The Compass Philosophy - Foundations & Mission
MATH-CS COMPASS is designed to bridge the gap between the rigorous logic of pure mathematics and the practical innovation of computer science.
In the modern view, mathematics is the study of structure. While pure mathematicians often prioritize formal rigor and computer scientists tend to focus on experimental results, I believe that understanding the underlying structure of a system enriches both perspectives. MATH-CS COMPASS offers an accessible yet principled look at the mathematical blueprints that power modern technology.
The Two Worlds of the Compass
Your journey here is anchored by Linear Algebra to Algebraic Foundations (Section I), which then guide you through two distinct conceptual domains:
- The Discrete World — The realm of countable structures and logical reasoning. Discrete Mathematics & Algorithms (Section IV) explores graph theory, combinatorics, and the theory of computation — from analyzing algorithmic complexity to understanding simplicial complexes and homology. Here, problems are solved through precise, step-by-step logic.
- The Continuous World — The realm of smooth change and infinite processes. Calculus to Optimization & Analysis (Section II) covers optimization, functional analysis, measure theory, and topology — providing the analytical machinery for understanding the spaces where modern algorithms live and converge.
The bridge between these realms is Probability & Statistics (Section III), where discrete counting meets continuous distributions, translating structure into the language of uncertainty.
Machine Learning (Section V) is where these foundations meet real-world application. But rather than a final destination, each topic in Section V serves as a viewpoint — a vantage point that reveals how mathematical tools come together, and simultaneously motivates the pursuit of deeper foundations. The curriculum targets three broad application domains: Geometric Deep Learning (symmetry and manifolds), Categorical Deep Learning (compositionality and abstraction), and Quantum Computation (Hilbert spaces and unitary evolution). Each emerges naturally when the right mathematical tools are in hand.
Linear Algebra to Algebraic Foundations
━━━━━━━━━
The Core"] %% Discrete World Group subgraph discrete["The Discrete World"] S4["Section IV
Discrete Mathematics
& Algorithms
━━━━━━━━━
Logic & Structure"] end %% The Bridge S3["Section III
Probability & Statistics
━━━━━━━━━
The Bridge"] %% Continuous World Group subgraph continuous["The Continuous World"] S2["Section II
Calculus to Optimization
& Analysis
━━━━━━━━━
Spaces & Convergence"] end %% Application Viewpoints S5["Section V
Machine Learning
━━━━━━━━━
Application Viewpoints"] %% Forward connections with labels S1 -->|structure| S4 S1 -->|foundation| S2 S1 -->|algebra| S3 S4 -->|algorithms| S5 S2 -->|optimization| S5 S3 -->|uncertainty| S5 %% Cross-bridges S4 -.->|combinatorics| S3 S2 -.->|measure| S3 %% Feedback loop: viewpoints motivate deeper foundations S5 -.->|deeper math| S1 %% Core styling style S1 fill:#1565c0,stroke:#0d47a1,color:#fff,stroke-width:3px,rx:10 style S3 fill:#00838f,stroke:#006064,color:#fff,stroke-width:3px,rx:10 style S5 fill:#ef6c00,stroke:#e65100,color:#fff,stroke-width:3px,rx:10 style S2 fill:#2e7d32,stroke:#1b5e20,color:#fff,stroke-width:2px,rx:8 style S4 fill:#6a1b9a,stroke:#4a148c,color:#fff,stroke-width:2px,rx:8 %% Subgraph styles style discrete fill:#f3e5f533,stroke:#6a1b9a,stroke-width:2px,rx:10 style continuous fill:#e8f5e933,stroke:#2e7d32,stroke-width:2px,rx:10 %% Links click S1 "Mathematics/Linear_algebra/linear_algebra.html" "Go to Section I" click S2 "Mathematics/Calculus/calculus.html" "Go to Section II" click S3 "Mathematics/Probability/probability.html" "Go to Section III" click S4 "Mathematics/Discrete/discrete_math.html" "Go to Section IV" click S5 "Mathematics/Machine_learning/ml.html" "Go to Section V"
Specific tools, libraries, and frameworks may become outdated quickly, but the mathematical foundations are eternal. My goal is to equip you with the insights necessary to not only use existing technology as a "black box" but to understand, adapt, and even create new approaches as the field progresses.
If you have any suggestions, requests, or inquiries, please feel free to use the contact form below. Your input is invaluable in helping improve this site for everyone.
I would like to express my deep appreciation to the students and faculty of the Western Washington University Mathematics and Computer Science Departments. Their passion and inspiration were the primary motivation for creating this platform following my graduation.
Thank you for visiting, and enjoy your exploration!
(*Please note that the content on this site is a continuous work in progress. More resources are frequently added as the Compass expands.)
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Interactive Knowledge Map
Navigate all topics visually with prerequisites and guided exploration.
✨ RecommendedI - Linear Algebra to Algebraic Foundations
Explore foundations of modern mathematics & computer science.
II - Calculus to Optimization & Analysis
Explore optimization techniques and mathematical analysis.
III - Probability & Statistics
Explore probability theory and statistical methods.
IV - Discrete Mathematics & Algorithms
Explore graph theory, combinatorics, the theory of computation, and algorithms.
V - Machine Learning
Explore machine learning ideas. (Most of the mathematical topics for ML will be covered by Section I - IV.)