Should I Become a Mathematician?

[mathwonk]

It may be of interest to some to peruse my vita on my web page at school:

http://www.math.uga.edu/~roy/There you will see a short synopsis of 30 years of activity. Note roughly the last half of it discusses tasks that are not related much to research, committee work, refereeing, exams, prelims, teaching numerous courses, of which only those numbered 800, 8000 or above relate to material one finds instructive as a professor. So you get an idea of how many hours were spent doing service related activity. Of course it is not bad to be of service to someone.

There are about 30 research papers listed there, roughly one a year, of which maybe a little over a third are ones I particularly would single out.

 

[mathwonk]

maybe try the career guidance section? (for comp sci.) there is also a computer section but I saw no career advice there. but there are readers of this thread who know about comp. sci. and they will eventually chime in I think. I'm sorry I don't know exactly who to suggest.

 

[dkotschessaa]

Looking at your CV reminds me of another question, if you don't mind. It concerns research areas. Your interests are in algebraic geometry. When did you find out that was your interest? How and when do I know what mine are, being so early in my undergraduate courses I don't even know what most of these topics are about yet? How soon do I *need* to know? Do you find the topic or does the topic find you, so to speak?

(Obviously I'm not looking for an answer to every question, just prompting).

-Dave K

 

[DeadOriginal]

Looking at your CV reminds me of another question, if you don't mind. It concerns research areas. Your interests are in algebraic geometry. When did you find out that was your interest? How and when do I know what mine are, being so early in my undergraduate courses I don't even know what most of these topics are about yet? How soon do I *need* to know? Do you find the topic or does the topic find you, so to speak?

(Obviously I'm not looking for an answer to every question, just prompting).

-Dave K

Hmm. This is something I would also like to know.

 

[homeomorphic]

It's nice to have a general direction your are heading in by the time you apply to grad school. You can put that in your application letter and it may help them see whether you would fit in in the department. Something like "topology" or "analysis" or "logic" is enough for that purpose, or even just narrowing it down to two or three areas like that is fine. I know a guy who started grad school with logic in mind and switched to algebraic topology (fairly big leap, to my mind). Depending on the program, you ought to have an adviser definitely within 3 years, so that means you have to narrow it down to the right subfield by then. Within two years would be better. Then, within maybe another year, you ought to be working on your thesis topic.

But lots of people switch fields later on in life. From a non-academic point of view, there's no requirement at all. I don't know what happens to professors if they switch fields on a whim. Like, if they hired you as topology professor, and then after 2 months there, you decide, "screw topology, I'm switching to set theory", I don't know if that would go over very well. Maybe if you got tenure, you could get away with it? But, still, you can probably get away with changing areas within reason, if it doesn't slow down your publications.

But, if you just do it as a hobby, you can do whatever you want, obviously.

 

[dkotschessaa]

Interesting. I've seen professors who do work that seems different than what they started with, but in a kind of natural transition that allowed them to bring something different to the field. A professor whose research I'm looking at right now at my university says the following on his web page "Research Areas

I was trained as a mathematical logician, with an emphasis on theoretical computer science. My specialty was Finite Model Theory, but I found myself working in combinatorial games and random structures as well. During the past few years, I have been working on geometry and its applications to materials science and what is often called nanoscience. ("Nanoscience" is probably a misnomer, since it refers to the "meso-scale" of microscopic physics -- from many Angstroms to about a micron -- in which quantum effects are usually minor.) Here are the areas ordered by my current level of attention."

Which to me sounds like a "big leap" that you described, but clearly he made a transition that seemed natural to him.

I myself have a lot of interests, which can be a problem in narrowing down to one, but if I know that I can get my foot in the door in one discipline and then use it to bring something new to another discipline, then perhaps it won't be so difficult. It's just finding that first thing...

-Dave K

 

[mathwonk]

Well I didn't have a clue when I applied to grad school. I liked a beautiful book by Hurewicz and Wallman on dimension theory and I mentioned that on my application, just to have something to say, and not knowing the subject was basically closed for decades.

Then in my first year at grad school I was amazed by my algebra teacher Maurice Auslander and wanted to work with him. But he was teaching algebra with an orientation towards algebraic geometry, and when i said i wanted to work in algebraic geometry he said he was not a specialist and i should work with Allan Mayer who was.

