How to augment mathematical education?

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P13808
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How to augment mathematical education?

Postby P13808 » Thu May 08, 2014 10:29 pm UTC

So I'm in that situation where I'm studying mathematics at the undergrad level and really quite enjoy it, but I'm limited by credit hours and what the university actually offers in terms of classes and such. I have the luxury of excessive free time, and I want to spend at least some of it filling in holes, especially with the strong possibility of aiming for higher education in mathematics.

As it stands, I'll be taking linear algebra, a class on tex and maths literature and becoming a "professional mathematician", a year of analysis, abstract algebra, combinatorics, each, three semesters of calculus (two down), and experimental maths (computer-generated proofs and such). Also doing some as-of-yet unknown computer science courses.

Right now I'm teaching myself some number theory and formal logic, since both seem useful. I doubt I'll have time for Diff Eq, especially since I'm more pure-oriented, and the class, according to my advisor, is aimed at people who will be applying it (like engineers). So I'll find a text and work through the problemsets as much as I can. Also working with the complex numbers. Statistical analysis and probability also seem useful, though I may be able to take one as a class. I do not know how much discrete mathematics (seemingly omnipresent subject) would be redundant with the other stuff already covered. I also don't know where else to go.

So, thread topic and all, what would you say are good/useful/interesting/fun things to learn in mathematics? And which, if only some can be done with the support of faculty, etc., are better done with such support than trying to self-learn?

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Qaanol
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Re: How to augment mathematical education?

Postby Qaanol » Thu May 08, 2014 11:33 pm UTC

There are several types of pure math: algebra, analysis, topology, and number theory are the big ones.

Algebra
You might find Galois theory useful and enjoyable. It only requires a basic understand of introductory abstract algebra, which you can probably pick up the bulk of on your own pretty quickly: groups and fields are straightforward objects, and you’ll want to get a real understanding of the isomorphism theorems.

Analysis
You could try to pick up some measure theory. The Lebesgue integral is the first real milestone here. The best-known analysis text is by Rudin. Another subject you could look at is functional analysis. Another book, more fun than Rudin, is Counterexamples in Analysis.

Number theory
There are a bunch of fun theorems relating to prime numbers that you can get to pretty quickly. To get a feel for some actual cutting-edge research, check out the expository articles about Polymath 8a/b, on bounded gaps between primes. At the beginning of last year, no one knew if there was any finite liminf for gaps between primes. Then in April 2013, Zhang established a bound of 70,000,000, which was quickly reduced to a few thousand. Last fall, Maynard cut it to 600, and as of last month it’s been lowered to 246.

Topology
I have not studied topology in any depth. Someone else might be able to chime in here.
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Farabor
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Re: How to augment mathematical education?

Postby Farabor » Sat May 10, 2014 3:10 am UTC

Since you have time on your hands, I'd recommend reading "Gödel, Escher, Bach: An Eternal Golden Braid". It's an amazing work that, among other things, will walk you through parts of the foundational crisis in mathematics and Gödel's incompleteness theorem.

(On a side note, I love that the autocorrect here puts the umlaut above the o when you type Gödel :))

P13808
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Re: How to augment mathematical education?

Postby P13808 » Sat May 10, 2014 5:51 am UTC

Thank you both..

Qaanol, are those all common subtopics of the main topics you listed (algebra, topology, etc.)? I'm taking Algebra and Analysis classes, so I'm presuming I needn't figure those out myself. The topics in number theory sound neat, though, so I'm going to be doing some Googling soon. (Also figuring out where topology fits into my uni's curriculum, since no class explicitly is listed as such.)

Farabor, weirdest thing; just yesterday I attended a (student's independent study's) presentation on that text. It sounds hella interesting. (I also had no idea it included the topics you listed; the entire presentation was surrounding "This statement is false", and considering the title I assumed that was the gist of it. Good to know.)

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PM 2Ring
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Re: How to augment mathematical education?

Postby PM 2Ring » Sat May 10, 2014 11:23 am UTC

P13808 wrote:Farabor, weirdest thing; just yesterday I attended a (student's independent study's) presentation on that text. It sounds hella interesting. (I also had no idea it included the topics you listed; the entire presentation was surrounding "This statement is false", and considering the title I assumed that was the gist of it. Good to know.)

