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Ask HN: How to self-study mathematics from the undergrad through graduate level?

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Re: Ask HN: How to self-study mathematics from the undergrad through graduate level?

#151

Post by gumby »

This is really great.

One specific point:

> One idea I had was to complete the MIT open courseware courses for the Applied and Pure math fields

A note about Open Courseware: they tend to be much of or all of the material handed out in a course (lecture notes, problem sets, etc). They aren't a "course" in the sense of Coursera, Khan Academy and the like.

A few years ago I needed to brush up on my thermodynamics and was able to read the material for the same course I'd taken as an undergraduate (well, same course number; I took thermo in 1983, and the prof and some of the material had changed). This was merely an undergraduate thermo class that I had already taken 30 years before and it was still quite hard.

I don't mean in any way to discourage you!!! This is an excellent idea. But OPenCourseware itself is more like a box of legos for someone who wants to teach a class in the subject. You may find a different source better, especially for topics that are new to you.

Re: Ask HN: How to self-study mathematics from the undergrad through graduate level?

#152

Post by throwawaymath »

13415 wrote: post #123
MattHeard wrote: post #115

Earlier quoted context omitted.

This is exactly the kind of advice I give to people who ask me about teaching themselves to "code". 1. Find a thing you want to make. 2. Find out how to make it. 3. Try to make it. 4. Learn the skills that previously prevented you from making it.

I don't think this is a good comparison. People who are skilled in both areas might not realize that, but learning how to program is a hundred times easier than working through any advanced topic in higher mathematics and coming up with your own proofs.

While I agree with what you've claimed here, in fairness the OP never said anything about proving novel theorems :). Learning advanced mathematics at the higher undergraduate or graduate levels is probably more difficult than programming (at least in my opinion), but it's much closer than proving something original. Likewise programming existing algorithms is significantly easier than designing novel algorithms, which is much closer to proving novel mathematics.

Re: Ask HN: How to self-study mathematics from the undergrad through graduate level?

#153

Post by joker3 »

adenadel wrote: post #101

I think this is something like what you're asking for How to Become a Pure Mathematician (or Statistician) http://hbpms.blogspot.com/

This might be a good roadmap for pure math, but don't use this for guidance on how to be a statistician.

Re: Ask HN: How to self-study mathematics from the undergrad through graduate level?

#154

Post by tprice7 »

throwawaymath wrote: post #150
tprice7 wrote: post #146

Earlier quoted context omitted.

Not necessarily. I left undergrad after four terms and self-studied to the point of having published research, giving invited talks, and even being a visiting researcher for three months with my expenses paid. Links: http://content.algebraicgeometry.nl/2017-2/2017-2-007.pdf https://projecteuclid.org/download/pdfview_1/euclid.ecp/1508... Look for my name (Thomas / Tom Price) on these pages: https://web.archive.org/web…

Wow, that's remarkable. As an aside, your work on numerical cohomology appears to have been useful for a new result pertaining to lattices. Given the authors of the followup work it's likely helpful for the study of lattices in post-quantum cryptography.

Are you referring to the "An Inequality for Gaussians on Lattices" paper? They cite my paper, but it's to give an example of an application of their result (which I use), not because they built on it. But anyway, I think it's very fascinating that the people who discovered a key result that I needed for that paper, which could probably be best classified as arithmetic geometry, are mainly computer scientists!

Re: Ask HN: How to self-study mathematics from the undergrad through graduate level?

#155

Post by throwawaymath »

joe_the_user wrote: post #118
abnry wrote: post #38

Here's my personal opinion about how you should approach this. It's all well and good to want to cover undergraduate math courses. When you are actually enrolled in a university, you will have enough inertia and motivation to complete the courses. However, when you are self-studying you are doing it all on your own. It's hard to be as thorough and cover everything. And so I ask, what really is your goal here? You don…

The OP did specify what he wanted - the basic undergraduate and starting graduate curriculum. That's a pretty well defined area: Algebra, Real Analysis, Geometry and Topology with maybe some complex analysis, number theory, statistics, CS or etc thrown in. I personally did work myself up to the graduate in math during the last two years of High School & first year of college. I was motivated by exploring ideas and ga…

> I personally did work myself up to the graduate in math during the last two years of High School & first year of college.