Then the next year I took a course with Alan Mayer in algebraic geometry and was blown away by it. I loved it. So for me, the professors showed me what I liked. It is hard to pick a research topic in undergrad, if like me you were still learning really old and/or elementary stuff.

So its sort of an exploration and at a certain point you go, Wow, I want be one of those guys, or I want to study that subject!

In my case I did not finish with Alan, even though he was very helpful, due to distracting influences from the vietnam war. I took a break and then a few years later Hugo Rossi kindly recruited me to Utah and taught me a lot of very valuable complex analysis of several variables, and then I met the brilliant C.Herbert Clemens, who put me back on the path I had been in love with of classical algebraic geometry of curves and Jacobians, and guided me patiently and generously to a thesis.

I was very lucky. All my career by the way I have mascaraded as an algebraic geometer but really functioned and thought as a several variable complex geometer thanks to the training from Hugo and Herb. I also benefited enormously from postdoc training with Philip Griffiths, David Mumford, and Heisuke Hironaka.

While hanging around with those guys, one meets also as a consequence, an incredible list of amazing people like Bernard Tessier, John Fay, Mori, Mattuck, Kleiman, Hartshorne, Kolla'r, Barry Mazur, David Kazhdan, Igusa, Freitag, Bott, Tate, Mike Schlessinger, Saul Lubkin, Johnny Wahl, Mike Artin, Miles Reid, Frans Oort, Białynicki-Birula, Eduard Looijenga, Steenbrink, Boris Moishezon, Serre, Dolgachev, William Fulton, Murre, Wolf Barth, Herbert Lange, David Gieseker, George Kempf, Nori, Andre Tyurin, ... it just goes on and on. I could write down a list so long of brilliant people who have helped me that it would easily exhaust the character limit of this post.

If you go to meetings as well as these top places, you also meet younger people, and brilliant students of these icons, Joe Harris, David Morrison, DeConcini, Ciliberto, Ziv Ran, Jim Carlson, Rob Lazarsfeld, Enrico Arbaello, Maurizio Cornalba, Fabrizio Catanese, Gerald Welters, van Geemen, van der Geer, Arnaud Beauville, Olivier Debarre, Ragni Piene, Rick Miranda, Bob Friedman, Ron Donagi, Robert Varley, Valery Alexeev, Elham Izadi, Werner Kleinert, Edoardo Sernesi, Igor Krichever,... and it blossoms for you as well.
And now there is another generation of people who have more recently helped me; Ravi Vakil, Sam Grushevsky,...

This is truly but a tiny fragment of the people who have kindly taught me this subject, and I apologize to the many I omit. Literally every time I close this post more names crowd to mind. But my point is not to list them all, but to show you that you are not alone, you get a LOT of help.
The benefit of speaking or even listening to these people is immeasurable.

The moral is: Work as hard as you can to acquire some skill and knowledge. Then go to some math meetings as soon as possible, and meet people who are active in the subject you are interested in. You will be glad you did. The more people you meet the more they will contribute to your own work. Once you get on your feet and to a point where you can benefit from their conversation, you will be amazed how much you learn from talking and listening to other people.

If you want to understand topology, talk to a topologist, if you want to understand resolution of singularities, ask Hironaka, ... you get the idea.

For an online version of these conversations, check out mathoverflow now and then. there are many very knowledgeable young and senior mathematicians there sharing their knowledge, but only on a level above what you should find in books yourself. But there is no penalty for reading answers to other peoples questions. And even if your question is too mickey mouse for them they will just close it.

 

[dkotschessaa]

I can't tell you how many hours I've spent just staring at my list of potential undergraduate courses and trying to figure out which way to go. But what you've said just re-enforces to me that that's probably the least of it. As one of my calculus professors told me "When you're done studying calculus, you'll be about 400 years behind in mathematics. When you're done with your undergraduate degree, you'll be about 200 years behind." (Perhaps not those exact numbers, but you get the idea).

I want to leverage somewhat my intuitive understanding of computers, and I'm finding myself drawn towards things like set theory and logic. So perhaps there's something there.

I'm really fascinated by Cantor - but is that the sort of glorified mystical stuff that is kind of oversaturated and over hyped? (like how everybody that studies physics wants to be an astrophysicist?)

I'm also finding that I don't like physics as much as I thought I did! (That was a *weird* thing to find out).

But for the most part I'm so mathematically illiterate it could just as well be anything.

