MATH-CS COMPASS: Curriculum Roadmap & Development Plan

Author: Yusuke Yokota
Last Updated: 4/5/2026 Website: https://math-cs-compass.com

Project Overview

MATH-CS COMPASS is an educational platform bridging pure mathematics and computer science, addressing the gap where CS students struggle with mathematical foundations while math students lack awareness of practical applications. The primary focus is providing rigorous mathematical foundations for modern AI/ML.

Three Application Domains

The curriculum serves three broad application domains. These are not terminal goals but recurring viewpoints — intermediate plateaus where accumulated mathematical tools illuminate a class of real-world problems, and simultaneously reveal deeper mathematics worth pursuing.

1. Geometric Deep Learning (GDL): Symmetry, invariance, and continuous geometry applied to network architectures. Draws on Lie groups, Riemannian geometry, and fiber bundles.
2. Categorical Deep Learning (CDL): Compositionality, structural abstraction, and functorial reasoning applied to network operations. Draws on category theory, quivers, and string diagrams.
3. Quantum Computation & Quantum Theory: Unitary evolution, spectral decomposition, and Hilbert space geometry applied to computation and physics. Emerges naturally from Fourier analysis, Lp completeness, and operator theory.

Design philosophy: Rather than building toward these domains as endpoints, we develop rigorous mathematical foundations and introduce application domains at the moment sufficient tools are in hand — the same way Natural Gradient Descent (ml-12) previewed Riemannian geometry before manifolds were formally defined. Each “viewpoint” page motivates further mathematical depth, creating a virtuous cycle: math → application preview → deeper math → richer application.

Architectural Principle

There is no isolated “Geometry” or “Physics” section. Geometric and quantum concepts are inherently distributed across foundational sections:

• Section I (Algebra): Lie groups, tensor products, representation theory
• Section II (Analysis): Smooth manifolds, Hilbert spaces, unitary operators, spectral theory
• Section IV (Discrete): Simplicial complexes, discrete differential geometry, quivers
• Section V (ML): Application viewpoints — GNN, equivariant networks, GDL intro, CDL intro

Current Coverage (as of 4/5/2026)

Section I: Linear Algebra to Algebraic Foundations (30 pages) ✅

• Core Linear Algebra (14 pages): Linear systems, transformations, matrix algebra, determinants, vector spaces, eigenvalues, orthogonality, least squares, symmetry/SVD, trace/norms, Kronecker, Woodbury, stochastic matrices, graph Laplacians.
• Abstract Algebra (8 pages): Groups, cyclic/permutation, structural group theory, classification of finite abelian groups, rings & fields, ideals & factor rings, polynomial rings, integral domains.
• Field Theory & Continuous Symmetry (8 pages): Extension fields, geometry of symmetry (Dₙ, SO(3), SE(3)), algebraic extensions, finite fields, Lie groups, matrix exponential, Lie algebras & bracket, Lie correspondence.

Section II: Calculus to Optimization & Analysis (30 pages) ✅

• Derivatives & Integration (9+3 pages): Gradients, Jacobians, Newton’s method, KKT, Duality, Riemann integration, measure theory, Lebesgue integration.
• Fourier Analysis (2 pages): Fourier series, Fourier transform & FFT.
• Metric Spaces & Topology (8 pages): Metric spaces, convergence, continuity, completeness, connectedness, compactness, metric equivalence (homeomorphism), topological spaces.
• Functional Analysis (7 pages): Banach & Hilbert spaces, bounded operators, dual spaces & Riesz representation, weak topologies & Banach-Alaoglu, spectral theory of compact operators, RKHS & kernel methods, Lp spaces & Riesz-Fischer.

Section III: Probability & Statistics (23 pages) ✅

• Foundations & Inference: Probability, distributions, covariance, MVN, MLE, hypothesis testing, linear regression.
• Bayesian & Stochastic: Bayesian inference, exponential family, Fisher information matrix, decision theory, Markov chains, Monte Carlo, importance sampling, Gaussian processes.
• Measure-Theoretic Probability (2 pages): Random variables as measurable functions, expectation as Lebesgue integral, convergence theorems, limit theorems & product measures.

