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 the mathematical foundations of Cryptography. 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 limits, derivatives, integrals, and the theory of convergence—providing the mathematical machinery for optimization algorithms and the analytical foundations essential for Machine Learning.
The bridge between these realms is Probability & Statistics (Section III), where discrete counting meets continuous distributions, translating structure into the language of uncertainty. All paths finally converge in Machine Learning (Section V), the grand synthesis of these disciplines.
Linear Algebra to Algebraic Foundations
━━━━━━━━━
The Core"] %% Discrete World Group subgraph discrete["The Discrete World"] S4["Section IV
Discrete Mathematics
& Algorithms
━━━━━━━━━
Logic & Complexity"] end %% The Bridge (intersection) S3["Section III
Probability & Statistics
━━━━━━━━━
The Bridge"] %% Continuous World Group subgraph continuous["The Continuous World"] S2["Section II
Calculus to Optimization
& Analysis
━━━━━━━━━
Change & Approximation"] end %% Convergence Point S5["Section V
Machine Learning
━━━━━━━━━
The Synthesis"] %% Connections S1 -->|structure| S4 S1 -->|foundation| S2 S1 -->|algebra| S3 S4 -->|algorithms| S5 S2 -->|optimization| S5 S3 -->|uncertainty| S5 S4 -.->|combinatorics| S3 S2 -.->|measure| S3 %% 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.)
Update Log
Explore Topics
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.)
References
Books
- Boyd, Stephen, and Lieven Vandenberghe. Convex Optimization. Cambridge University Press, 2004.
- Bronstein, Michael M., Joan Bruna, Taco Cohen, and Petar Veličković. Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges. arXiv preprint, 2021.
- Cormen, Thomas H., et al. Introduction to Algorithms. 4th ed. MIT Press, 2022.
- David C. Lay. Linear Algebra and Its Applications. 4th ed., Pearson Education, Inc., 2012.
- Diestel, Reinhard. Graph Theory. 5th ed., Springer, 2017.
- Durrett, Rick. Probability: Theory and Examples. 5th ed. Cambridge University Press, 2019.
- Gallian, Joseph A. Contemporary Abstract Algebra. 11th ed. Chapman and Hall/CRC, 2025.
- Horn, Roger A., and Charles R. Johnson. Matrix Analysis. 2nd ed. Cambridge University Press, 2013.
- Lee, John M. Introduction to Smooth Manifolds. 2nd ed. Graduate Texts in Mathematics, Springer, 2012.
- Menezes, Alfred J., Paul C. van Oorschot, and Scott A. Vanstone. Handbook of Applied Cryptography. CRC Press, 1997.
- Merris, Russell. Combinatorics. 2nd ed., Wiley, 2003.
- Murphy, Kevin P. Probabilistic Machine Learning: An Introduction. The MIT Press, 2022.
- Murphy, Kevin P. Probabilistic Machine Learning: Advanced Topics. The MIT Press, 2023.
- O'Searcoid, Mícheál. Metric Spaces. Springer Undergraduate Mathematics Series, Springer, 2006.
- Sipser, Michael. Introduction to the Theory of Computation. 3rd ed., Cengage Learning, 2013.
- Stein, Elias M., and Rami Shakarchi. Fourier Analysis: An Introduction. Princeton University Press, 2003.
- Stillwell, John. Naive Lie Theory. Undergraduate Texts in Mathematics, Springer, 2008.
Online Courses
- Automata, Computability, and Complexity in MIT Open Course
- Fundamentals of Probability in MIT Open Course
- Graph Theory and Additive Combinatorics in MIT Open Course
- Matrix Calculus for Machine Learning and Beyond in MIT Open Course
- Probabilistic Methods in Combinatorics in MIT Open Course
- Stanford Engineering Everywhere (SEE), Convex Optimization
- CSE446 Machine Learning in UW
- CSE447 Natural Language Processing in UW