 

[homeomorphic]

I can't tell you how many hours I've spent just staring at my list of potential undergraduate courses and trying to figure out which way to go.

I would recommend talking to the professors or taking a look at textbooks. Most of the stuff you have to study is just standard stuff that everyone has to know, until later in undergrad.

But what you've said just re-enforces to me that that's probably the least of it.

It just kind of happens at some point. You shouldn't worry about it too much. When I was applying for grad schools, I spent quite a bit of time reading graph theory and logic, just to try a couple things I hadn't done much of before. Ending up sticking with what I thought I was going to do, which was topology.

As one of my calculus professors told me "When you're done studying calculus, you'll be about 400 years behind in mathematics. When you're done with your undergraduate degree, you'll be about 200 years behind." (Perhaps not those exact numbers, but you get the idea).

Beyond that, it just keeps getting worse since the growth of math is exponential. You start being able to catch up only in narrower and narrower areas.

I want to leverage somewhat my intuitive understanding of computers, and I'm finding myself drawn towards things like set theory and logic. So perhaps there's something there.

Yeah, maybe theoretical computer science.

I'm really fascinated by Cantor - but is that the sort of glorified mystical stuff that is kind of oversaturated and over hyped? (like how everybody that studies physics wants to be an astrophysicist?)

Maybe. That stuff has a place in today's math, but set theory isn't a very big research area these days.

I'm also finding that I don't like physics as much as I thought I did! (That was a *weird* thing to find out).

Ah, but maybe you're wrong about that. I went through a similar experience, except that I was aware that it didn't have to be that way. After many years of studying math and physics, all my suspicions about my classical mechanics class were proven to be correct. I never had any doubts that they were making the subject ugly when it didn't need to be, so I am not the least bit surprised about this. I knew I was going to have my way with the subject from the beginning. It was only a matter of time. As Hardy said, "there is no permanent place for ugly mathematics."

I think it's true for physics, too. Or should be true.

 

[mathwonk]

that remark of your professor was meant as a joke. (if not, he was just being negative.)

when you know advanced calculus and linear algebra well, you already know most of useful mathematics.

 

[dkotschessaa]

that remark of your professor was meant as a joke. (if not, he was just being negative.)

when you know advanced calculus and linear algebra well, you already know most of useful mathematics.

Well, of course he was being funny, but it seems there's some truth to that. So you think calculus and linear algebra are the essential "language" in which the rest of the more specialized math is spoken?

 

[dkotschessaa]

I would recommend talking to the professors or taking a look at textbooks. Most of the stuff you have to study is just standard stuff that everyone has to know, until later in undergrad.

Well, for example, here's our requirement flowhcart: http://i40.tinypic.com/1zwge54.jpg

Once I get out of the core reqs (that grey box) I only need 5 more courses. (Well, 18 credits, but it amounts to about that).

You can take vector calculus rather than Intermediate analysis, but as someone told me "you haven't really done a math degree if you haven't taken analysis." The COP course takes care of a non-major requirement as well as the degree requirement, so that's more or less what math majors take. Then there's the "Would love to take" courses and the "but I probably should take" courses and I get a bit confused. :)

Yeah, maybe theoretical computer science.

This is one I'm considering more and more. It seems to me that it's a field that appeals to those of us that are drawn towards "pure math," i.e. number theory, etc. but which has a potential applied side to it. Does that sound like a correct statement?

Maybe. That stuff has a place in today's math, but set theory isn't a very big research area these days.

Yeah, I'll get through abstract next semester and I'll have a better perspective here. Maybe get something out of my system.

Ah, but maybe you're wrong about that. I went through a similar experience, except that I was aware that it didn't have to be that way. After many years of studying math and physics, all my suspicions about my classical mechanics class were proven to be correct. I never had any doubts that they were making the subject ugly when it didn't need to be, so I am not the least bit surprised about this. I knew I was going to have my way with the subject from the beginning. It was only a matter of time. As Hardy said, "there is no permanent place for ugly mathematics."

I think it's true for physics, too. Or should be true.

I do suspect that it's somewhat just the nature of undergraduate coursework. I don't have many problems with the concepts in physics, or even the math, but somewhere where you have to bridge the two (given a situation and set up the problem) is something I find very hairy and unpleasant.

There's also mathematical physics, which I believe deals with the less ugly side of things. I think?

