self study syllabus - Obsidian Publish

## Introduction This is the background for self-directed learning I'm doing in Mathematics, as well as sharpening my skills in programming and computer science. ## Mathematics ### Basis This framework was inspired by several YouTube videos: 1. [How to self-study pure maths](https://youtu.be/byNaO_zn2fI) by Aleph 0 2. [Arongil math self-study playlist](https://youtube.com/playlist?list=PLapqQU8bF_-8waAcQcTLf0ypp0511qWjX&si=yV-kgE_sXYnUbvyb) 3. [My course recommendations for studying mathematics](https://www.youtube.com/watch?v=nUELlHrJMyM) by "Struggling Grad Student" Additional resources have been included to fill gaps in knowledge and provide a solid foundation. ### Step 1: Basics (DONE) 1. [Algebra Basics](https://www.khanacademy.org/math/algebra-basics) on Khan Academy 2. Introductory Trigonometry 3. Lang, S. (2012). *Basic Mathematics*. Springer. 4. [Precalculus](https://www.khanacademy.org/math/precalculus) on Khan Academy ### Step 2: Foundational Mathematics #### Stats and Probability (Intro) 1. [Stats and Probability](https://www.coursera.org/learn/machine-learning-probability-and-statistics) section of the DeepLearning.AI Mathematics for ML specialization 2. Khan Academy probability and statistics courses 3. [3Blue1Brown Probabilities](https://youtube.com/playlist?list=PLZHQObOWTQDOjmo3Y6ADm0ScWAlEXf-fp) 4. [3Blue1Brown Central Limit Theorem](https://youtube.com/playlist?list=PLZHQObOWTQDOMxJDswBaLu8xBMKxSTvg8) 5. [LibreText Stats Library](https://stats.libretexts.org/) #### Linear Algebra (Intro) 1. [Linear Algebra for Machine Learning and Data Science](https://www.coursera.org/learn/machine-learning-linear-algebra) section of the DeepLearning.AI Math for ML specialization 2. [Linear Algebra](https://www.khanacademy.org/math/linear-algebra) on Khan Academy 3. [Essence of Linear Algebra](https://youtube.com/playlist?list=PLZHQObOWTQDPD3MizzM2xVFitgF8hE_ab) 4. [Linear Algebra for Programmers](https://coffeemug.github.io/spakhm.com/posts/01-lingalg-p1/linalg-p1.html) #### Calculus 1. [Calculus](https://www.coursera.org/learn/machine-learning-calculus) section of the DeepLearning.AI Math for ML specialization 2. [Essence of Calculus](https://youtube.com/playlist?list=PLZHQObOWTQDMsr9K-rj53DwVRMYO3t5Yr) 3. Stewart, J. (2020). *Calculus*. Cengage Learning. - [Accompanying playlist](https://www.youtube.com/watch?v=RpEmexWu_9o&list=PL4MvRz3GM4bjey2id6ougaoxP2RyoOe_V) 4. Khan Academy Calculus sequence: - [Calculus 1](https://www.khanacademy.org/math/calculus-1) - [Differential Calculus](https://www.khanacademy.org/math/differential-calculus) - [Integral Calculus](https://www.khanacademy.org/math/integral-calculus) - [Multivariable Calculus](https://www.khanacademy.org/math/multivariable-calculus) #### Other Topics 1. [College Algebra](https://www.khanacademy.org/math/college-algebra) on Khan Academy 2. Intro to number theory 3. Intro to proofs 4. Complex numbers 5. Additional resources: - [Workbooks video](https://www.youtube.com/watch?v=vuvcOXH4Z5Q) - McMullen, C. (2012). *Trigonometry Essentials Practice Workbook with Answers*. CreateSpace Independent Publishing Platform. - [Paul's Online Math Notes](https://tutorial.math.lamar.edu/) ### Step 3: Differential Equations, Mechanics, and Modeling #### Differential Equations 1. [MIT 18.03: Differential Equations](https://ocw.mit.edu/courses/18-03-differential-equations-spring-2010/) - [playlist](https://www.youtube.com/watch?v=76WdBlGpxVw&list=PL64BDFBDA2AF24F7E) 2. Edwards, C., & Penney, D. (2003). *Elementary Differential Equations with Boundary Value Problems*. Prentice Hall. This book is extraordinarily overpriced for what is a book of average value. Buy it if you need to for a course or if you can get it second hand. There is a Dover book and some books from India that are like 1/5 of the cost or something and are just as good or better. 