Section IV: Discrete Mathematics & Algorithms (15 pages)

• Computation & Graph Theory (11 pages): Graph theory, combinatorics, automata, Boolean logic, context-free languages, Turing machines, time complexity, Eulerian/Hamiltonian, P vs NP, network flow, trees.
• Discrete Topology Arc (4 pages) ✅: Planar graphs (Euler’s formula), incidence structure, simplicial complexes, intro to homology (Betti numbers, Euler-Poincaré).

Section V: Machine Learning (12 pages) ✅

• Foundations & Deep Learning: Intro to ML, regularized regression, classification, SVM, neural networks, automatic differentiation, deep NN (CNNs/Transformers).
• Unsupervised & Physical AI: PCA & autoencoders, clustering, reinforcement learning, natural gradient descent, variational autoencoders (VAE).

Total: 110 pages.

Dependency Map

Rather than a linear convergence diagram with terminal goals, the curriculum forms a web of mutual reinforcement. Each arrow represents prerequisite knowledge; each application viewpoint (marked with ◈) motivates return to deeper foundations.

SECTION I (Algebra)          SECTION II (Analysis)          SECTION III (Probability)     SECTION IV (Discrete)
═══════════════════          ═════════════════════          ═════════════════════════     ═════════════════════
Groups & Rings               Metric Spaces (calc-16~22)✅    Foundations & Inference       Graphs & Combinatorics
(linalg-15~22)✅                   │                          (prob-1~13) ✅                (disc-01~11)✅
     │                             │                             │                            │
Geometry of Symmetry✅        Functional Analysis            Bayesian & Stochastic         Planar Graphs & Euler
(linalg-24: Dₙ,SO(3),SE(3)) (calc-23~28: FA Block) ✅     (prob-14~21) ✅               (disc-12)✅
     │                             │                             │                            │
     │                       ┌─────┴──────────────┐              │                       Incidence & ∂₁
     │                       │                    │              │                       (disc-13)✅
     │                calc-29: Topological   calc-30: Lp         │                       Simplicial Complexes
     │                Spaces ✅              Spaces ✅            │                       (disc-14)✅
     │                       │                    │              │                            │
     │                       │                    ├──────── prob-22&23: Measure-         Intro to Homology
     │                       │                    │         Theoretic Prob. ✅✅          (disc-15) ✅
     │                       │                    │              │                            │
     │                       │               calc-31:            │                            │
     │                       │               Fourier in          │                            │
     │                       │               Hilbert Spaces      │                            │
     ├───────────────────────┤                    │              │                            │
     │                       │                    │              │                            │
Lie Group Series        calc-XX (~3 pp):          │              │                       disc-XX: Quivers
(linalg-27~30) ✅       Smooth Manifolds          │              │                       disc-XX: Categories
     │            ◀────▶ (Atlases, Tangent         │              │                            │
     │                   Spaces, Vector Fields)    │              │                            │
     │                       │                    │              │                            │
     │                  calc-XX (~2 pp):           │              │                       disc-XX: Discrete
     │                  Riemannian Metrics         │              │                       Ext. Calculus (DEC)
     │                  & Curvature                │              │                            │
     │                       │                    │              │                            │
linalg-XX (~3-4 pp):        │                    │              │                            │
Representation Theory        │                    │              │                            │
     │                       │                    │              │                            │
     ▼                       ▼                    ▼              ▼                            ▼
┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄

                           SECTION V: Application Viewpoints
                           (each ◈ motivates return to deeper foundations)

  ◈ ml-13: Graph Neural Networks ← disc-15, linalg-14 (Graph Laplacian)
       Introduces GDL principle via discrete symmetry; foreshadows manifold series
  ◈ ml-14: Equivariant Neural Networks ← linalg-27~30, linalg-XX (Rep. Theory), ml-13
  ◈ ml-XX: GDL Overview ← calc-XX (Riemannian), linalg-27~30, ml-14
  ◈ ml-XX: CDL Overview ← disc-XX (Category Theory), ml-13
  ◈ Quantum computation topics ← calc-31 (Fourier/Hilbert), calc-27 (Spectral)