I just know that I'm not as "fascinated" by "real" things as I am with ideas and concepts. I went on a trip with my local SPS (Society of Physics Students) chapter, and we actually visited Fermilab and Argonne labs. Here I was looking at a particle accelerator (cool, right?) and thinking it was neat and all, but not as impressed as I should have been. For some reason I randomly thought of Euler's identity, and thought "I find that little equation more impressive than this gigantic particle smashing machine. What's wrong with me?"

-DaveK

 

[dcpo]

There's a lot of attractive mathematics connected to computer science. Many alternative logics, for example, find uses in modelling the behaviour of programs, and there are some very nice algebraic and topological approaches to soundness and completeness proofs for these.

 

[Dr. Physics]

I am very much interested in Computer Science and Mathematics. Which program to choose at undergraduate level. I heard of Applied Mathematics, Computational Science and Mathematics, also Computer Science. Which is best among all of these.
I want to go from Undergrad to Doctorate.
Thanks :)

 

[dkotschessaa]

There's a lot of attractive mathematics connected to computer science. Many alternative logics, for example, find uses in modelling the behaviour of programs, and there are some very nice algebraic and topological approaches to soundness and completeness proofs for these.

So even topology is applicable? (Sorry for super basic questions, but it's where I am.

 

[dcpo]

A sketch of how it works in this case is that certain logics have algebraic counterparts (usually based on Boolean algebras or lattices), and there are various ways, largely based on Stone and Priestley dualities, for interpreting these algebraic structures as topological spaces (possibly equipped with some extra structure, like the ordering in Priestley duality), and the maps between them as continuous functions (usually with additional properties) with domain and range switched. I know that these dualities have been used to prove completeness results for various logics, though I can't give details off the top of my head.

 

[bublik13]

Many thanks for the input, mathwonk.
Also, I noticed the many books listed in the beginning of the thread. I found previews of some of them on the internet, and they seemed a little too complex for my high-school level skills. Can you recommend any mathematics-related books that would be interesting for someone who is passionate about math without much knowledge of university math? What I'm looking for is not a textbook, but something that can be read (analyzed and worked-on) recreationally as well. I just want to develop skills beyond the curriculum, and know things that don't require a mere substitution into, or use of, a formula (so essentially, skills which would benefit me in a contest-type problem). So I'm not looking for a textbook, and nor am i looking for a novel, but a mix of the two. My skills extend to enriched (AP) grade 12 calculus (canadian curriculum) So any recommendations would be greatly appreciated, and thanks once again for your help on the previous question I posted.

 

[A. Bahat]

Try What is Mathematics? by Courant and Robbins.

 

[mathwonk]

exactly my choice.

 

[Cod]

Try What is Mathematics? by Courant and Robbins.

A great book. I've read through it and Stewart's Concept of Mathematics numerous times. I highly recommend both for individuals jumping into higher maths.

 

[bublik13]

Wow, "What is Mathematics?" by Courant and Robbins it is. Thank you all very much for the suggestion. I have also been recommended "Lessons in Geometry" by Jacques Hadamard (ISBN 0821843672). In the book description, it states "The original audience was pre-college teachers, but it is useful as well to gifted high school students and college students, in particular, to mathematics majors interested in geometry from a more advanced standpoint." so I think that this book would be suitable for me.
Can any of you confirm the good things I heard about this book? :)
Thanks in advance.

 

[thrill3rnit3]

Any knowledgeable folks in here have any idea about the algebra group at UCLA? I'm looking at it as one of my potential graduate schools and am wondering if their algebra research is thriving..

 

[Monocles]

I am not at UCLA but I went to the prospective grad students open house a few weeks ago and talked to a few of the algebraists there. I asked Haesemeyer about algebraic K-theory and he said that UCLA might be the best place to do K-theory, since besides the K-theorists at UCLA there are strong K-theorists nearby at USC as well. So there is a lot of interaction between the two departments. I also talked to an algebra grad student who said that there were plenty of people interested in algebra and algebraic geometry (including more than one person working in motives). In any case it seemed like a great place to do algebra (which I began considering much more strongly after talking to two extremely enthusiastic algebra professors, even though I have always been more interested in geometric/topological things).

 

[A. Bahat]

bublik13, while I am not familiar with the book you mentioned, it looks good. Hadamard was a great mathematician (he was the first* to find a proof of the prime number theorem) so I would expect anything by him to be valuable.

*de la Vallée Poussin discovered a proof independently at the same time.

