This is the program's identity, filed the way a registrar would file it. From a distance it will read as machine learning applied to geometry; up close the dependency runs the other way. It is the honed combination of computational geometry, information theory, machine learning, information dynamics, and dynamical systems — a Machine Learning / Artificial Intelligence research program built on a Programming-Languages-style geometric substrate, with the mathematics supplying the invariants: state, cost, dynamics, and global realizability. Every class below is earned by a named artifact in the record, not by affinity.
| Full name | Code | Earns its place by |
|---|---|---|
| Computational Geometry / Graphics | Medial axis, frame fields, quad topology, the modeler application | |
| Algebraic Topology | Betti/Euler admission, Morse–Smale and Neumann domains, persistence-as-sensor | |
| Spectral Theory | Laplacian spectra, operator families, spectral certificates | |
| Symbolic Computation | The operator registry, symbolic regression over a typed algebra, grammar growth | |
| Optimization and Control / Analysis of PDEs | Transport-as-optimization, proximal evolution; optimal transport straddles both | |
| Differential Geometry | Finsler/Randers constitutive metrics, curvature-set transport | |
| Dynamical Systems | Koopman operators, generators, admissible evolution | |
| Mathematical Physics | The GENERIC/metriplectic formalism | |
| Algebraic Geometry | Divisors, meromorphic quartic differentials, Abel–Jacobi realizability | |
| Functional Analysis | The norm the gate must use: Banach completeness versus Hilbert projection, reproducing kernels and the Moore–Aronszajn constant, and the proof that an L2 residual bounds no supremum without a declared hypothesis | |
| Numerical Analysis | The discrete certificate: cotangent Laplacian and the discrete maximum principle, mass-matrix amplification, maximum-norm finite-element estimates, and the exactness of the nodal supremum for piecewise-linear output | |
| Operator Algebras / High Energy Physics – Theory | Structure only, and marked as such. Spectral triples and the gluing pairing supply the shape of a composition law — not a bound. The Atiyah state space of the relevant theory is one-dimensional, so nothing analytic crosses into the gates |
| Code | Full name |
|---|---|
| Artificial neural networks and deep learning | |
| Logic in artificial intelligence | |
| Theory of compilers and interpretersTHE DISTINCTIVE ONE | |
| Differentials on Riemann surfaces | |
| Jacobians and Prym varieties | |
| Persistent homology and applications, topological data analysis | |
| Spectral problems; spectral geometry | |
| Local differential geometry of Finsler spaces | |
| Optimal transportation | |
| Optimization of shapes | |
| Numerical treatment of dynamical systems | |
| Irreversible thermodynamics (the GENERIC home) | |
| Hilbert spaces with reproducing kernels (the sharp sup-norm constant) | |
| Sobolev spaces and embedding theorems (the s > d/2 threshold) | |
| Asymptotic distribution of eigenvalues (the local Weyl law) | |
| Finite element methods for boundary value problems | |
| Noncommutative geometry — structure only | |
| Topological field theories — structure only |
And machine learning asks the sixth question — which admissible construction is worth proposing next — which is why it is primary without being sovereign.
This program will read, from a distance, as “machine learning applied to geometry.” Up close the dependency runs the other way: the deepest object in the architecture is the typed state-transition language connecting every layer; the mathematics gives that language its notions of state, cost, dynamics, and global realizability; the compiler gives it syntax, semantics, and execution; and learning is the mechanism by which the language grows. The predominant academic lineage is therefore a Machine Learning / Artificial Intelligence research program built on a Programming-Languages-style geometric substrate — and the substrate, not the network, is what the other disciplines are holding up.