3. Other resources: - [3Blue1Brown Differential Equations](https://youtube.com/playlist?list=PLZHQObOWTQDNPOjrT6KVlfJuKtYTftqH6) - good for intuition but won't actually teach you how to solve ODEs - [Professor Leonard playlist](https://www.youtube.com/playlist?list=PLDesaqWTN6ESPaHy2QUKVaXNZuQNxkYQ_) - good if you want to go slow with lots of explanation but enough repetition for me to find it infuriating at times - [Trefor Bazett playlist](https://www.youtube.com/watch?v=B5IjsTONKkw&list=PLHXZ9OQGMqxde-SlgmWlCmNHroIWtujBw) - Medium pace - [Steve Brunton engineering math playlist](https://youtube.com/playlist?list=PLMrJAkhIeNNTYaOnVI3QpH7jgULnAmvPA&si=RksW4jqergBN6fav) - exceptionally clear and covers a lot of useful material and includes a lot of modelling in both python and matlab - [Michael Penn playlist](https://www.youtube.com/watch?v=30CVhA6FV_I&list=PL22w63XsKjqxWUzDlkrbtifIx-XbKYnge) - ok great. That's a good place to stop. - [Khan Academy Differential Equations](https://www.khanacademy.org/math/differential-equations) - McMullen, C. (2022) *Differential Equations. Essential Skills Practice Workbook with Answers.* Zishka Publishing. #### Mechanics 1. [MIT 8.012: Classical Mechanics](https://ocw.mit.edu/courses/8-012-physics-i-classical-mechanics-fall-2008/) - [playlist](https://www.youtube.com/watch?v=F3N5EkMX_ks&list=PLUl4u3cNGP61qDex7XslwNJ-xxxEFzMNV) 2. Kleppner, D., & Kolenkow, R. (1973). *An Introduction to Mechanics*. McGraw-Hill. #### Modeling 1. [Maxima by Example](https://home.csulb.edu/~woollett/mbe.html) 2. Torrence, B., & Torrence, E. (2019). *A Student's Introduction to Mathematica and the Wolfram Language*. Cambridge University Press. #### Multivariable Calculus and Vector Calculus 1. Stewart, J. *Calculus* 2. Additional Resources 1. Michael Penn playlists 1. [vector-valued functions](https://www.youtube.com/watch?v=ML7v1HvBxiM&list=PL22w63XsKjqxHk45H_ZDYVUY4XeZv7ZtK&pp=iAQB) 2. [multiple integrals](https://www.youtube.com/watch?v=YHtzHVOIypE&list=PL22w63XsKjqz033oE59Vwc1lOIg_detQj&pp=iAQB) 3. [multivariable functions](https://www.youtube.com/watch?v=1OtTUdo5_4Q&list=PL22w63XsKjqyurOw5_v_xFu7XHNtmh7x1&pp=iAQB) 4. [vectors for multivariable calculus](https://www.youtube.com/watch?v=93vCDrPj_QE&list=PL22w63XsKjqyC3bWVd5EjGEVN9pqq55yF) 5. [multivariable calculus](https://www.youtube.com/watch?v=93vCDrPj_QE&list=PL22w63XsKjqz4R-2yzZDbxRuioaOjGml3&pp=iAQB) ### Step 4: Pure Mathematics #### Advanced Calculus/Analysis 1. Spivak, M. (2008). *Calculus*. Publish or Perish. - [Study program](http://alpha.math.uga.edu/%7Epete/MATH2400F11.html) 2. [The Matrix Calculus You Need for Deep Learning](https://explained.ai/matrix-calculus/) 3. Alcock, L. (2017). *How to Think About Analysis*. Oxford University Press. 4. Cummings, J. (2023). *Real Analysis: A Long-Form Mathematics Textbook*. LongFormMath. 5. Abbott, S. (2015). *Understanding Analysis*. Springer and this set of [lectures by Francis Su](https://www.youtube.com/playlist?list=PL0E754696F72137EC) 6. [MIT 18.100a: Real Analysis](https://ocw.mit.edu/courses/18-100a-real-analysis-fall-2020/pages/syllabus/) [playlist](https://www.youtube.com/playlist?list=PLUl4u3cNGP61O7HkcF7UImpM0cR_L2gSw) 7. Lebl, J. (2021). *Basic Analysis: Introduction to Real Analysis*. CreateSpace Independent Publishing Platform. 