The sentence "This sentence is false." is known as the liar paradox, and Gödel's theorem (to be more precise, his first incompleteness theorem) is very closely related to the liar paradox except that truth is replaced by provability. Gödel's big insight was basically how to express the liar paradox in any formal system that's sufficiently powerful, i.e any system capable of representing elementary arithmetic.

From Gödel's incompleteness theorems - Relation to the liar paradox.
Wikipedia wrote:The liar paradox is the sentence "This sentence is false." An analysis of the liar sentence shows that it cannot be true (for then, as it asserts, it is false), nor can it be false (for then, it is true). A Gödel sentence G for a theory T makes a similar assertion to the liar sentence, but with truth replaced by provability: G says "G is not provable in the theory T." The analysis of the truth and provability of G is a formalized version of the analysis of the truth of the liar sentence.



FWIW, I was given a copy of Gödel, Escher, Bach not long after it was originally released, and I've re-read various bits & pieces over the years. I re-read the whole thing from cover to cover about 4 years ago and probably enjoyed it as much then as I did the first time through. A 20th anniversary version was released in 1999 but it's identical to the original (apart from an updated preface).

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Qaanol
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Re: How to augment mathematical education?

Postby Qaanol » Sat May 10, 2014 8:44 pm UTC

I was suggesting ideas on how you can get to some fun stuff quicker, and as a side benefit make the classes you end up taking much easier because you’ll already know the basics.

If you are looking for obscure yet accessible things, maybe check out the Eulerian numbers, the Faulhaber polynomials, and the Bernoulli numbers. They never came up at all in my maths classes, but they’re worth knowing about.

Learn about how to solve recurrence relations with generating functions and/or partial summation.

If you want to go down a rabbit hole, read about how e and π were proved to be irrational (I assume you already know how to prove √2 is irrational: if not, learn it. And then look into √2√2.) If you find it interesting, the next level deeper is how e and π were proved to be transcendental.

How much experience do you have with proofs? If you’re going into pure maths, you’ll need a solid foundation in rigor. Do you know about proof by induction and proof by contradiction? Can you prove there are infinitely many prime numbers? Can you prove the Pythagorean theorem? How about the quadratic formula? The fact that indivisible integers greater than 1 are prime (and incidentally, the distinction between prime and indivisible)?

You might be interested in Heron of Alexandria, especially Heron’s method to calculate square roots, and Heron’s formula for the area of a triangle given the lengths of its three sides (but none of its angles).

A lot of math history gets elided in school, but there is a wealth of good stuff there: Archimedes’ proofs that a cone has one third the volume of the cylinder with the same base and height, while a sphere has two thirds the volume of the cylinder (the proof method is the neat part); Tartaglia’s solution of the cubic, Ferrari’s solution of the quartic, and the role Cardano played; everything about Évariste Galois; and a whole lot more. Heck, the fact that Gauss knew the Cooley-Tukey algorithm is pretty neat.

If you want some basic topology, check out the Euler characteristic of a surface, and also how to invert a sphere.

To bring in some computer programming, try making a fractal viewer program, and look into the different ways of coloring the result (and how to do so quickly).
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z4lis
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Re: How to augment mathematical education?

Postby z4lis » Tue May 13, 2014 5:37 pm UTC

Qaanol wrote:If you want some basic topology, check out the Euler characteristic of a surface, and also how to invert a sphere.


If you like topology and geometry, this book by Thurston is really cool. I can't say whether or not you'll be able to really understand what's going on in it, but if you work through the problems and use it as a guide for further reading, you might have a lot of fun.
What they (mathematicians) define as interesting depends on their particular field of study; mathematical anaylsts find pain and extreme confusion interesting, whereas geometers are interested in beauty.

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doogly
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Re: How to augment mathematical education?

Postby doogly » Wed May 14, 2014 7:10 pm UTC

Though his lecture notes on which the book are based are both better and more free!

http://library.msri.org/books/gt3m/
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