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Whilst this question can potentially be considered off topic I still believe that it is relevant and can benefit the site as a whole.

I've recently completed all of the mathematics courses in my school and still have a keen thirst for more knowledge as I really enjoy mathematics. I'm looking to pursue a dual degree of mathematics and computer science as an undergraduate degree for university, with a large focus on machine learning and it's applications. Whilst not entirely sure I think i'm leaning towards majoring in applied mathematics.

The topics that I've completed at school are:

Arithmetic and geometric sequences, functions and graphs, counting and probability, logarithmic and trigonometric functions

Calculus - Differentiation of continuous functions using the chain, product and quotient rule, Basic integration using U - substitutions. We've also covered basic applications in a 2 dimensional sense such as optimizing variables such as time.

Probability - Normal distributions, probability density functions, binomial distributions, sampling and estimation, confidence intervals.

Obviously my undergrad degree will be quite math intensive, but there is about 6 or so months until that starts and I don't think I can go that long without learning any new topics.

So, my question to all experienced mathematicians out there is: What are some math topics I should learn in preparation for studying mathematics and computer science at university?

Many thanks in advance.

LogLogLogLogx
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    You will need linear algebra and multivariable calculus for machine learning. Besides that, do you have a solid foundation in reasoning and proof writing? That's probably the main difference between high school maths and university maths. – Taladris Sep 01 '21 at 12:15
  • @Taladris My own curiosity has caused me to briefly explore methods such as proof by induction and direct proof – LogLogLogLogx Sep 01 '21 at 12:18
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    Math is a loooot of proving so "briefly" is good to start with, but probably there is a lot more to learn :D – LegNaiB Sep 01 '21 at 12:20
  • @LegNaiB thank you for your advice! Proofs and reasoning have always really interested me so that's good to here i'll be doing them in the future. – LogLogLogLogx Sep 01 '21 at 12:24
  • Machine Learning is probably more applied than other parts of math - so you will also calculate something. But proving is THE thing math is based on – LegNaiB Sep 01 '21 at 12:26
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    Also important are new definitions and what they really mean. It's not like with a foreign language really, but one could phrase your question a bit like "best grammar and language courses to be prepared for the big literature in this new language". Well, take some basic grammar (math) book and start doing it! Fundamentals are calculus and linear algebra (used as often in mathematics as the English words "and" , "or", "because" in English literature....). – Dietrich Burde Sep 01 '21 at 12:41
  • In university you must supply your own motivation to work. For example, in high school the teacher takes attendance and makes you pay attention. In university it's more like "You chose to be here. It's up to you." – DanielWainfleet Sep 01 '21 at 13:10
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    I've had a look at some of the questions you've asked to judge roughly where you are now. Also, something suggests to me that you're in the U.K. or Australia, where your first-year courses in math will be at a higher level than they would typically be in the U.S. I would suggest that you would benefit from: (1) reading an "introduction to proofs" book like Journey Into Mathematics by Rotman, (2) working through problem books (with solutions) on elementary material, where the problems might be a bit harder than the ones you've done. I'd suggest Problems in Calculus of One Variable by – Mike Sep 01 '21 at 20:15
  • Maron, and Problems in Elementary Mathematics by Lidsky. The problems in this last book are a bit tough. All the books I've mentioned can be found on Libgen. Have a look at the preface and table of contents of the book by Rotman to see what it's about. – Mike Sep 01 '21 at 20:16
  • They can also be found on archive.org, though to view the book by Rotman, you'll need to set up a free account. The answer book to Rotman is on Libgen. – Mike Sep 01 '21 at 20:23
  • @Mike Wow, thank you. I'll be sure to have a look at all of those – LogLogLogLogx Sep 01 '21 at 23:47
  • Does this answer your question? How do I self-learn undergraduate math? – user1147844 May 30 '23 at 07:22

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I think not the topic itself but the thinking in university is more important to learn. Math at school is completely different than math at university. You're proving so many things, in school you only apply some formulas.

However, I really like Algebra and I think that this is the best point to start with. You learn basic concepts (groups, rings, fields) and getting used to math notation, proofs and the mathematical thinking which is necessary to get a degree in maths.

For Machine Learning especially Analysis is of greater importance, but IMO this topic is harder to understand for a beginner than Algebra is. One reason is that in Algebra you can sometimes draw, but in Analysis you often work with infinity or infinitely small things which is not easy to handle (at first). I would really concentrate on getting this mathematical meta-skills how to proof things to prepare for university.

It depends on your interests. If you want to visit some lectures, you should have a look at MIT. All courses are publicly available and have high quality.

LegNaiB
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