Masters in Pure Mathematics (Geometry, Topology) before Theoretical Physics Phd?

[Functor97]

I am a Mathematics and Physics double major, currently in my second year. I really enjoy both subjects, but my interests are progressing towards Theoretical physics/mathematical physics. My academic goal is to improve my understanding of how the universe works and thus I would like to pursue a Phd in the area of High energy Physics or the mathematical basis of these theories.

Despite this, I do enjoy the rigour and proofs of pure mathematics and would like to pursue a masters in a geometric area (such as differential geometry or Topology) before starting my Phd. Is this unrealistic? I am aware that this will take time, but i feel that it will be beneficial. My main motivation for doing this is to gain an understanding of the mathematical proofs and framework, rather than just taking a mathematician's word for it and using them as tools in physics. I would also like to gain the experience of working in a field of pure mathematics just for self fulfillment. For instance John Baez is a mathematical physicist, but has quite a broad understanding of mathematics, and i wish to emulate this.

I have been told that mathematical physicists are better off being trained in mathematics departments, and this leaves me at a loss. I understand that no one can attain an understanding of all the areas of mathematics and physics, let alone research expertise, but i would enjoy being a theoretical Physicist with a deep understanding of the mathematical objects i am working with, maybe even using the intuition i develop to guide pure mathematicians (as Witten does).

 

[Functor97]

If no one understood question, i will rephrase it as; Is this an unusual thing to get a pure masters before working in theoretical physics?

 

[Nano-Passion]

If no one understoof question, i will rephrase it as, Is this an unusual thing to get a pure masters before working in theoretical physics?

Its worded fine, let's just wait for the more capable people to answer. =D

 

[nlsherrill]

Looks like this person did what you are thinking of:

http://www-personal.umich.edu/~jbourj/

If you look on that page, it says he got a masters in mathematics 1 year after he graduated with a B.S. in physics and math. Also, if you look at the solutions to physics problems link on the left side of that page, then go to the quantum field theory course homework, he claims to have taken quantum field theory as an undergraduate sophomore.

This probably explains how he got his mathematics masters in 1 year. He had already completed all the mathematics masters courses by the time he got his B.S.(I am guessing), and spent 1 year on research. Seems like a fine idea too me. You wouldn't want to be away from physics for too long...

Emailing him would be a good idea.

 

[micromass]

The problem with your plan is of course that you're going to take 1 or 2 years without doing physics. How much of physics will you have forgotten after that 2 years?? It's something to think about.

Doing a masters in math is cool, I'm sure it is very useful to whatever field you go into. But the risk is that you might forget a lot of physics in the meanwhile...

 

[nonequilibrium]

This probably explains how he got his mathematics masters in 1 year.

No, it's because he did the math tripos III in Cambridge which is a one-year taught math master. (and damn does that site make me feel like an under-achiever)

@ OP: good question, I'm thinking about the same! (and I'm applying to universities soon enough, so I have to make up my mind too) I'm actually tending to the opposite: first physics, then math, cause then I'd better know what math is crucial in upper end physics (not that I'd only do math for physics, of course, but then again, it helps to have a guiding line)

 

[SophusLies]

I would look into some schools because I have seen more than a handful require a graduate student to take 2-3 classes outside of their program, i.e. math, engineering.

As micromass pointed out, remembering physics might be a struggle if you're just doing math. I spent 4 years in industry before going back to grad school and it took a lot of self studying to get back up to a comfortable level.

 

[Functor97]

Sophus, what area is your Phd in?

Thanks for the advice guys. Personally i lean towards a mathematical approach rather than a more phenomenology based one, thus my interest in mathematics graduate school rather than physics. All courses in a mathematics grad programme look interesting, whereas Field theory and strings, Relativity and Quantum electrodynamics are the major interests in a physics degree. The problem being that many programmes require a wide subject selection to supplement your chosen field, and personally i find areas such as Astronomy, Optics, Plasmas and Superconductors of less interest.

Furthermore I have been studying journals from each subject, and I have noted that physics journals (even the more mathematical ones) are more calculation intense, rather than focused upon rigour and logic. The "hand wavy" approach certainly does not appeal to me.

