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How to Learn Advanced Mathematics Without Heading to University

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Re: How to Learn Advanced Mathematics Without Heading to University

#21

Post by ska »

The main thing to remember is that nobody learns mathematics to any significant depth by reading. The only way to learn it is by doing it. Doing it carefully, and in full detail, not falling into the trap of "and I understand it from there".

So the biggest barriers to doing it on your own aren't source material (there is lots of that) it's a good source of correction. There is also the usual problem of self study, in that you don't have a roadmap and can waste time easily.

That said though, one advantage if you are diligent is that you probably by necessity learn techniques of checking your work (formally and informally) earlier than typical students, which is a good thing.

Re: How to Learn Advanced Mathematics Without Heading to University

#22

Post by ska »

graycat wrote: post #9

An early place where the OP dropped the ball: > and some good experience manipulating continuous functions and their derivatives. Nope. If a function is differentiable, then it is continuous, but continuity is not sufficient for differentiability. So, we can't talk in general about the derivatives of continuous functions. E.g., each sample path of Brownian motion is almost surely differentiable nowhere. Just f(x) = |…

If you do study (continuous) mathematics, one of the things you build up is a little stable of strange functions to help you test your intuition.

A very common "right of passage" exercise in introductory analysis is to come up with the Weierstrass function or something similar (usually with a little coaching). This is an everywhere continuous and nowhere differentiable function that then goes into your little toolkit.

Re: How to Learn Advanced Mathematics Without Heading to University

#23

Post by graycat »

nmrm2 wrote: post #17
graycat wrote: post #9

An early place where the OP dropped the ball: > and some good experience manipulating continuous functions and their derivatives. Nope. If a function is differentiable, then it is continuous, but continuity is not sufficient for differentiability. So, we can't talk in general about the derivatives of continuous functions. E.g., each sample path of Brownian motion is almost surely differentiable nowhere. Just f(x) = |…

>> and some good experience manipulating continuous functions and their derivatives. > Nope 1. "good experience manipulating differentiable functions and their derivatives" sounds weird in prose. 2. Some continuous functions are differentiable. Those ones have derivatives you can manipulate. In fact knowing when a function is not differentiable is a pretty useful skill. > The interview was in their computer group, bu…

> Think of this as "calculus for engineers vs. analysis", and imagine how well a civil engineering interview would go if you talked about different types of integrals instead of talking about how to use the basic stuff to build good bridges.

That analogy shouldn't apply to quantitative trading on Wall Street: That challenge needs more than just engineering math approaches if only to read the literature.

E.g., apparently broadly the first cut way to evaluate exotic options is to use the Brownian motion solution to the Dirichlet problem, that is, the subject of Markov processes and potential theory. The subject is awash in measure theory, e.g., stopping times, the strong Markov property, regular conditional probabilities, of course conditioning and the Radon-Nikodym theorem. This isn't advanced calculus for engineers. E.g., the work of Marco Avellaneda at NYU Courant, Steve Shreve at CMU, no doubt the work of E. Cinlar at Princeton.

Re: How to Learn Advanced Mathematics Without Heading to University

#24

Post by graycat »

ska wrote: post #22
graycat wrote: post #9

An early place where the OP dropped the ball: > and some good experience manipulating continuous functions and their derivatives. Nope. If a function is differentiable, then it is continuous, but continuity is not sufficient for differentiability. So, we can't talk in general about the derivatives of continuous functions. E.g., each sample path of Brownian motion is almost surely differentiable nowhere. Just f(x) = |…

If you do study (continuous) mathematics, one of the things you build up is a little stable of strange functions to help you test your intuition. A very common "right of passage" exercise in introductory analysis is to come up with the Weierstrass function or something similar (usually with a little coaching). This is an everywhere continuous and nowhere differentiable function that then goes into your little toolkit…

Ah, a favorite is a function that is differentiable but its derivative is not Riemann integrable!

Recall, a function is Riemann integrable if and only if it is continuous everywhere except on a set of measure 0. So, the derivative has to be discontinuous on a set of positive measure. Now, to construct one of those!

Here's another favorite: For positive integer n and the set R of real numbers, suppose C is a closed subset of R^n with the usual topology. Then there exists a function f: R^n --> R that is 0 on C, strictly positive otherwise, and infinitely differentiable. So, for C, take, say, a sample path of Brownian motion, the Mandelbrot set, a Cantor set of positive measure, etc. Can use that function to settle an old question in constraint qualifications for the Kuhn-Tucker conditions in optimization.

