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

12 points by chillpenguin ↗ HN
I am a typical computer science person, having studied the typical math topics an undergraduate would study. The topic area I spent the most time on (besides computer science) would be calculus. I have been out of college for a few years, but I find myself missing mathematics. What are some topics/areas of math that I didn't know I needed in my life? Doesn't have to be practical, could be purely for intellectual interest. But practical/useful topics would be welcome too of course.

Here are some things I have already studied:

  - 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)
Recommend me something new!

16 comments

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euix
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

SOTGO
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.
chillpenguin
You are right, I haven't studied topology at all. I have heard the word before, but I don't know what it is. I'll look into it, thanks!
hnthrowaway0315
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.

jay-c
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).

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

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

The linked pages from that DDG search are mostly the reading list for BEFORE you attend and the reading list per class is not found like this.

On the other hand you can go directly to video lectures a MIT open courseware, and others, and most recently a great resource for pure mathematics is:

https://courses.maths.ox.ac.uk/course/view.php?id=1051

wherein the courses are split by the old term names/tripos:

Michaelmas Hilary Trinity

https://courses.maths.ox.ac.uk/course/index.php

AprilPhoenix
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.

jstx1
Game theory is reasonably interesting with fairly straightforward math.
ggr2342
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.

bordercases
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.