 

[deRham]

My academic goal is to improve my understanding of how the universe works and thus I would like to pursue a Phd in the area of High energy Physics or the mathematical basis of these theories.

I do encourage you to go on to physics grad school. I think your plan to get a master's in pure math is fine, as long as you stay in touch with physics during that period - take a class or two, do some research, read the important papers, etc.

I think it would be a slight shame if you abandoned physics grad school, because from your posts, you seem ripe for being "sucked into the pure math world," and while that's fine overall, you'll find the community of pure mathematicians, even those who do things in mathematical physics, often don't really do physics. I think you may "lose the will" to do physics. And when you have such a fire to learn about the universe, that would at least strike me as a shame. The thing is that pure math is an attractive way of thinking about things, and to be honest, someone like you will find you'll have to learn a TON of both worlds. This can make just sitting and doing mathematics and feeling like you "almost" did physics an attractive option. But you might regret that later.

I say go for taking some time to study geometry/topology deeply, but make sure you stay grounded in physics (and for the sake of avoiding too many excursions, let's define physics as broadly as you want it...)

I think while you may make the argument that physics "almost is mathematics" as you have done before, we can probably both agree it's quite possible to lose track of physics while doing mathematics, and that is, to be clear, what I caution about.

 

[deRham]

are more calculation intense, rather than focused upon rigour and logic.

In my world, there is no such thing as calculation without rigor :) I mean, the whole point of "calculating" is to give some details right?

However, I do see how it may not be so in all worlds.

 

[nonequilibrium]

Ah shoot this thread has made me doubt my own plan! Now I'm also considering doing a master in math first...

 

[EternityMech]

just do a masters in theoretical physics instead...

 

[nonequilibrium]

or mathematical physics

 

[Functor97]

I do encourage you to go on to physics grad school. I think your plan to get a master's in pure math is fine, as long as you stay in touch with physics during that period - take a class or two, do some research, read the important papers, etc.

I think it would be a slight shame if you abandoned physics grad school, because from your posts, you seem ripe for being "sucked into the pure math world," and while that's fine overall, you'll find the community of pure mathematicians, even those who do things in mathematical physics, often don't really do physics. I think you may "lose the will" to do physics. And when you have such a fire to learn about the universe, that would at least strike me as a shame. The thing is that pure math is an attractive way of thinking about things, and to be honest, someone like you will find you'll have to learn a TON of both worlds. This can make just sitting and doing mathematics and feeling like you "almost" did physics an attractive option. But you might regret that later.

I say go for taking some time to study geometry/topology deeply, but make sure you stay grounded in physics (and for the sake of avoiding too many excursions, let's define physics as broadly as you want it...)

I think while you may make the argument that physics "almost is mathematics" as you have done before, we can probably both agree it's quite possible to lose track of physics while doing mathematics, and that is, to be clear, what I caution about.

derHam, I appreciate the response! I think however that the physics community could manage without my meager "fire". I know we have discussed the mathematics-physics relationship before, so let it suffice to say that I still find both options attractive; pure mathematics and mathematical physics. I understand a choice must be made, and i am trying to adopt a realistic point of view.
By calculations i meant to say that physicists tend to take mathematics as a tool, as an engineer would use basic calculus as a tool, the process of proof is what i see as a validation of truth, and natural empiricalism/induction is the basis of physics, over logical proof...and personally i see myself as a poor experimenter...

However your point about mathematicians and physics is poignant. I often feel that they neglect many physical aspects of their work, and purify the other aspects to an extreme extent.

 

[deRham]

I think however that the physics community could manage without my meager "fire".

Sure, but I mean from the standpoint of doing what would be fulfilling for you. You might find your fire is kept alive if you keep the right company!

Yes I know what you meant by calculation :) I was just fooling around since what mathematicians call a calculation is often really what would be considered a very theoretical proof in another land.Glad what I said made some sense!

I agree someone like you probably should try a combo of theoretical and mathematical physics, but be sure never to leave the physics community. The thing is simply that mathematicians often seem to find it hard to have the energy to do physics work.

