from Hacker News

What are some interesting areas of math to study beyond calculus?

by chillpenguin on 5/20/23, 6:57 PM with 16 comments

  - Calculus
    - Differential, Integral, Sequences and Series, Vector (partial differential equations, gradients, Lagrange multipliers)
  - Statistics and Probability basics
    - normal distributions, hypothesis testing ("rejecting null hypothesis"), combinations/permutations
  - Linear Algebra
    - matrices, vector spaces
  - Abstract algebra concepts from my studies in Functional Programming
    - monads, monoids, higher order functions
  - Discrete Structures and other CS stuff
    - proof theory, set theory basics, logic, boolean algebra, graph theory
  - Computer Science
    - Deterministic Finite Automata, Grammars, Turing Machines (and Chomsky hierarchy of languages)
    - Fractals, Cellular Automata
    - Algorithms and analysis of algorithms
    - alternative number system: base 2 number system (binary)

• by ggr2342 on 5/23/23, 1:48 PM

  You many want to study how to make educated guesses. This are sometimes called back of the napkin calculations or Fermi estimation problems.

  There is one course on MIT Opencourseware that teaches it: https://ocw.mit.edu/courses/18-098-street-fighting-mathemati...

  The book used in the course is also very good. The author is the same as the course instructor Prof Sanjoy Mahajan.

• by hnthrowaway0315 on 5/20/23, 11:14 PM

  Number theory can hook you up for many lives. I believe Discrete Mathematics in CS program touches the surface, but you can always audit an elementary Number Theory and see if you enjoy it. Once done you can just go from there.

  Mathematical Logic also looks interesting, especially from a historical perspective.

  Please also check out "Gödel, Escher and Bach" if you haven't done so.

  Just a note that studying Mathematics, especially tough topics needs a lot of discipline, time and energy, so be prepared mentally and physically. I don't think it's something one can do one hour daily and accumulate. Without many hours of study the more difficult topics just don't stick.

• by AprilPhoenix on 5/21/23, 9:18 AM

  One thing I'd recommend is tying in the calculus you've learned with the language of differential forms. It really gives you a lot more power in visualizing what is going on (line integral-you aren't just projecting that dot product in the integral onto an "infinitesimally small element", it is a map that does something simple but profound), and also extends your reach with visualization.

  Also, if you are interested in more sophisticated ways of capturing quantum physics, geometric algebra is really cool.

• by SOTGO on 5/20/23, 7:35 PM

  I just finished my math undergraduate program, and the class that I enjoyed most that you don't seem to have studied at all is topology. We used Munkres, which seems to be pretty standard as far as undergrad topology is concerned, and if you're interested in getting into some more advanced pure math topics topology is super useful. I also found that learning the topological definitions of common properties like continuity to be very insightful and often much more natural than the definitions usually given in an analysis class.

• by jay-c on 5/20/23, 11:38 PM

  Most CS undergraduate mathematics stops just short of proof-based courses, which is a shame.

  A proof-based linear algebra course would be the next course that is typical in the progression. If you've already done that, then an introductory real analysis course.

  If you want something a bit more practical then maybe a probabilistic modeling course? ( think Gaussian mixture models, bayesian networks, plate models).

• by AtlasBarfed on 5/21/23, 5:35 AM

  Computer Graphics can be a nice integration of several aspects of math (linear algebra + algorithms) with some rewarding visual feedback.

• by beyondCritics on 5/21/23, 5:40 AM

  In general i would ask the pro's, e.g.: https://duckduckgo.com/?q=Mathematics+Reading+List+Cambridge

  My personal recommendation would be to cover functional programming from the ground up. I consider the benefits of learning this as enormous for me.

• by deterministic on 5/22/23, 4:20 AM

  If you are a software developer then this is the #1 choice:

  https://softwarefoundations.cis.upenn.edu/

  It will not only make you a better software developer (by improving how you think) it will also show you how to prove software correct. And it is fun!

• by euix on 5/20/23, 7:11 PM

  Have you ever looked Ulam spiral? I find a lot of ideas in number theory to be magical - I would never have the talent for it professionally, but I know enough to appreciate the discoveries when they are explained to me.

  https://en.wikipedia.org/wiki/Ulam_spiral

• by rahimnathwani on 5/22/23, 4:58 AM

  Maybe go deeper into Linear Algebra using Axler?

  See page 15 of this PDF for his treatment of eigenvectors: https://linear.axler.net/Eigenvalues.pdf

• by bordercases on 5/23/23, 6:12 PM

  I find that anchoring my mathematics to something in the world motivates me more to learn math than math for its own sake. In this case it's quantum computing. Now I love spinors.

• by jstx1 on 5/21/23, 11:03 AM

  Game theory is reasonably interesting with fairly straightforward math.