Or, any closed set can be the level set of an infinitely differentiable function.

Sure, really fun reading for such things is:

Bernard R. Gelbaum and John M. H. Olmsted, Counterexamples in Analysis, Holden-Day, San Francisco, 1964.

Re: How to Learn Advanced Mathematics Without Heading to University

#25

Post by pmalynin »

namelezz wrote: post #6
steQ wrote: post #5

I know that many learn programming themselves but I'm curious if anyone have learn advanced mathematics this way.

What are the best ways for a non-genius, normal person to learn advanced mathematics?

Notes from the intro math course at my University are pretty good.

https://www.math.ualberta.ca/~xinweiyu/117-118.14-15/Math117...

Re: How to Learn Advanced Mathematics Without Heading to University

#26

Post by ska »

graycat wrote: post #24
ska wrote: post #22

Earlier quoted context omitted.

If you do study (continuous) mathematics, one of the things you build up is a little stable of strange functions to help you test your intuition. A very common "right of passage" exercise in introductory analysis is to come up with the Weierstrass function or something similar (usually with a little coaching). This is an everywhere continuous and nowhere differentiable function that then goes into your little toolkit…

Ah, a favorite is a function that is differentiable but its derivative is not Riemann integrable! Recall, a function is Riemann integrable if and only if it is continuous everywhere except on a set of measure 0. So, the derivative has to be discontinuous on a set of positive measure. Now, to construct one of those! Here's another favorite: For positive integer n and the set R of real numbers, suppose C is a closed su…

Fun book indeed!

Your first example you won't typically run into until an introductory measure theory course.

Re: How to Learn Advanced Mathematics Without Heading to University

#27

Post by kafkaesq »

namelezz wrote: post #6
steQ wrote: post #5

I know that many learn programming themselves but I'm curious if anyone have learn advanced mathematics this way.

What are the best ways for a non-genius, normal person to learn advanced mathematics?

First off, "genius" talent is by no means required.

Second, it should be something you really, really like doing. Like music is to a musician (or an audiophile), cooking (and watching people get off on your creations) is to a chef, sports training is to an athlete, etc.

And third, like anything else of true value in this life -- it will take a significant amount of time; in particular devoted to practice (and very importantly, play), especially solving (often obscure-seeming) problems on your own, just to scratch an itch, or to know that you can.

Easily a few thousand hours to attain what's called "mathematical maturity"[1], and probably somewhere on the order of the fabled 10,000 to obtain what might be called true expertise in the field. Which should (by itself) be no obstacle, if it's something you're really, really, really into.

Just like any other field, basically.

[1] https://en.wikipedia.org/wiki/Mathematical_maturity

Re: How to Learn Advanced Mathematics Without Heading to University

#28

Post by kafkaesq »

bmer wrote: post #10
namelezz wrote: post #6

Earlier quoted context omitted.

What are the best ways for a non-genius, normal person to learn advanced mathematics?

It is very easy to get most mathematics textbooks online, through slightly unsavoury means. It is far more difficult to figure out which textbooks are worth reading---this is a process that requires trial and error, and browsing through recommendations (math.stackexchange and mathoverflow.net have many good textbook recommendation questions, with many excellent answers). Also, it is very easy to audit courses at univ…

"Unsavoury", as in?

Re: How to Learn Advanced Mathematics Without Heading to University

#30

Post by chatmasta »

ska wrote: post #21

The main thing to remember is that nobody learns mathematics to any significant depth by reading. The only way to learn it is by doing it. Doing it carefully, and in full detail, not falling into the trap of "and I understand it from there". So the biggest barriers to doing it on your own aren't source material (there is lots of that) it's a good source of correction. There is also the usual problem of self study, in…

This is probably the reason why there are so many more autodidact programmers than mathematicians. When programming, you know if you made an error as soon as you try to run your code. The program, by definition, must be "correct" in order to execute. The "source of correction" is the error detection built into any programming language runtime or compiler.

No such "source of correction" exists for mathematics, and that makes it an inherently more difficult subject to teach yourself, because any errors you make will "fail silently" unless you are capable of detecting them yourself, which by definition you cannot do without experience. This is why a mentor/professor makes learning mathematics so much easier; he/she plays the role of mathematical compiler.

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