Math (Self-Study)
Journey

Started in Algebra 1, ended in Algebraic Topology!

Overview


When I didn't have the resources to continue my love for math as a low-income student, I tried an unconventional route of learning that quickly took off. Starting in the second half of 8th grade, I mastered algebra 1 all the way through differential calculus in just 6 months. Then I exhausted community college courses (multivariable calculus, linear algebra and differential equations). There was a huge hurdle after this, but I clung on to learning complex analysis by auditing classes in 9th grade. Auditing also became unrealistic, so I began self-studying smooth manifolds (while catching up on real analysis, algebra, and other topics).
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Luckily in COVID summer, I was mentored by Ryan Thorngren to study Yukio Matsumoto's Morse Theory book. The next semester, I got lucky again and talked to Yair Minsky at Yale, leading to me remotely auditing graduate level Algebraic Topology at Yale.

This whole experience taught me so much more than math—I learned about learning, community and giving back, resilience and forging a path for myself despite all of the things in my way, and so much more. I'm incredibly grateful for all of the amazing mentors and friends who supported me this whole time.
  • Timeline
  • Jan 2011 - Dec 2020
  • Skills
  • Learning
    Math
    Resillience
    ...
  • Team
  • So many people
    that inspired
    and helped me
    along the way!
  • Tools
  • Books + Notes
    Khan Academy
    Yale Math 544
    Cal Math 185
    BCC & CCC

Context

Coming from a first-generation low-income background, I always wanted to go to academic summer camps like my friends but couldn't afford the prices. I mainly learned high school math on Khan Academy (so good), and supplemented with problems from textbooks and online. After exhausting community college classes, I had to take classes at UC Berkeley, which was already impractical schedule wise but the $5000/class price tag was not an option.

How could I further my love for math without the resources to pay for tutors, college classes, or summer camps?

Process

Initial Takeoff

It was an ordinary day in 8th grade, when the second semester just started. I saw my older sister struggling with quadratics and my dad struggling to help her, I had a thought of "what if I try to learn this and see how it goes?". My parents were laughing at me at first, as their perception of education was limited to a classroom, but I had so much free time I didn't care and want to show them I can do it.
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What started as an initial "I'll prove you wrong" quickly turned into an obsession for learning. In 4 months, I finished everything from Algebra 1 to Precalculus. I was acing the problems on Khan Academy, but people were doubting my mastery so I supplemented it with textbook problems and home works online. I was shocked when I took a placement test at community college and got placed into Calculus—I knew then my hard work paid off and everything was real.

Community College (CC)

In the summer of 8th grade, I took Calculus 1 with Christopher Nagel. His lectures were so incredible he even inspired me to become a teacher and tutor. The community he fostered was also incredible it made me feel accepted despite being young, and this made me fall in love with math more. I continued with Calculus 2, and became a math tutor for 4 years!

I took multivariable calculus 9th grade spring with Salvador Garcia, who introduced me to the beauty of proofs and higher level math in a rather unconventional class. I asked him what I should do after finishing CC classes, he encouraged me to continue learning whatever I want and handed me some books. This is when I started learning whatever was interesting (complex analysis at the time), and just picked everything else up as I went (real analysis, point set topology, algebra and etc). Thanks to Professor Garcia, that summer I audited math 185: Complex Analysis at UC Berkeley.

Self Study

After auditing the summer class at UC Berkeley, I realized this won't be realistic as school picks up and I find other engagements. Instead of giving up, I decided to go back to the un-engaging textbooks. This became extremely difficult as I would get stuck on problems for days at times without help, making it difficult to progress. Nonetheless, I started collecting a collection of math book pdfs, and clung on to learning whatever I found interesting. I was obsessed with Lee's smooth manifold book, but my knowledge wasn't there, so I went back to the fundamentals and moved slowly.

Morse Theory and
Algebraic Topology

When COVID hit, I saw an opportunity to continue my love for math (the textbook workflow was getting pretty unbearable)—I could reach out to people and audit classes on Zoom!

I was lucky to connect with Ryan Thorngren, a postdoc at Harvard at the time, through a co-worker. He was incredibly patient and mentored me as we worked through Yukio Matsumoto's Morse Theory book. Seeing Morse theory come together ignited my love for math again, and I absorbed everything like a sponge. As the summer came to an end and I was looking into colleges, I emailed Yale's math department about their toplogy offerings. To my surprise, Yair Minsky responded and we chatted over Zoom about math. I was excited about algebraic topology having gotten a taste of it in Morse Theory, so I asked and he said I could audit the class at Yale if I wanted to. It was an incredible class and the professor helping me as if I was an official student (thanks Subhadip)!

Reflections & Takeaways

Wow! What an incredible journey. My 8th grade self would have never seen how this unfolded. I learned so much along the way it'd take pages for me to fully flush out everything, but thanks for hearing my story and making this far!

Thanks for stopping by!

Ryan Lian ©2023

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