Algebra, Topology, Differential Calculus, and Optimization Theory for Computer Science and Machine Learning (Gallier & Quaintance)
A 2,204-page manuscript by jean-gallier and jocelyn-quaintance at Penn's Department of Computer and Information Science, served as a single PDF from Gallier's "Books in Progress" page. It develops, from first principles and in one volume, most of the mathematics the rest of this spoke holds one book per subject for — and then keeps going, through convex sets and linear programming to support vector machines and Lasso.
It is the largest single text in the corpus by an order of magnitude: lal-algebra-1 and stirzaker-elementary-probability are ~500 pages each; this is over four times the two combined.
What it contains
Eight parts, plus two prefatory chapters (an introduction and a chapter on groups, rings and fields that the rest leans on):
| Part | Subject | Where it starts |
|---|---|---|
| I | Linear algebra — vector spaces, matrices, determinants, Gaussian elimination and LU/Cholesky, norms and condition numbers, duality, Euclidean and Hermitian spaces, QR, eigenvalues, spectral theorems, graph Laplacians, SVD and pseudo-inverses | p. 47 |
| II | Affine and projective geometry | p. 791 |
| III | The geometry of bilinear forms — Cartan–Dieudonné, Witt's theorem | p. 955 |
| IV | Algebra — polynomials and PIDs, primary decomposition, UFDs and Noetherian rings, tensor and exterior algebras, modules over a PID, rational canonical form | p. 1053 |
| V | Topology and differential calculus (with a detour on fractals) | p. 1319 |
| VI | Preliminaries for optimization — extrema, Newton's method, quadratic optimization, Schur complements | p. 1459 |
| VII | Linear optimization — convex sets and cones, linear programs, the simplex algorithm, LP duality | p. 1541 |
| VIII | Nonlinear optimization — Hilbert spaces, nonlinear optimization, subgradients, dual ascent and ADMM, positive definite kernels, soft-margin SVMs, ridge regression / Lasso / elastic net | p. 1647 |
The order is the argument: algebra and geometry first, optimization last, machine learning only as the final chapters of the optimization part. Kernels and SVMs arrive on p. 1913 and after, once Hilbert spaces, convexity and duality are already proved.
Why this matters to the corpus
Three things are new here.
A single spine under several subjects. Every other text here teaches one subject and names the others as prerequisites; linear-algebra in particular "sits under every text and is the subject of none" was this spoke's founding complaint. This book supplies linear algebra, abstract-algebra and topology itself and then uses them, so the dependency the corpus records as a gap between books is here a chapter boundary inside one.
A second answer to "the application picks the mathematics." nelson-essential-math-for-ai (routed 2026-09-09) organizes by application — "the maths for convolution." Gallier and Quaintance have the same destination (machine learning is in the title) and the opposite organization: subject order, proofs first, applications as consequences 1,900 pages in. Two books, one stated audience, contradictory pedagogy. See synthesis.
Topology and optimization theory arrive as subjects. Neither is named in this spoke's five families. Optimization is the closest boundary the spoke has — see the routing note below.
Level and rigour
Graduate and proof-complete, and it does not pretend otherwise: definitions, propositions and proofs run continuously, sections marked ⊛ flag the harder optional material (transvections and dilatations, subgradients), and chapters end with Summary and Problems. It assumes mathematical maturity the way hefferon-linear-algebra deliberately does not. On this spoke's rigour spectrum it sits at the top with osborne-rubinstein-course-in-game-theory and above every first-course text here.
Access and status
Free to download, and explicitly not free to redistribute: the terms allow educational reproduction and copying for classes at cost, forbid commercial reproduction, and forbid posting the file at any other web location — links to the page only. That is a fourth position on this spoke's access axis, alongside copyleft (hefferon-linear-algebra, judson-abstract-algebra), CC0 (grinberg-graph-theory) and paywalled (lal-algebra-1): free as in price, reserved as in rights, with the author holding the sole distribution point.
"Book in Progress" is a status, not a disclaimer about quality — the manuscript is complete enough to be taught from — but it means there is no edition to cite and the file changes under its URL.
Dates disagree, and the page is the older witness
The landing page says "Approx. 2192 pages (2023)" and is signed 2017-04-15. The PDF's own cover
page says April 14, 2025, and the file was last modified 2025-04-15; it is 2,204 pages. The
published: field records the manuscript's own date; the page count here is counted, not quoted.
The page's numbers are stale, which is what a hand-maintained landing page for a moving file does.
Why T1
A full professor of computer science and of mathematics at Penn (jean-gallier), developing the subject from first principles with complete proofs, published first-party. Not peer-reviewed as a book and not final — but the tier grades what the source is, and this is a primary mathematical development, not a survey.
Connections
- authors jean-gallier · jocelyn-quaintance; institution university-of-pennsylvania
- subjects: linear-algebra · abstract-algebra · topology · graph-theory · matrix-calculus
- contrast in organization: nelson-essential-math-for-ai; contrast in entry level: hefferon-linear-algebra
- cross-spoke: the optimization half meets
../optimization-algorithms-wiki(convex optimization, linear programming) and../machine-learning-wiki(kernels, SVMs, ridge/Lasso) — the maths is the subject here, a claim about a model is theirs