The Geometry of Forbidden Directions8 parts · 5 appx 29 min

The Geometry of Forbidden Directions: A Research Program

When a physical field forbids a direction of motion — not merely slows it — the configuration space stops being ordinary and becomes sub-Riemannian. This post scopes a program built on that distinction, states the identity it is organised around (Q = d + k + 2 — a dimensional count, verified numerically), and draws the sharp line between the physics that qualifies (magnetic fields, rotation) and the physics that only looks like it does (anisotropic transport, gravity).

Mathematics
sub-riemannian magnetic-fields holonomy connections
The Geometry of Forbidden Directions: A Research Program
From Caustics to Groups4 parts · 6 appx 60 min

From Caustics to Groups: A Research Program

A caustic is where a flow of paths focuses — the bright edge in a coffee cup, the cusp in a gravitational lens, the shell-crossing wall of the cosmic web. This post scopes a falsifiable research program for the reverse map: given an observed field of caustics, infer the hidden sub-Riemannian (Lie-group) geometry that produced it. The program is built around a hard obstruction — most caustics are provably group-blind — and the three places where the discriminating information actually lives.

Mathematics
sub-riemannian caustics lie-groups singularity-theory
From Caustics to Groups: A Research Program
Geometry of the Cosmic Web3 parts · 5 appx 86 min

The Geometry of the Cosmic Web: A Research Program

Matter in the Universe collapses into a web of filaments. This post scopes a falsifiable research program: lift the cosmic web onto the position–orientation manifold $\mathbb{R}^3 \times S^2 \cong \mathrm{SE}(3)/\mathrm{SO}(2)$ — the same machinery the visual cortex uses to complete contours — and test with public simulation and survey data whether sub-Riemannian geodesics trace, and perhaps shape, the filaments.

Mathematics
sub-riemannian SE3 cosmic-web filaments
The Geometry of the Cosmic Web: A Research Program
Geometry of Seeing4 parts · 5 appx 101 min

The Visual Cortex as a Contact Manifold

Your brain fills in contours that do not exist. Petitot's neurogeometry models oriented cortical states by a contact bundle; this article separates the free sub-Riemannian geodesics from their smooth Euler-elastica siblings.

Mathematics
sub-riemannian SE2 visual-cortex elliptic-functions
The Visual Cortex as a Contact Manifold
Software 9 min

Foundation Models, The AI Frontier

A four-year-old child processes roughly one petabyte of visual information — fifty times more than GPT-4 has ever read — and learns effortlessly. Closing that gap is what's pushing AI past sequential token prediction toward predictive world models, retrieval-augmented memory, and Karpathy's Software 3.0 paradigm: the LLM as operating system.

machinelearning deeplearning
Foundation Models, The AI Frontier