Taking cross disciplinary subjects during Math graduate school

[Group_Complex]

Hello. While I am not heading off to graduate school for a little bit, I am interested in persuing a phd in pure mathematics (Geometry is a broad interest of mine). I would also be interested in taking one or two mathematically inclined graduate courses from the Physics department (String Theory, Quantum Field Theory or such) due to both an interest in theoretical physics and a feeling that it may help me understand the applications of Geometry to physics. I am aware it will depend on the school, but is this a possibility, will I still be able to take graduate courses in Pure Mathematics as well as one or two courses from the Physics department? Is this unusual?

Also out of curiosity, do people learn much from Graduate courses, or did you find them more of a chore, something to be finished as quickly as possible? Does there come a point where teaching yourself is more effective than attending courses or seminars?

 

[Monocles]

No, that is not unusual at all. In fact many math departments offer courses in mathematical/theoretical physics, and some programs with a 'minor' requirement will accept physics courses as your minor. But of course there is no general rule.

 

[homeomorphic]

I actually used two physics classes for my "breath requirement" (I'm now ABD).

It took me a very long time to get to the point where I could really connect what I knew about math to what I knew about physics in a very substantial way, and I'm still just beginning that process. Learning physics from a very mathematical point of view is harder than you'd think. Or at least harder than *I* thought. It's easy to say, "geometry has applications to physics", but actually doing research on it is another story. I wanted to do research on applications of topology to physics, and, my thesis is sort of close to that (it appears I was even half-way scooped by some string theorists), but it ended up being pretty pure math.

 

[Group_Complex]

This is good to know. Previously my aim was to go to graduate school for theoretical physics, yet the more physics I study the more I feel it lacks rigor, and sometimes even the beauty you find in pure mathematical fields such as Number Theory. To me, mathematics feels closer to the "truth" whatever that may be.

So while I am interested in learning more mathematical physics, I am also considering research in a pure field such as Algebraic Geometry or Number theory. Would taking some Physically inclined subjects detract from study in fields such as these? I am aware that most grad schools have a breadth component, and I may consider writing a minor thesis in a more physically relevant field (Mathematics of string theory or something similar).

Also, I am only a freshman at the moment, but I intend to start taking graduate level courses next year, as I have almost exhausted the undergraduate ones. However, some fields (Algebraic number theory) are weaker at my school, so would a text suffice to learn this?

 

[Robert1986]

Well, we HAVE to take some classes outside of the Math Department for the Math Ph.D. (I think about 3 to 5.)

But, how have you exhausted the undergrad courses? You said you are a freshman, are you in your first year? This seems absolutely implausible to me that you have taken almost all the undergrad math courses AND your school has a Math grad program.

The reason that I mention this is that I wouldn't rush into taking grad courses if you have only 1 year of math under your belt. At best, this is 1 (school) year of doing proof-based math. I really don't think people can gain the mathematical maturity it takes to do well in grad courses in one year.

So, if you have only one year of proof-based math classes, I'd wait another year before you start taking grad-level courses.

 

[Group_Complex]

Well, we HAVE to take some classes outside of the Math Department for the Math Ph.D. (I think about 3 to 5.)

But, how have you exhausted the undergrad courses? You said you are a freshman, are you in your first year? This seems absolutely implausible to me that you have taken almost all the undergrad math courses AND your school has a Math grad program.

The reason that I mention this is that I wouldn't rush into taking grad courses if you have only 1 year of math under your belt. At best, this is 1 (school) year of doing proof-based math. I really don't think people can gain the mathematical maturity it takes to do well in grad courses in one year.

So, if you have only one year of proof-based math classes, I'd wait another year before you start taking grad-level courses.

I took calc 1-3, differential equations, linear algebra in High School.

 

[Group_Complex]

Although I took math olympiads in High School which were proof based, I do agree that my mathematical maturity is not as strong as it should be.
I am currently taking Real Analysis, Abstract Algebra and Complex analysis (It is a bit odd to do both Real and Complex at the same time, but the course convenor said I could), as well as Classical mechanics (Lagrangian, Hamiltonian) for physics. I may focus on Physics courses in the next year.

