Papers
Working papers and long-form breakdowns — probability, category theory, and the math underneath real systems.
The Seven Hurwitz Points of M23
One Rational Point, Maximal Mixing
Six mathematicians settle the last sporadic case of the inverse Galois problem and say they have no conceptual explanation for the point that did it. Seven genus-4 curves computed to a hundred digits, a mirror that swaps two pairs, a sextic with full S6 symmetry, and a prime where the special point sits alone — the exact shape of the symmetry arithmetic breaks.
- inverse galois
- hurwitz spaces
- arithmetic geometry
Categorical Probability
Probability Maps as First-Class Arrows
A sensor hands a noisy estimate to a filter, the filter hands a belief to a controller, and every handoff is a distribution rather than a value. Categorical probability treats each step as a Markov kernel — an arrow you compose like a function — with Wasserstein distance as the ruler between beliefs. Giry, Fritz, and Perrone, read with an engineer's eyes.
- category theory
- markov kernels
- wasserstein
Topos-Theoretic Probability
Descent as Epistemic Coherence
Two thermometers in adjacent rooms disagree about the doorway they share. A drone and a ground unit file different reports on the same intersection. Sheaf descent makes that overlap the whole story: keep every belief local, price the mismatch where instruments both look, and glue what survives into one coherent picture.
- sheaf theory
- bayesian
- sensor fusion