8. Additional resources 1. Michael Penn real analysis [playlist](https://www.youtube.com/watch?v=L-XLcmHwoh0&list=PL22w63XsKjqxqaF-Q7MSyeSG1W1_xaQoS&pp=iAQB) #### Linear Algebra 1. Treil, S. (2017). *Linear Algebra Done Wrong*. [Online Book](https://www.math.brown.edu/streil/papers/LADW/LADW.html) 2. [MIT 18.06SC: Linear Algebra](https://ocw.mit.edu/courses/18-06sc-linear-algebra-fall-2011/) 3. Axler, S. (2015). *Linear Algebra Done Right*. Springer. 4. [MIT 18.700: Linear Algebra](https://ocw.mit.edu/courses/18-700-linear-algebra-fall-2013/) #### Abstract Algebra 1. Intro 1. Pinter, C. C. (2010). *A Book of Abstract Algebra*, Dover Books. 2. . [MIT 18.703: Modern Algebra](https://ocw.mit.edu/courses/18-703-modern-algebra-spring-2013/) Group theory Lecture [playlist](https://www.youtube.com/playlist?list=PLelIK3uylPMGzHBuR3hLMHrYfMqWWsmx5) by Benedict Gross (not actually for the MIT course but similar syllabus, following a Michael Artin Algebra book) 3. Fraleigh 4. Hien, Marco 5. Reference: 1. Lang, S. *Algebra*, Springer. 2. Herstein, I. N. *Topics in Algebra* 3. Dummit, D. S., Foote, R. M. *Abstract Algebra* 6. Additional resources: 1. Arongil Self-study [advice](https://www.youtube.com/watch?v=2ihiR8hJlMs&list=PLapqQU8bF_-8waAcQcTLf0ypp0511qWjX&index=5&t=153s) for Abstract Algebra 2. Michael Penn [playlist](https://www.youtube.com/watch?v=m4yYeTGe-ic&list=PL22w63XsKjqwN7sHsEiy0yqkcjQfXAuVb&pp=iAQB) on the basics of groups and [this one](https://www.youtube.com/watch?v=nKxl9Qtfjsk&list=PL22w63XsKjqw3ZFydd_LqOCUBtWYg5svI&pp=iAQB) on group theory exercises 2. Socratica [playlist](https://www.youtube.com/playlist?list=PLi01XoE8jYoi3SgnnGorR_XOW3IcK-TP6) 3. Ben1994 [playlist](https://www.youtube.com/playlist?list=PLAvgI3H-gclb_Xy7eTIXkkKt3KlV6gk9_) 3. Herstein, I. N. (1975). *Topics in Algebra*. John Wiley and Sons. 4. McLarty, C. (2003). _Elementary Categories, Elementary Toposes_. OUP. 5. van der Waerden, B. L. (1930), _Moderne Algebra_ (_Algebra I_ and _Algebra II_ in English editions). Springer #### Other Topics 1. Cummings, J. (2021). *Proofs: A Long-Form Mathematics Textbook*. LongFormMath. 2. [Feynman Lectures on Physics: Chapter 22](https://www.feynmanlectures.caltech.edu/I_22.html) 3. Halmos, P. *Naive Set Theory*. Springer. ### Step 5: More Advanced Mathematics #### PDEs 1. [MIT 18.152: Introduction to Partial Differential Equations](https://ocw.mit.edu/courses/18-152-introduction-to-partial-differential-equations-fall-2011/) 2. Constanda, C. _Solution Techniques for Elementary Partial Differential Equations_. CRC Press 3. Haberman, R. (2012) *Applied Partial Differential Equations with Fourier Series and Boundary Value Problems*. Pearson. 4. Salsa, S., Verzini, G. (2010). *Partial Differential Equations in Action: From Modelling to Theory*. Springer. 5. [Vector Calculus and PDEs playlist](https://www.youtube.com/watch?v=Jt5R-Tm8cV8&list=PLMrJAkhIeNNQromC4WswpU1krLOq5Ro6S) 6. Pivato, M. (2010). *Linear Partial Differential Equations and Fourier Theory*. Cambridge University Press. 7. Evans, L. C. (2010) *Partial Differential Equations*. American Mathematical Society. 