 

[Robert1986]

I was going to add to my above post something like "it will be hard unless you are VERY talented." I see now that you might actually have a great deal of natural talent, if you managed those classes in high school.

I still caution against taking grad-level courses. Not because you are not talented enough, but because I still think it is hard (even for an uber-mensch) to have developed the level of mathematical maturity it takes to do grad-level work. Because, you have probably taken 4 or so proof-based math classes, right? (I still don't see how this can possibly almost exhaust your undergrad courses, but perhaps you took way more than 4 classes.) The grad-level courses are much, much different. They aren't just harder (but they are harder) they are different in a way that I can't explain.

I'm an undergrad now, but I took the grad level Algebra course last semester (well, I had to drop because stuff changed at my job, but I wasn't doing too good in it, either, but anyway) and it was tough.

 

[Group_Complex]

I was going to add to my above post something like "it will be hard unless you are VERY talented." I see now that you might actually have a great deal of natural talent, if you managed those classes in high school.

I still caution against taking grad-level courses. Not because you are not talented enough, but because I still think it is hard (even for an uber-mensch) to have developed the level of mathematical maturity it takes to do grad-level work. Because, you have probably taken 4 or so proof-based math classes, right? (I still don't see how this can possibly almost exhaust your undergrad courses, but perhaps you took way more than 4 classes.) The grad-level courses are much, much different. They aren't just harder (but they are harder) they are different in a way that I can't explain.

I'm an undergrad now, but I took the grad level Algebra course last semester (well, I had to drop because stuff changed at my job, but I wasn't doing too good in it, either, but anyway) and it was tough.

No, I do not think I have a great deal of natural talent at all (I did terrible in the math competitions I mentioned), I just had good opportunities in high school. I took the Putnam math competition last year and got about 10, while Evan O'dorney a freshman at Harvard was a putnam fellow (probably the highest score), I find that level of talent beyond me.

I have not taken all the undergraduate courses in mathematics, but I was thinking of taking Graduate level Algebra or Analysis to see how it was. There are many courses that are for both "graduates and undergraduates", such as Differential geometry, which I also intend to take (I have been doing some reading on the subject and it is fascinating).
However I am also interested in physics, but I do not do as well in physics as I would like, I am not sure why, I manage to get A's, but it is quite tough developing the intuition, and very often it seems I develop the wrong intuition, which frustrates me greatly. Classical mechanics, being more mathematical is much better, but I was quite slow in previous physics courses (I took level freshman physics in HS), often applying the wrong principle in exams, despite thinking I understood the principle perfectly. I am not sure if this is due to a lack of preperation (I was quite lazy when it came to study in HS, not even doing all the textbook problems, only those set for homework) or a lack of talent in physics.

 

[homeomorphic]

This is good to know. Previously my aim was to go to graduate school for theoretical physics, yet the more physics I study the more I feel it lacks rigor, and sometimes even the beauty you find in pure mathematical fields such as Number Theory. To me, mathematics feels closer to the "truth" whatever that may be.

I always regret not being in close enough touch with reality, myself. I am really a physicist at heart, albeit a very mathematical one. In the end, reality ought to matter more than a mathematical fantasy world. The mathematical fantasy world can be fun and applicable to reality, though. Math is overwhelmingly huge; physics is overwhelmingly huge. But math just leaves me wondering what the point of doing something so endless and hard is if it doesn't connect back up to reality at some point. Even if it's interesting. It's easy to argue that mathematicians don't need to worry about applications, but an abstract argument, even if you agree with it completely, is not always completely psychologically convincing. You may not care about these things now, but after doing math, math, math, math, for several years, the issue can come up. At least with physics, you know that you are dealing with reality (probably, you have similar issues in physics).

So while I am interested in learning more mathematical physics, I am also considering research in a pure field such as Algebraic Geometry or Number theory. Would taking some Physically inclined subjects detract from study in fields such as these? I am aware that most grad schools have a breadth component, and I may consider writing a minor thesis in a more physically relevant field (Mathematics of string theory or something similar).