Completed Work

Phase 1 — Completed Early (March 2026)

• calc-29 (Topological Spaces)
• calc-30 (Lp Spaces & Riesz-Fischer)
• prob-22 & prob-23 (Measure-Theoretic Probability)

Phase 2a — Lie Group Series (April 2026) ✅

Originally planned as 2 pages (linalg-27 & 28), expanded to 4 pages during drafting:

Lesson learned: Topics requiring new conceptual paradigms (here, the passage from discrete to continuous symmetry) consistently expand beyond initial estimates. This pattern is expected to recur for smooth manifolds, representation theory, and category theory.

Phase 2b: Current & Near-Term Work

calc-31: Fourier Analysis in Hilbert Spaces (1 page)

• Goal: Elevate Fourier analysis from engineering tools (calc-14, calc-15) to pure functional analysis. Capstone page that synthesizes existing machinery rather than introducing new foundations.
• Topics: Plancherel’s theorem (Fourier as unitary operator on L²), Riemann-Lebesgue lemma, Heisenberg’s uncertainty principle as a mathematical theorem.
• Connections: calc-27 (spectral theory) provides the abstract framework; this page instantiates it concretely. Insight Boxes connect to quantum mechanics (unitary evolution, measurement) and signal processing (Nyquist-Shannon as a Hilbert space projection).
• Why 1 page: All heavy prerequisites are done (calc-23 Hilbert spaces, calc-25 Riesz, calc-27 spectral theory, calc-30 Lp completeness, prob-22 DCT). The content is convergence of existing tools, not new foundation-building. Group-theoretic Fourier analysis (Peter-Weyl) is deferred to after representation theory.

ml-13: Graph Neural Networks (1 page)

• Goal: Show how discrete symmetry (permutation invariance) constrains network architecture. Introduces the GDL principle in its simplest form.
• Prereqs: disc-15 (Homology — graph-level features), linalg-14 (Graph Laplacian — spectral convolution).
• Key insight: Message passing = local, permutation-equivariant aggregation. This is the simplest instance of the GDL principle.
• Critical role: This page plants forward references to the manifold series (calc-32+). When the reader asks “what about continuous symmetries? curved spaces?” — the manifold pages answer that question. Must be written with this foreshadowing in mind.

Phase 3: Smooth Manifolds (Summer 2026)

The original plan (1 page for calc-32) was recognized as inadequate during the Lie group expansion. Lee’s Introduction to Smooth Manifolds Ch. 1–3 and Ch. 8 require at minimum 3 pages. Page IDs will be assigned when drafting begins.

calc-XX: Smooth Manifolds & Atlases (~1 page)

• Topics: Topological manifolds, charts (local coordinate systems), atlases, transition maps, smooth structure, smooth maps between manifolds.
• CS angle: A manifold is a data structure that is locally Euclidean — local tensor operations apply, but global consistency requires the atlas machinery.
• Payoff: calc-29’s Hausdorff and second countability conditions are revealed as specifications that exclude pathological spaces.
• Prereqs: calc-29, linalg-27~30.

calc-XX: Tangent Spaces & The Pushforward (~1 page)

• Topics: Tangent vectors (equivalence classes of curves; derivations), tangent space T_pM, differential/pushforward (dF or F_*), Jacobian matrix in local coordinates.
• CS angle: The pushforward of a map between manifolds is the Jacobian of neural network forward propagation. The dual (pullback) corresponds to backpropagation.
• Payoff: T_I G from the Lie group series (linalg-29) is revealed as a special case of the abstract tangent space.
• Prereqs: Previous manifold page.

calc-XX: Vector Fields, Flows & The Tangent Bundle (~1 page)

• Topics: Tangent bundle TM, vector fields, integral curves and flows, Lie bracket of vector fields [X, Y].
• CS angle: ODE solvers on manifolds. Neural ODE and fluid simulation as flows on manifolds.
• Payoff: The matrix commutator [A, B] = AB − BA from linalg-29 is revealed as the matrix representation of the Lie bracket of vector fields — a higher level of abstraction.
• Prereqs: Previous manifold page.

calc-XX: Riemannian Metrics & Beyond (~2 pages, scope TBD)

• Topics: Inner products on tangent spaces, metric tensor g, geodesics, Levi-Civita connection, curvature, Laplace-Beltrami operator Δ_g.
• Prereqs: Manifold series above, calc-25 (Riesz — musical isomorphisms extend to manifolds).
• Note: Likely splits into at least 2 pages (metrics & geodesics vs. curvature & Laplace-Beltrami). Final structure determined during drafting.