...how in the world did you manage to do this? Did you actually self-study, or were you placed in a gifted program? Self-studying all of undergraduate mathematics is more impressive than actually studying all of it in a four year classroom setting. Doing so as a teenager is amazing.

The most gifted person I've ever known in mathematics is actually a physicist who entered Harvard at 16. He was already taking undergraduate math courses at 13/14 and had mastered all the undergraduate material by the first semester of his undergraduate degree. For the remaining years of his undergraduate degree he took graduate math courses.

But in order to do that he needed to not only be gifted; he was placed in a program for extremely gifted kids sponsored by Stanford University from his preteen years. I don't think your anecdote is a great comparison for the OP's expectations (or for calibrating advice they'd benefit from).

Re: Ask HN: How to self-study mathematics from the undergrad through graduate level?

#156

Post by oefrha »

jcora wrote: post #127
13415 wrote: post #123

Earlier quoted context omitted.

I don't think this is a good comparison. People who are skilled in both areas might not realize that, but learning how to program is a hundred times easier than working through any advanced topic in higher mathematics and coming up with your own proofs.

This thread seems bizarre to me. There's one guy claiming he "worked himself up to graduate level math" in the last two years of high-school. So either he's a literal IMO-gold-medal-level genius, or people here don't quite understand what undergraduate math studies actually involve. Even if he is that intelligent there's no way he actually did the amount of work required on such a broad number of topics.

Can confirm. I’m not an IMO gold medalist, but I come from a very competitive nation that fetch five or more gold medals on most years, and I almost made the national team, twice (failed at TST, which selects 6 out of ~30). And I graduated with a math degree from one of the top institutions. There’s no way I would have completed undergrad plus entry level grad math in the last two years of high school — those took me three years in college (of course I was doing other things, but still).

Unlike programming, mathematics isn’t something you can pick up in a weekend.

EDIT: Now that I think about it, you can probably bang your head against, say, Lang, for two years, “digesting” a big chunk of it, earning you the bragging right of “working yourself up to graduate level math”. That won’t give you the breadth of a good bachelor of mathematics, and it certainly won’t prepare you for quality research.

Re: Ask HN: How to self-study mathematics from the undergrad through graduate level?

#157

Post by nabla9 »

You need to do lot's of exercises.

Shaum's Outlines has lots of cheap math books with exercises and solutions. You can use them to complement other books

https://www.mhprofessional.com/9780071623667-usa-schaums-out...

https://www.mhprofessional.com/9780071635349-usa-schaums-300...

https://www.mhprofessional.com/9780071383929-usa-schaums-out...

https://www.mhprofessional.com/science-math/math/topology

Re: Ask HN: How to self-study mathematics from the undergrad through graduate level?

#158

Post by nicoburns »

throwawaymath wrote: post #50
nicoburns wrote: post #8

The University of Oxford has all of their notes and exercises available online https://courses.maths.ox.ac.uk/overview/undergraduate If you get stuck you could try asking on https://math.stackexchange.com/ Although, if you can, finding someone else to work with, and someone else who already knows some math to ask occasional questions would probably help you a lot.

Many of the courses listed there have no course materials available or only very incomplete materials.

Oh really? The ones I cliked through to did!

Re: Ask HN: How to self-study mathematics from the undergrad through graduate level?

#159

Post by throwawaymath »

tprice7 wrote: post #154
throwawaymath wrote: post #150

Earlier quoted context omitted.

Wow, that's remarkable. As an aside, your work on numerical cohomology appears to have been useful for a new result pertaining to lattices. Given the authors of the followup work it's likely helpful for the study of lattices in post-quantum cryptography.

Are you referring to the "An Inequality for Gaussians on Lattices" paper? They cite my paper, but it's to give an example of an application of their result (which I use), not because they built on it. But anyway, I think it's very fascinating that the people who discovered a key result that I needed for that paper, which could probably be best classified as arithmetic geometry, are mainly computer scientists!

Ah, thanks for the clarification. The computer scientists who work on quantum computational complexity and post-quantum cryptography tend to be much more mathematical than the norm :)
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