Writing something meaningful about string theory is pretty hard, much more so if you're not a specialist in it. This is what you are up against, and this is just the math side:

http://superstringtheory.com/math/index.html

I know all the math there, except maybe 3 subjects towards the very end, and it would still take me considerable effort to learn any string theory, let alone write something about it.

Then, on the physics side, you have to know a little GR and QFT. QFT is extremely daunting. Picking up string theory on the side while you do a PhD is nearly impossible in my judgement, if you are doing something demanding like algebraic geometry. You have to specialize pretty drastically--that's something I've learned the hard way.

That's what I'm talking about when I say it's harder than you'd think. I have a number theory grad student friend who is interested in interactions between number theory and QFT, though. Seems to work for him. I'm not sure if that's his main research or just a diversion.

Graduate classes are not all uniformly difficult. Depends what university you are at, who the prof is, and so on. It's a good idea to take a couple (and more than a couple if you want to get into one of the top schools) to prepare yourself for grad school.

 

[Group_Complex]

Thanks for the advice Homeomorphic. May I ask why you did not attend graduate school in theoretical physics if you are a physicist at heart?
I think physical models are mathematical in nature, mathematics allows us to understand this vast set of possible worlds, while experiment is used to pinpoint which world is most akin to our "reality". At least that is my limited view on the issue. Would it be easier for a pure mathematician to understand work done in fields such as Quantum gravity, than it would be for a physicist to understand Algebraic Geometry? Some people are of the view that string theorists are not physicists but rather mathematicians, and if you look at Ed Witten for instance, his work seems more mathematical than physical in nature (at least from what I understand of his work). This is why I love mathematics, it is behind everything, the beauty of pure mathematics helps us understand our reality better! I don't think I would be able to turn my back on pure mathematics to work soley in theoretical physics, which is why I was interested in gaining some physical understanding for my mathematical studies.

 

[homeomorphic]

May I ask why you did not attend graduate school in theoretical physics if you are a physicist at heart?

I said a very mathematical one. Therefore, physicists usually do things in a way that doesn't really satisfy me. In undergrad, when subjects used to be simple, the way math classes were taught appealed to my independent-minded style. I don't like to have to take anyone's word for things. Unfortunately, sometimes taking someone's word for something gets the job done faster. So, in the end, it turned out, most mathematicians also do things in a way that doesn't really satisfy me. A lot of the math and physics out there is just a mess, conceptually, in my opinion. Therefore, I'm more interested in straightening out the mess than in establishing new results, physically or mathematically. It would be different if mathematicians and physicists hadn't made such a mess of things. This is part of my problem in both fields. I am so arrogant as to think I know better than most of the mathematics and physics community, not in terms of what's true, but in terms of style. The reason I can maintain this arrogance is simply that my version of math and physics is beautiful and theirs, all too often (but not always), dull and lifeless. At least to me. Call it a matter of taste. A healthy disrespect for authority is a trait required of any serious thinker.

I think physical models are mathematical in nature, mathematics allows us to understand this vast set of possible worlds, while experiment is used to pinpoint which world is most akin to our "reality". At least that is my limited view on the issue.

Yes, but what kind of math? You don't always need PhD level math to get the job done--in fact, I would think that is the exception, rather than the rule.

Would it be easier for a pure mathematician to understand work done in fields such as Quantum gravity, than it would be for a physicist to understand Algebraic Geometry?

Depends on which physicist or which mathematician. Those are just labels, after all. It appears physicists have been faster to learn math than the reverse.

Some people are of the view that string theorists are not physicists but rather mathematicians, and if you look at Ed Witten for instance, his work seems more mathematical than physical in nature (at least from what I understand of his work). This is why I love mathematics, it is behind everything, the beauty of pure mathematics helps us understand our reality better! I don't think I would be able to turn my back on pure mathematics to work soley in theoretical physics, which is why I was interested in gaining some physical understanding for my mathematical studies.

I wouldn't turn my back on either physics or pure math.

 

[SophusLies]

But math just leaves me wondering what the point of doing something so endless and hard is if it doesn't connect back up to reality at some point.