 

[morphism]

Any knowledgeable folks in here have any idea about the algebra group at UCLA? I'm looking at it as one of my potential graduate schools and am wondering if their algebra research is thriving..

What kind of algebra? If it's with a number theoretic bent (algebraic number theory, galois representations, etc.), then UCLA would be great for that.

 

[mathwonk]

One thing I do to research the best regarded people in math is to look at the invited speakers to the ICM. The one in Hyderabad in 2010, featured Paul Balmer, algebraist from UCLA.

http://www.icm2010.in/scientific-program/invited-speakers

 

[Sina]

I can safely that towards the end of your undergraduate or atleast during your Ph.D there should be some topics that excites you (in the sense that you feel passionate towards learning, thinking and asking questions about that topic). It need not necessarily be your Ph.D topic as not all people have the chance to work exactly on their topic of interest (but something related). Basically at your Ph.D stage you should atleast heave dreams about studying a certain topic when you become a faculty :p

 

[Monocles]

One thing I do to research the best regarded people in math is to look at the invited speakers to the ICM. The one in Hyderabad in 2010, featured Paul Balmer, algebraist from UCLA.

http://www.icm2010.in/scientific-program/invited-speakers

This was one of the professors I spoke to. He works on tensor triangulated categories which, if I understood correctly, allows you to prove things about algebraic geometry, motives, noncommutative geometry, symplectic geometry, and more, all at once. Crazy powerful stuff. This is his survey on the topic: http://www.math.ucla.edu/~balmer/research/Pubfile/TTG.pdf

 

[qspeechc]

Hi everyone.

Sorry to cut in on your discussion like this and change the topic.

I graduated a few years ago with my bachelors in maths, and have been working since, and recently I have been reviewing the maths I did at university. I have worked through Herstein's algebra book, and I wanted to know if I should work through Artin, since everyone talks so highly of it. My aim is eventually to read grad-level books (my interest isn't in algebra, but everyone needs to do graduate algebra, right?).

But here's the thing. I really don't have money to spare, and even used copies of Artin are expensive (for me at least). Instead of getting another book on undergrad algebra, which I already know, I'd rather spend the money on a book on another topic, maybe even Lang's algebra book.

So, do you think Artin is really worth getting, or should I get some other book?

 

[mathwonk]

well Artin's book is better than Herstein's in my view, but if you are poor, why not take a look at my free notes for math 843-4-5 on my page

http://www.math.uga.edu/~roy/\\

I am not in Artin's league, but my book has helped some pretty good people.

 

[qspeechc]

well Artin's book is better than Herstein's in my view, but if you are poor, why not take a look at my free notes for math 843-4-5 on my page

http://www.math.uga.edu/~roy/\\

I am not in Artin's league, but my book has helped some pretty good people.

Well, there's no need to put it quite like that

What I meant was, if you think it's really worth it, then I guess I'll save up for Artin, and I'll just have to postpone on getting some other book.

In the mean time I'll take a look at your notes, thanks.

 

[Nano-Passion]

I;m taking a calculus II course. Partial fractions seem very unmotivated and ugly to me. But I'm sure there has to be some beauty behind it. Can anyone link me to the underlying theory of it all?

 

[Robert1986]

I agree that partial fractions are ugly in the sense that they can be a pain. But, I don't get the unmotivated part. Aren't you decomposing a complicated quotient into the sum of several easier quotients that you can integrate? That is the motivation.

As for underlying theory, I really think it is just algebraic manipulations, like partial fractions or something.

 

[Nano-Passion]

I agree that partial fractions are ugly in the sense that they can be a pain. But, I don't get the unmotivated part. Aren't you decomposing a complicated quotient into the sum of several easier quotients that you can integrate? That is the motivation.

As for underlying theory, I really think it is just algebraic manipulations, like partial fractions or something.

Well unmotivated because they seem to just come out of nowhere. The book I'm using says do this and this and you will get this. But I don't blame it, deriving it seems tricky-- you need a bunch of clever manipulations that aren't so straightforward.

 

[DrummingAtom]

I;m taking a calculus II course. Partial fractions seem very unmotivated and ugly to me. But I'm sure there has to be some beauty behind it. Can anyone link me to the underlying theory of it all?

I agree with Robert1986 that the motivation is simply just decomposing it into a useful form. If you want a more general form of it check out wikipedia:

http://en.wikipedia.org/wiki/Partial_fraction