniformly random plane instead of rotating. The estimate still converges to the exact value, and the accumulated shadows trace out the shape they average to. -
Matroids: independence in two guises
One matroid on six elements, presented twice: as vectors, where independence means linearly independent, and as a graph, where it means cycle-free. Pick any subset and the two pictures give the same answer, on all sixty-four of them.
Each simulation opens in a new tab and runs entirely in your browser.