That's the reason I'm in physics grad school and not math. I was an algebra/logic junky during my undergrad but in my last quarter I hit a wall. I felt like that stuff would never be useful and nearly failed that course in my final quarter from loss of motivation. In retrospect, I think I pursued more algebra/logic related classes because I felt like that deep abstraction would eventually pay off (somehow) in physics. Unfortunately, I was wrong. I should have been taking more topology and geometry classes but ended up only taking one course in those subjects. I think things would have worked out quite differently if that were the case.

By the way, homeomorphic, what is your research area? I'm guessing it's something topology related from your handle.

 

[homeomorphic]

That's the reason I'm in physics grad school and not math. I was an algebra/logic junky during my undergrad but in my last quarter I hit a wall. I felt like that stuff would never be useful and nearly failed that course in my final quarter from loss of motivation. In retrospect, I think I pursued more algebra/logic related classes because I felt like that deep abstraction would eventually pay off (somehow) in physics.

Abstraction might pay off in physics. The big area here where that seems like it might be about to happen is in topological quantum field theory, which has strong ties to string theory, loop quantum gravity, and from the more useful side, anyonic condensed matter systems. However, I think that in order to process information effectively, our brains can't handle too much abstraction, so the abstraction may be there, but someone needs to bring it down to Earth in order to make it easier to learn and think about.

Unfortunately, I was wrong. I should have been taking more topology and geometry classes but ended up only taking one course in those subjects. I think things would have worked out quite differently if that were the case.

Maybe. But I think logic might have some implications for quantum computing, even though it's a bit different from classical logic. It's interesting to think of physical processes as computations. From that point of view, logic does seem like it could be relevant.

By the way, homeomorphic, what is your research area? I'm guessing it's something topology related from your handle.

Yes, topological quantum field theory at the moment. In the future, my interests are heading towards topological quantum computation, developing a deeper mathematical understanding of QFT, and more topological quantum field theory stuff.

 

[Group_Complex]

Homeomorphic, in your opinion, do you think most ground breaking theoretical physics, such as String Theory, is done by people who are considered more mathematicians than physicists? I realize the title is quite arbitary, but when I look at the work of say Ed Witten it seems much more focused on the mathematics rather than physical observation. I was surprised to learn that Ed Witten went to graduate school in physics rather than mathematics, considering most of his work seems to have strong connections with pure mathematics. To what degree has Witten mastered pure mathematics or rather has he provided physical intuition for pure mathematical topics?
If one were interested in the very deeply mathematical physics Witten does, would it be better to attend math or physics grad school? The former I would have thought, but I can find only a few individuals who seem to describe themselves as mathematicians who also have what one would call a deep understanding of theoretical physics...

 

[homeomorphic]

Homeomorphic, in your opinion, do you think most ground breaking theoretical physics, such as String Theory, is done by people who are considered more mathematicians than physicists?

String theorists are more like mathematicians. However, I don't know if I would consider string theory to be groundbreaking as physics. Too hard to test it, as of now, and it doesn't look promising, but I'm not an expert on either string theory or particle physics. As math, it's groundbreaking. And some of that math is trickling into other areas of physics.

I realize the title is quite arbitary, but when I look at the work of say Ed Witten it seems much more focused on the mathematics rather than physical observation. I was surprised to learn that Ed Witten went to graduate school in physics rather than mathematics, considering most of his work seems to have strong connections with pure mathematics. To what degree has Witten mastered pure mathematics or rather has he provided physical intuition for pure mathematical topics?

Witten knows math better than most mathematicians and physics better than most physicists.

If one were interested in the very deeply mathematical physics Witten does, would it be better to attend math or physics grad school? The former I would have thought, but I can find only a few individuals who seem to describe themselves as mathematicians who also have what one would call a deep understanding of theoretical physics...

My instinct is to say math. Most physics grad students don't know what a manifold is, in my experience, which is pretty basic for most math people. I guess they sort of work with manifolds without realizing it. But it's kind of a dual track to do string theory. You can get a phd in mathematical physics. Very few people do it because it's almost like getting a degree in both.