8. [Commutant playlist](https://www.youtube.com/playlist?list=PLF6061160B55B0203) List of book resources by [Mathematical Toolbox](https://www.youtube.com/watch?v=0sIzAZLtfKc) He recommends 1. for problem solving- Wazwaz, A. *Partial Differential Equations and Solitary Waves Theory*. Springer 2. for theory- Pivato above #### Mathematical Statistics/Inference 1. [Mathematical Statistics playlist](https://www.youtube.com/playlist?list=PLLyj1Zd4UWrPZH-fknPLak0tlUpUISBZR) 2. Wasserman, L. (2004). *All of Statistics: A Concise Course in Statistical Inference*. Springer. 3. Casella, G., & Berger, R. L. (2002). *Statistical Inference*. Duxbury. 4. Murphy, K. P. (2022). *Probabilistic Machine Learning: An Introduction*. MIT Press. #### Complex Analysis 1. Wegert, E. (2012) *Visual Complex Functions: An Introduction with Phase Portraits*, Springer 2. Stein, E. M., & Shakarchi, R. (2003). *Complex Analysis*. Princeton University Press. #### Stochastic Calculus and Financial Maths ##### Probability prerequisites 1. Blitzstein, J. K., & Hwang, J. (2019). *Introduction to Probability* (2nd ed.). Chapman & Hall/CRC. 2. Ash, C. (1993). *The Probability Tutoring Book: An Intuitive Course for Engineers and Scientists (and Everyone Else!)*. IEEE Press. 3. Song, I., Park, S. R., & Yoon, S. (2022). *Probability and Random Variables: Theory and Applications*. Springer. (1 or 2, then 3) ##### Stochastic Differential Equations 1. Calin, O. (2015). *An Informal Introduction to Stochastic Calculus with Applications*. World Scientific. 2. Evans, L. C. (2013). _An Introduction to Stochastic Differential Equations_. American Mathematical Society 3. Klebaner, F. C. (2012). *Introduction to Stochastic Calculus with Applications* (3rd ed.). Imperial College Press. 4. Albin, P., Hamza, K., & Klebaner, F. C. (2024). *Problems and Solutions in Stochastic Calculus with Applications*. World Scientific. 1 or 2, then 3 or 4. ##### Financial Maths 1. Hirsa, A., Neftci, S. (2013). *An Introduction to the Mathematics of Financial Derivatives*. Academic Press 2. Saari, D. (2019). *Mathematics of Finance: An Intuitive Introduction*. Springer 3. Stefanica, D. (2011). *A Primer for the Mathematics of Financial Engineering*. FE Press 4. Shreve, S. E. (2004). *Stochastic Calculus for Finance II: Continuous-Time Models*. Springer. 5. Chin, E., Nel, D., & Ólafsson, S. (2014). *Problems and Solutions in Mathematical Finance: Stochastic Calculus* (Vol. 1). Wiley. ## Other Mathematical Topics (Review once the background is in place) 1. [How to self-study pure maths](https://youtu.be/byNaO_zn2fI) 2. Discrete Maths 1. Graham, R. L., Knuth, D. E., & Patashnik, O. (1994). *Concrete Mathematics: A Foundation for Computer Science*. Addison-Wesley. 2. Lovász, L. (2005). [Discrete Mathematics lecture notes](https://cims.nyu.edu/~regev/teaching/discrete_math_fall_2005/dmbook.pdf) 3. Point Set Topology 1. [MAT327 Course Notes](http://www.math.toronto.edu/ivan/mat3...) 4. Differential Geometry 1. TODO- add some resources 6. Number Theory 1. Maybe Andrews, G. E. (1994) *Number Theory*. Dover. 2. [The Erdős Problems Repository](https://www.erdosproblems.com/) 7. Fractals 1. [Santa Fe Institute MOOC: Fractals and Scaling](https://www.complexityexplorer.org/courses/169-fractals-and-scaling-2023) 2. [Yale Fractal Resources](https://users.math.yale.edu/public_html/People/frame/Fractals/) ## Computer Science/Software Engineering 1. Expanding programming skills - Python data science and visualization - Rust (see separate learning resources) - TypeScript (on hold) - Practical ML - Mathematica (for math self-study aid) 2. [The Missing Semester of Your CS Education (MIT)](https://missing.csail.mit.edu/) 3. AI and Machine Learning (separate syllabus to be curated)