What Does One Learn in Theoretical Math?

[StonedPanda]

Theoretical math, that is. What does one learn. What is the path one takes?

I think I want to go into medecine, however, I doubt I'll be able to go far enough in Mathematics. I just finished my junior year of H.S and got an A in AP Calc BC.

 

[fourier jr]

With biology it's easy to explain what someone does for a PhD because you probably deal with things that people are kind of familiar with or have at least heard of like pollen, DNA, animals, etc etc. Similar with physics, chemistry, maybe literature, etc. With math it's not really like that though; it's a lot harder (for me anyway) to explain what, say, topology, is to someone who hasn't done a lot of math. All I can say is that high school is only a very specific, narrow part of algebra; calculus is a very specific, narrow concrete part of analysis and topology. Think about this: the real numbers (that would be the values of 'x' you plug into a function f(x) ) are very very special and no other numbers have all the properties that the reals have. That's the sort of thing you look at starting around 3rd/4th year.

 

[StonedPanda]

What is required to get a Phd in math? Yeah, it's kind of scary that I'm of the most advanced in mathematics with respect to my peers yet we know so very very little. I do plan to major in mathematics (or at least do a buch of mathematics), thogh.

 

[matt grime]

Research mathematics is impossible to describe, in the majority, to a lay person. Fortunately University level mathematics, especially in the US, has nothing to do with research. So don't worry about it. Major in maths if you want, and as long as you understand that you're supposed to think for yourself abstractly then you won't have a problem - it's about applying rules you're taught to different examples, that's all.

 

[selfAdjoint]

What you do to get a PhD in math is

1. Take the first year grad courses and concentrate on mastering them. They will span the branches of mathematics.

2. From this, you should find out what direction your talents lie in. Are you an algebraist? an analyst? Does topology float your boat? Talk to your peers and the older grad students. Look for a niche you can not only fill, but enjoy filling.

3. Based on this, seek out an advisor. He/She must be an active researcher in your chosen field, and ready to help you.

4. What happens next depends on the advisor. Some of them will give a lot of help, others feel honor bound to let you sink or swim on your own. Attend seminars, even try to present at seminars, build up your expertise in your chosen niche. After a while it should become natural.

5. Sort of naturally out of this activity you should come up with a problem to solve for your thesis. Some advisors will suggest or even assign a problem. In any case your advisor should approve your problem.

6. Start proving results in and around your problem. Part of your conception of it will be an idea of how to go about proving it and of what it will accomplish when proved. Work things out as best you can and approach your advisor if you get stuck.

7. It should take about a year, then you do the orals and the committee shakes your hand. Congratulations new doctor!

 

[fourier jr]

I think he was asking what mathematicians do, or what advanced math is like.

 

[e(ho0n3]

I don't know about you guys, but as I progress through maths., things are just becomming duller and duller. All of the books I read have the same general layout: Definitions, Axioms, Theorem, Example, Proposition, Lemma, Theory, Theory, Example, etc., etc. This doesn't bother me so much as I'm used to reading dull things (like reading source code).

 

[fourier jr]

really? I think it's gotten more interesting. There isn't as much mindless drilling & computation like in the lower-level stuff.

 

[Icarus]

I suppose it depends on what you see. I, like Fourier jr I suspect, see mathematics as a means of exploring new worlds of thought. Those definitions and lemmas were not dry facts of no worth outside of themselves, but rather clues to the solving of old mysteries, and the uncovering of new ones.

It appears on the other hand that e(ho0n3 was too busy looking at the trees to notice the forest.

 

[Gza]

I only study the minimum amount that's required to learn physics. It's only at the interface of reality and mathematics I'm interested in. To me, studying pure math for the sake of it seems a little pointless, but that is just my opinion. Not that anyone asked for it.

 

[matt grime]

Good job some of us disagree with you, Gza, otherwise no coding, cryptography, enigma, RSA, string theory, TQFT, loop quantum gravity, quantum computing...

 

[Gokul43201]

Did Hardy not express relief that the "uselessness" of pure-math (or number theory ?) is what will keep it safe from the touch of the common man ?

And along came Rivest, Adleman and Shamir...

 

[e(ho0n3]

really? I think it's gotten more interesting. There isn't as much mindless drilling & computation like in the lower-level stuff.

I don't like computations either (which makes me wonder why I'm studying computer science, but anywho...).

I suppose it depends on what you see. I, like Fourier jr I suspect, see mathematics as a means of exploring new worlds of thought. Those definitions and lemmas were not dry facts of no worth outside of themselves, but rather clues to the solving of old mysteries, and the uncovering of new ones.

In the end, this all comes down to a question of values. I like maths. a lot since it makes me think constructively (although I get headaches sometimes).

It appears on the other hand that e(ho0n3 was too busy looking at the trees to notice the forest.

Interesting...

 

[Gza]

Good job some of us disagree with you, Gza, otherwise no coding, cryptography, enigma, RSA, string theory, TQFT, loop quantum gravity, quantum computing...

Interesting you should give all examples of applied math. Perhaps these branches of math did at one time start off as what we would call "pure" math, but it was the necessity of their physical application in the real world that gives them purpose. It wasn't the mathematics that inspired the application, it was vice versa, which is the way it should be. To say that string theory and quantum computing wouldn't exist without the study of pure math is like saying the Hubble telescope wouldn't be created without the invention of the wrench. If we have the ability to ponder the enormously complex concept of the Hubble telescope, we sure as hell have the ability to develop such a relatively simple tool as a wrench.

 

[matt grime]

I don't think from your reply, Gza, that you understood my point. The fact is that the mathematics in them is pure maths, and without people studying pure maths those subjects wouldn't have almost any of the results that are in them. As you are predetermined to insult pure mathematics as being useless, surely any use I give for it would make it automatically applied and therefore no longer pure?

If no one had studied category theory, topology and come up with homotopy theory, then the idea of n-categories wouldn't exist and they are fashionable right now.

If people didn't know about Hecke algebras and Serre correspondence then there are lots of results in mathematical physics that wouldn't be understood.

Fermions are *just* tracial elements in a matrix tensor algebra... good job people studied those as absract objects without needing to find uses for them before hand.

 

[fourier jr]

Did Hardy not express relief that the "uselessness" of pure-math (or number theory ?) is what will keep it safe from the touch of the common man ?

I haven't heard that, but it sounds possible. Hardy used to brag that he hadn't done anything "useful" in his entire life.

 

[StonedPanda]

Fourier, selfadjoint was doing pretty much what I was asking. but I'm interested in that too!

SA: How do you know what to prove for your PhD? Hasn't anything that you're ready to prove already been proven?

 

[Gza]

You do have a point matt. Plus judging from your other posts, I can tell that you probably have a better perspective on the issue, since you know a whole lot more math than me, and can see it's relevence in physical application more readily. I think that once I get into grad school and start taking more classes on math methods for physicists and such, I will gain an appreciation for math in itself.

 

[seeking knowledge]

hey, I'm tryin 2 help a m8 with standard diviations, but it's been a wile since i'v done it. A formula would really help. thanx

 

[Gokul43201]

hey, I'm tryin 2 help a m8 with standard diviations, but it's been a wile since i'v done it. A formula would really help. thanx

Wrong place for this post. Anyway, it's spelled 'standard deviation'. Google it.

Also : http://mathworld.wolfram.com/StandardDeviation.html should help.

 

[fourier jr]

Fourier, selfadjoint was doing pretty much what I was asking.

oh