The other thing is that beneath the slick and pretty language, to do real mathematics, you have to do some equivalent of what I call "calculation" - to actually show facts about a structure, reduce it to certain cases. So perhaps as long as you tie things back to physics a lot (especially the areas you're into), perhaps you can minimize the kind of wild, mass-application of formulas.

Caution : once you no longer care what the terms in a formula mean in relation to what the physics community cares, I think you've sunk into the state of mathematicians who simply don't do physics, and pretend they do! It's fine to build things up with mathematical precision, but you still should care about some of the questions of the theoretical physics community. Hopefully, at least.I think not leaving the physics community is certainly not the same as not joining the mathematics community.

 

[Functor97]

Sure, but I mean from the standpoint of doing what would be fulfilling for you. You might find your fire is kept alive if you keep the right company!

Yes I know what you meant by calculation :) I was just fooling around since what mathematicians call a calculation is often really what would be considered a very theoretical proof in another land.


Glad what I said made some sense!

I agree someone like you probably should try a combo of theoretical and mathematical physics, but be sure never to leave the physics community. The thing is simply that mathematicians often seem to find it hard to have the energy to do physics work.

The other thing is that beneath the slick and pretty language, to do real mathematics, you have to do some equivalent of what I call "calculation" - to actually show facts about a structure, reduce it to certain cases. So perhaps as long as you tie things back to physics a lot (especially the areas you're into), perhaps you can minimize the kind of wild, mass-application of formulas.

Caution : once you no longer care what the terms in a formula mean in relation to what the physics community cares, I think you've sunk into the state of mathematicians who simply don't do physics, and pretend they do! It's fine to build things up with mathematical precision, but you still should care about some of the questions of the theoretical physics community. Hopefully, at least.


I think not leaving the physics community is certainly not the same as not joining the mathematics community.

True, calculation and proof are very similar, differing via intent rather than content. A proof aims to demonstrate something, whereas a calculation aims to "find" something.

While i find the beauty of pure mathematics highly appealing, i do feel a certain something is missing, and it saddens me to think that this is the case. I feel as though pure mathematics is artistic rather than scientific in focus (not necessarily a bad thing). I mean it certainly is scientific, but the focus of pure mathematics is mathematics for the sake of mathematics, which sometimes leads me to question the purpose of my studies. Sometimes i feel access to a mathematically platonic realm, but this is often when i see the links between areas or the "reason" something is true. I often wonder if i simply am lacking access to the higher beauties of pure mathematics, which others are so quick to forsake "reality" for.

The other thing is, my personality is that of a questioner. Since i was a young boy I would pester my elders with the dreaded "why?". That is what i believe has led me down the scientific path. As a boy i read the popular science books, but i think that warped my view of the scientific process, which i am only beginning to comprehend. While I was good at mathematics, i saw it as a competitive sport or something to apply in physics. I didn't start to recognise the pure aspect until i was about 14-15 (after reading E.T Bell and G H. Hardy of all people, both of whom admonish applied mathematics as unworthy and lacking). In my studies it is always "why?" or "how?" that have sent shivers down my spine, but in pure mathematics we have these drated things called Axioms which you cannot question by definition. The aim of mathematics is to demonstrate the properties which extend from these axioms, so we can ask "why are these two lines parallel?", but we cannot ask "why do two parallel lines never meet (in Euclidean geometry)?". I am aware of the practice of "reverse mathematics" (deriving axioms for demonstrations), and that we can create new axioms and change old ones, but to me it "feels" that we are always avoiding the deepest heart of the matter, the perfect truth if you will. Contrast this with Physics; we can ask questions and create postulates, and like the axioms of mathematics they cannot be questioned, but the underlying reality is still there. We might not yet know why certain constants take there values, or understand quantum gravity, but we cannot make axioms and say "just accept that this is the case". The mystery does not feel artificial like commanding an army, whereas pure mathematics feels like chess, a beautiful art.