 

[Group_Complex]

Witten knows math better than most mathematicians and physics better than most physicists.

I don't doubt Witten knows quite a bit of mathematics in fields related to theoretical physics, but I think it would be unsubstantiated to claim he knows math better than most mathematicians. How much Number theory would he know? How much Algebraic Geometry? Category theory? I don't know, he very well may know these areas quite well (There are applications for Algebraic Geometry in Physics are there not?), but I do not think that it is possible (even for the most mathematically talented individual since Newton) to learn that much mathematics whilst simultaneously working in theoretical physics (afterall he attended Physics graduate school and was a history major in college, so nearly all his free time would have to have been spent studying pure mathematics).

 

[chiro]

I don't doubt Witten knows quite a bit of mathematics in fields related to theoretical physics, but I think it would be unsubstantiated to claim he knows math better than most mathematicians. How much Number theory would he know? How much Algebraic Geometry? Category theory? I don't know, he very well may know these areas quite well (There are applications for Algebraic Geometry in Physics are there not?), but I do not think that it is possible (even for the most mathematically talented individual since Newton) to learn that much mathematics whilst simultaneously working in theoretical physics (afterall he attended Physics graduate school and was a history major in college, so nearly all his free time would have to have been spent studying pure mathematics).

You also have to be aware that his father was a physicist with particular attention to General Relativity which I'm sure would have helped intuition of physics and subsequent mathematics greatly to say the least.

 

[Group_Complex]

You also have to be aware that his father was a physicist with particular attention to General Relativity which I'm sure would have helped intuition of physics and subsequent mathematics greatly to say the least.

I don't deny that, but I think it is a bit unsubstantiated to claim Witten knows more Mathematics than most Mathematicians.

 

[chiro]

I don't deny that, but I think it is a bit unsubstantiated to claim Witten knows more Mathematics than most Mathematicians.

The quote wasn't directed at the first part of your post, but rather the last part of your post.

 

[homeomorphic]

I don't doubt Witten knows quite a bit of mathematics in fields related to theoretical physics, but I think it would be unsubstantiated to claim he knows math better than most mathematicians. How much Number theory would he know? How much Algebraic Geometry? Category theory? I don't know, he very well may know these areas quite well (There are applications for Algebraic Geometry in Physics are there not?), but I do not think that it is possible (even for the most mathematically talented individual since Newton) to learn that much mathematics whilst simultaneously working in theoretical physics (afterall he attended Physics graduate school and was a history major in college, so nearly all his free time would have to have been spent studying pure mathematics).

It's far from unsubstantiated. Witten was awarded the fields medal. He has made massive contributions to topology that are more significant than those made by most topologists. He talked to Atiyah and learned a lot that way. That's one of the best ways to learn. Find an expert and talk to them. Much more efficient than reading.

There is a story about Michael Larsen, who is kind of a big shot, himself now. Witten told Faltings (another fields medalist) that he wanted to learn about such and such. Faltings told him to talk to his grad student, Larsen. Like a vampire, Witten sucked everything that Larsen knew about the subject out of him in a very short period of time. Larsen is an algebra/number theory guy, by the way.

 

[Group_Complex]

It's far from unsubstantiated. Witten was awarded the fields medal. He has made massive contributions to topology that are more significant than those made by most topologists. He talked to Atiyah and learned a lot that way. That's one of the best ways to learn. Find an expert and talk to them. Much more efficient than reading.

There is a story about Michael Larsen, who is kind of a big shot, himself now. Witten told Faltings (another fields medalist) that he wanted to learn about such and such. Faltings told him to talk to his grad student, Larsen. Like a vampire, Witten sucked everything that Larsen knew about the subject out of him in a very short period of time. Larsen is an algebra/number theory guy, by the way.

Very ineresting, I suppose Witten was talking to Larsen about things unrelated to mathematical physics, out of sheer curiosity? If that is the case, I admire Witten even more than I did before.

