A primer · read this first, then the series

The Mathematics of the Work, Told as One Story

Two ways of looking do most of the lifting — and the texts that open each.

Almost everything across these volumes and papers is a single move in two disguises. Take a domain — language and meaning, an economy, a body of law, cognition, taste — and do one of two things with it. Either place it on a manifold, a curved space in which distance and direction carry the domain's own logic; or measure it by how well it compresses, by how little information a faithful description needs. Geometry is the first lens. Information theory is the second.

Beauty is compressibility. Meaning has holonomy.

Those two slogans are the two lenses pointed at art and at language. Learn to switch between them and most of the mathematics stops looking like separate subjects. Two foundations sit under both, and then the work splits into two trails that occasionally rejoin.

The base · both trails need it

Probability and linear algebra

Probability is the grammar of anything uncertain; learn it for intuition before rigor, from Blitzstein & Hwang, Introduction to Probabilityfree PDF. Then read MacKay, Information Theory, Inference, and Learning Algorithmsfree — it teaches probability, inference, and compression as one story, exactly the posture the aesthetics work takes. Linear algebra is the grammar of anything high-dimensional; every embedding, PCA, and rotation is a matrix acting on a space, and Strang's linear-algebra course (open at MIT) is the gentlest way in.

Trail one · the Geometric Series

From curvature to holonomy

Start with the geometry you can see — curves, surfaces, and the single most important word, curvature — in Pressley, Elementary Differential Geometry. Then generalize a surface to a manifold, where geodesics replace straight lines: Tu is gentle, Lee the reference. This is where “meaning lives on a manifold” becomes literal — hyperbolic and SPD embeddings are just manifolds chosen so distance means the right thing.

The subtlest idea here is holonomy: carry a vector around a loop on a curved space and it returns rotated, and the rotation measures the enclosed curvature. In the bond-space work that rotation is semantic distortion, and the symmetry allowed to act on it — the gauge group — is what “gauge-invariant meaning” quotients away. For that, read groups and symmetry from scratch (Armstrong), then connections and holonomy — Baez & Muniain is the kindest route, Nakahara the reference.

Trail two · the systems and ML papers

Chains, concentration, control

The admission-control work is built from four pieces. A Markov chain models the neighbour's load (Norris), and its mixing time and spectral gap — how fast it forgets — are the whole ballgame, since the central law says feedback is worth exactly the autocorrelation of capacity across the sensing delay (Levin, Peres & Wilmerfree). Concentration explains why estimators from dependent samples still converge (Vershyninfree) — the same mathematics that makes a random rotation distribution-free in the codec. Convexity and duality give the achievable-region “kink” the impossibility theorem turns on (Boyd & Vandenberghefree). And feedback and regret close it: AIMD is a control loop on a delayed signal (Åström & Murrayfree), and the adaptive controller's guarantee is a regret bound (Hazanfree). One volume touching the high-dimensional geometry, the SVD, and random projection at once is Blum, Hopcroft & Kannan, Foundations of Data Sciencefree.

Where the trails rejoin

Probability is geometry

Two ideas belong to both. Information geometry (Amari) shows the space of probability distributions is itself a curved manifold — the cleanest statement of why the two lenses were never really separate. And topological data analysis (Ghristfree) asks not where the data sits but what shape it has. Compressibility closes the loop: a good coordinate system is one that compresses, so choosing the right manifold and choosing the shortest description are the same act.

How to read it

Don't go top to bottom. Take the two foundations, then follow one trail as far as the chapter or paper in front of you demands — the Geometric Series wants trail one, the systems and ML papers want trail two, and the aesthetics and observer work sits on the bridge and rewards a little of both. Every book is chosen to be the gentlest correct entry to its idea, not the most complete.