Phase 4: Representation Theory & Equivariance (Autumn 2026)

linalg-XX: Representation Theory (~3–4 pages)

• Topics: Group representations, subrepresentations, irreducibility, Maschke’s theorem, Schur’s lemma, character theory (finite groups), Lie group representations and Peter-Weyl.
• Prereqs: linalg-27~30 (Lie group series), linalg-22 (classification of finite abelian groups).
• Connection: Peter-Weyl theorem bridges back to calc-31 (Fourier as a special case of harmonic analysis on groups).
• Note: Page count estimated at 3–4 based on the Lie group experience. Finite group representations alone (Maschke, Schur, character tables) fill at least 2 pages; Lie group representations add 1–2 more.

ml-14: Equivariant Neural Networks

• Goal: Generalize from permutation invariance to continuous group equivariance (SO(3), SE(3)).
• Prereqs: linalg-27~30 (Lie groups), linalg-XX (representation theory), ml-13 (GNN).
• Key insight: “The architecture encodes the symmetry” — once this is clear, the reader naturally asks for a unifying framework, which is GDL.

At this point, readers have seen permutation equivariance (GNN), rotation/translation equivariance (Equivariant NN), and Riemannian structure (NGD). The unifying language is GDL — which becomes not a lesson to teach but a pattern the reader has already experienced.

Phase 5: Categorical Foundations & Viewpoints (Future)

Section IV — Discrete Differential Geometry & Categories

Page IDs deferred. Estimated scope:

• Quivers & Directed Graphs (~1 page) — Prereqs: disc-01, linalg-15.
• Category Theory (~2–3 pages) — Categories, functors, natural transformations, adjunctions. Prereqs: quivers as motivating example, familiarity with algebraic structures from Sections I–IV.
• Discrete Exterior Calculus & Hodge Theory (~1–2 pages) — Discrete differential forms, Hodge star, Hodge decomposition. Prereqs: disc-14 (simplicial complexes), calc-XX (Riemannian metrics).

Section V — Ultimate Viewpoints

• GDL Overview ← calc-XX (Riemannian), linalg-27~30, ml-14
• CDL Overview ← disc-XX (Category Theory), ml-13

Lee Preparation: Gap Analysis for calc-29

What calc-16~22 already cover vs. what Lee’s Introduction to Smooth Manifolds requires:

Filename Registry (Forward Links)

Completed Pages

Planned Pages (ID assigned)

Planned Pages (ID deferred — assigned at drafting time)

Reference Map: Books × Pages

Which books serve which pages. All references are listed on the site-wide index; this map tracks primary usage for development planning.

Currently on index.html (18 books + 3 new additions)

New additions to index.html

Not yet on index.html (acquire when triggered)

Deferred Items (Non-Blocking)

These topics are explicitly deferred — not forgotten, but not on the critical path for 2026.

Key Learnings & Development Principles

1. Notation Consistency is Non-Negotiable:
   • Calligraphic letters for spaces: (\mathcal{X}, \mathcal{Y}, \mathcal{Z}, \mathcal{H}, \mathcal{X}^*, \mathcal{X}^{**}).
   • Functionals as (\varphi); operator norm as (|\varphi|_{\mathcal{X}^*}).
   • All new pages in Section II must match the notation established in calc-23~28.

2. Cross-Page Linking: Links must be verified against actual curriculum filenames and anchor IDs before inclusion.

3. No In-Body Citations: References are handled in a site-wide reference index only.

4. Critical AI Usage: Third-party AI suggestions must be evaluated critically, not accepted wholesale.

5. Zero Tolerance for Proof Gaps: Circular reasoning, incomplete definitions, and missing logical bridges must be explicitly addressed.