 

[chiro]

I don't doubt Witten knows quite a bit of mathematics in fields related to theoretical physics, but I think it would be unsubstantiated to claim he knows math better than most mathematicians. How much Number theory would he know? How much Algebraic Geometry? Category theory? I don't know, he very well may know these areas quite well (There are applications for Algebraic Geometry in Physics are there not?), but I do not think that it is possible (even for the most mathematically talented individual since Newton) to learn that much mathematics whilst simultaneously working in theoretical physics (afterall he attended Physics graduate school and was a history major in college, so nearly all his free time would have to have been spent studying pure mathematics).

Also one thing that I wanted to say that was that these are different times in education.

Look at what happens with things like the internet and Physics Forums. You can ask an expert a question and get an answer very quickly. It's not necessarily the same as speaking face to face with an expert over a few hours but this is what is happening.

Because of this, I imagine that in the future, you will have some bright, motivated, hard working people that are in an environment where they have access to information that has been so refined, so clarified, in such a way that people can learn more in a week than they may have learned in half a year.

Back a little over a hundred years ago, you could have been an expert with the equivalent of a bachelors, maybe even with a masters degree at the very most. But even though we have an explosion of information and knowledge nowadays, we also have an explosion in the way all of this ends up being organized and refined and it's this particular attribute (the refinement and organization) that will allow people to learn things in a way that we could never anticipate right now.

When that happens, I agree that those people will never learn everything in a particular field, but they will learn a hell of a lot more both in depth and in speed than we learn now.

 

[homeomorphic]

Very ineresting, I suppose Witten was talking to Larsen about things unrelated to mathematical physics, out of sheer curiosity? If that is the case, I admire Witten even more than I did before.

No, my guess is that it probably was relevant to string theory in some way and that's why he needed to know it. Not to say that Witten doesn't have mathematical curiosity. Lately, he's been working on Khovanov homology. I saw him give a talk about it last summer. It may be physically interesting, but in a video I saw, he seemed to indicate that he was interested in it mathematically. He wants to explain things more conceptually than a lot of the mathematicians. So, when I heard that, my admiration was increased.

 

[homeomorphic]

Look at what happens with things like the internet and Physics Forums. You can ask an expert a question and get an answer very quickly. It's not necessarily the same as speaking face to face with an expert over a few hours but this is what is happening.

Very true. My own view of physics and math would be VERY different if it were not for John Baez's website. I basically consider myself to be one of Baez's disciples. He's influenced me more than my own adviser.

 

[Group_Complex]

homeomorphic, as you seem very knowledgeable in this regard, could you recommend some good graduate schools which have a strong focus on Geometry, Topology and how they relate to mathematical physics?
Despite being the top grad schools in the US Harvard and Princeton don't seem to have much in the way of this, at least as far as I can find, both seem to have stronger focus on other areas of mathematics, but i very well may be wrong. I have heard Stony Brook is quite good in this regard, and I have not done much research into MIT. Of course I am by no means assured places at any of these schools, but I would be lying if I said they were not a goal.

 

[homeomorphic]

homeomorphic, as you seem very knowledgeable in this regard, could you recommend some good graduate schools which have a strong focus on Geometry, Topology and how they relate to mathematical physics?

That's a narrow enough target, you could narrow it down to looking for specific people, rather than departments. But it's good to have a plan B if you target a specific person to work with.

Berkeley has a string theorist in the math department. UC Davis has some TQFT people. University of Washington has a couple string theorists, I think. Crane and Yetter are at Kansas State. Baez is coming back to UC Riverside, but he sort of quit that kind of physics to help fight global warming. Still, might be worth looking into. Louis Kauffman is at UIC. UC San Diego has Justin Roberts. I think Stanford has some people. UCSB might be an option. That's not an exhaustive list or anything.

Despite being the top grad schools in the US Harvard and Princeton don't seem to have much in the way of this, at least as far as I can find, both seem to have stronger focus on other areas of mathematics, but i very well may be wrong.

There seems to be some activity in TQFTs at Harvard. Jacob Lurie seems to have solved the Baez-Dolan conjecture, which is huge. His style seems very, very abstract, though. You can look at his webpage. He's more into algebraic topology, it seems, but that stuff is what allowed him to solve the conjecture.