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What kind of topics do you prefer to learn?

11 Answers

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I, personally, would NOT recommend learning a new topic over the summer unless you have something very specific in mind. I spent my summers over undergrad doing problems from classes I have already taken (especially doing the harder ones). I think it's far more interesting to deepen your understanding of stuff you already know. It's also largely beneficial to strengthen your problem-solving abilities by solving harder problems.

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Like you, I am also interested in graph theory (and more generally combinatorics). Here are some texts that have some great problems that you can spend some time on.

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1) 'A Course in Combinatorics' by van Lint and Wilson. I'd recommend the first 5 chapters. I think the problems are really, really good and some of them are definitely quite fun.

2) 'Graph Theory' by Diestel. The first 3 chapters have more than enough problems to keep you busy over the summer. It's a well-written text and, since you've already taken a graph theory course, a nice reference.
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I like my math easy. I like a little bit of graph theory and a little bit of matrix analysis.
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Algebra. Without a doubt. It's part of what drove my interest into maths.

However, without someone to guide you through a textbook and bounce issues you're having off of it's hard to teach yourself maths. It's definitely doable, but I'd suggest reviewing graph theory. You already have a handle on it, and it's easier to retread old ground than break new ground.
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Good question. When just starting out, I leaned toward anything new that would broaden my horizons the most. Now that there are a huge selection of topics available to study, and that I know my tastes better, I tend towards ones that already interest me or are relevant to my main topics.
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I actually like to study stuff that have financial return at this point of my life, but I really like to pick up a new field and study a bit to see the its landscape. And imo always is good have on specific field to dive down and somewhat specialize.
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Haha, why not study both! Jk, I’m doing this right now, but both of my courses serve as foundational prerequisites to study other things I want to study. I think to answer your question, right now I’m focused on trying to get a more firmer deeper understanding of foundational topics before I take on what I really want to study. Here’s my experience.

My area of interest is Bayesian statistics. I first started to read a well known book on the topic, only to realize I needed to brush up on some probability theory topics I had forgotten. Okay, did this, then started reading again, then I realized, to be a good Bayesian one needs to be an even better frequentist. So I decided I needed to go back and look at some mathematical statistics but at a deeper level. So I ended up putting down the Bayesian book and picking up a more rigorous foundational text in mathematical statistics that was harder than the class I took in undergrad. The material is interesting, so I can have motivation to get through it, but I know deepening my foundational skills after this book will make researching and understanding what I really want to read less painful. For statistics especially, probability theory is like the one foundational topic you should always review and reread and sharpen, because my understanding of this will determine my understanding of advanced material.

So as a long way to answer your question, study the foundational matter, as it can benefit you when you study something that builds on top of it.
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This is a tough one: I’d normally go for fundamentals because most research topics rely on either analysis or algebra anyway and there are great self-studying books for either subject. The sooner you get through the fundamentals the sooner you can start meaningful research. However, graph theory doesn’t really require that much prereqs for research and is one of the few areas where undergrad publications are common. So if you have identified a great advisor to supervise your graph theory research, it might help you more to delve deep into this topic. If you are unsure you want to do your undergrad research in graph theory, I’d recommend algebra using Pinter.
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In my opinion, it would be better to deepen your understanding of something you've already learned. The main reason for this is that you'll read more advanced proofs that don't necessarily lay out everything in detail. This is a key skill.
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I would recommend learning the new topic. As an undergrad, you should get rudimentary exposure to as much as you can. Abstract algebra is very cool and will change how you look at math forever.
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I studied analysis and think you could get a great leg up by first taking a vacation, and then building familiarity and comfort with something that pleases you. I think you could get a lot of miles from something like "Calculus on Manifolds" by Spivak, "Singular Integrals and Differentiability Properties of Functions" by Stein, if you want the historical approach "Mathematics for the Physical Sciences" by Schwartz, or the new approach "The theory of distributions" by Friedlander & Joshi.

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