6. Forward Links: For unwritten pages, use descriptive text only (no <a href>). Convert to actual links only when the target page is created.

7. Application Viewpoint Philosophy: Use Insight Boxes and the knowledge map’s Tessera to connect abstract theorems to applications without breaking formal proofs in the main text. Application domains (GDL, CDL, quantum) are introduced when tools are ready — not forced prematurely.

8. Fisher Information vs. Hessian Distinction: The real distinction is reparametrization invariance (Čencov’s theorem) vs. loss-dependence and non-guaranteed positive definiteness — not “global vs. local.”

9. Theorem Numbering in LaTeX: When environments share a counter, hardcoded numbers in Appendix/Discussion are fragile — use \Cref{label} throughout.

10. Page Count Estimation: Topics requiring new conceptual paradigms consistently expand to 3–4× initial estimates. Plan for expansion and defer ID assignment until drafting begins. The Lie group series (2 → 4 pages) is the baseline calibration.

Changelog

• 4/5/2026: Major roadmap restructuring.
  • Reflected Lie group series expansion: linalg-27~30 (4 pages) now complete. linalg-27 (Lie Groups), linalg-28 (Matrix Exponential), linalg-29 (Lie Algebras), linalg-30 (Lie Correspondence).
  • Adopted deferred ID convention: unassigned future pages use XX instead of premature numbered IDs.
  • Planned manifold series split: calc-32 (Smooth Manifolds & Tangent Spaces) → calc-XX (~3 pages: Atlases, Tangent Spaces/Pushforward, Vector Fields/Flows). Riemannian metrics estimated at ~2 pages.
  • Representation theory (formerly linalg-29) deferred to linalg-XX (~3–4 pages), after manifold series.
  • Revised implementation order: linalg-27~30 → calc-31 → ml-13 → calc-XX (manifolds) → linalg-XX (rep. theory) → ml-14.
  • ml-13 elevated to pre-manifold position: introduces GDL principle and plants forward references to manifold series.
  • Added “Page Count Estimation” to Key Learnings.
  • Added Variational Inference to Deferred Items.
  • Updated page counts: 110 total (was 102).
• 3/31/2026: Finished prob-22 and prob-23. Updated Schedule & Sprint Plan.
• 3/29/2026: Finished calc-29 and calc-30.
• 3/28/2026: Major roadmap revision following strategic review.
  • Replaced “three pillars / terminal goals” framing with “three application domains as viewpoints” philosophy.
  • Removed QML as a named destination; quantum computation emerges naturally from Fourier/Lp/spectral track.
  • Promoted calc-30 (Lp Spaces) and calc-31 (Fourier in Hilbert Spaces) to core curriculum.
  • Added prob-22 (Measure-Theoretic Probability) to Section III — bridges calc-11/12 (measure theory, Lebesgue) to prob-1~21 (classical probability). Scheduled concurrently with calc-30 in June.
  • Added Section V bridge pages: ml-13 (GNN), ml-14 (Equivariant NN) before GDL overview.
  • Restored Lee Gap Analysis table from March roadmap.
  • Restored and expanded Deferred Items with explicit triggers (added conditional expectation, continuous-time processes).
  • Replaced linear convergence diagram with web-of-mutual-reinforcement dependency map (now includes Section III).
  • Added Filename Registry for forward links.
  • Introduced linalg-27 (Lie Groups) into Phase 2 interleaved with analysis pages.
  • Added Reference Map: full book × page mapping for existing 18 references + 3 new additions (Edelsbrunner & Harer, Fong & Spivak, Leinster) + 2 future acquisitions (Amari, Nielsen & Chuang).
• 3/27/2026: Verified curriculum to confirm 102 completed pages. Marked disc-12~15 complete.
• 3/20/2026: Completed FA block (calc-24~28). Added Lee gap analysis and deferred items.
• 3/03/2026: Consolidated Intro to Functional Analysis into calc-23. Added linalg-25, 26.
• 3/01/2026: Added Section I-25, 26 (Algebraic Extensions, Finite Fields).