I certainly love Pure mathematics, but i am not sure i can commit to the purist attitude of some mathematicians, repudiating applications and claiming that mathematics is independent of the human mind, it doesn't feel scientific or ironically, logical. However all of classical physics was so easily applied via pre existing mathematical models, that i do wonder if the purists might have a point. We use mathematical formulas for centripital acceleration to explain gravitational orbits and we prove the formula via simply calculus. Electromagnetism is a swathe of vector fields, surface integrals and complex numbers, and to explain the natural phenomena we cannot separate these purely mathematical aspects from reality, it seems reality is mathematics.

 

[MathematicalPhysicist]

I just can give my two cents tip, choose what interest you more, are you more a math-guy or a pratical-physics guy?

BTW, even if you go the pure math way you can always keep touch with the theoretical physics research which interest you, just keep in mind that this time will be taken off from spending time with your girlfriend and other mundane activities.

 

[Mépris]

One thing that I find pretty cool with Physics is if it so happens that after 3-4 years, I don't like Physics anymore, I can go do a master's in CS, Applied Math or even Neuroscience, if that's what tickles my fancy then.

I just can give my two cents tip, choose what interest you more, are you more a math-guy or a pratical-physics guy?

BTW, even if you go the pure math way you can always keep touch with the theoretical physics research which interest you, just keep in mind that this time will be taken off from spending time with your girlfriend and other mundane activities.

 

[Functor97]

I just can give my two cents tip, choose what interest you more, are you more a math-guy or a pratical-physics guy?

BTW, even if you go the pure math way you can always keep touch with the theoretical physics research which interest you, just keep in mind that this time will be taken off from spending time with your girlfriend and other mundane activities.

are all physicists practical though? I do not think the difference between a theoretical physicist and pure mathematician is that the physicist is more "practical". It may even be the reverse.

 

[MathematicalPhysicist]

It really depends where you do your research.

String Physics is really quite odd from other physics stuff, (I read that it incorporates stuff from almost any branch in pure mathematics, does it mean it's physically meaningful, I don't have a clue).

One difference is that a physicist won't always need to provide a rigorous sound mathematical proof for his claims, as long as it's physical valid.

If you're interested in string theory, you just need to know that both departments have people working in it, everyone with his own interest.

 

[MathematicalPhysicist]

Just telling it as it is.

 

[SophusLies]

Sophus, what area is your Phd in?

I just started my 2nd year so I haven't quite decided, but AMO theory looks interesting. I'm looking for a computational theory area because I have a strong background in programming. I have no desire to stay in academia so I want to grab skills and work on something neato then get back into industry after I'm done.

 

[Functor97]

It really depends where you do your research.

String Physics is really quite odd from other physics stuff, (I read that it incorporates stuff from almost any branch in pure mathematics, does it mean it's physically meaningful, I don't have a clue).

One difference is that a physicist won't always need to provide a rigorous sound mathematical proof for his claims, as long as it's physical valid.

If you're interested in string theory, you just need to know that both departments have people working in it, everyone with his own interest.

Yes, that is the primary reason I am interested in entering a mathematics department rather than a purely physical one. I think that the entirety of pure mathematics is more congruent to my interests rather than the subject matter of a physics graduate school and department. I prefer the abstract over the concrete, the theoretical over the experimental, thus i feel mathematics to be the best choice.

 

[nonequilibrium]

But maybe a theoretical/mathematical physics master is a good start: if one sees he is still not getting the rigour he longs for, one can then do a supplementary math master? (asking myself)

 

[nlsherrill]

Just remember, if you go the math direction you will understand the math more, but the physics less. Try not to get caught up in the idea that like, for example, if you understand differential equations, then you have any idea idea of how classical mechanics really works. Mathematics is certainly necessary in physics as a language, but every hour you spend on learning how to prove some arbitrary relation, you miss an hour trying to understand how angular momentum works...

Its a great idea to combine as much math and physics as possible if you plan on going the theoretical physics direction(as I plan on doing myself). They require different skills, and its good to be well rounded in both.

 

[bpatrick]

If you are interested in staying with Physics to the point of getting a PhD, I would abandon the idea of a masters in pure math. Assuming you can land a fellowship (most ideal), and have a very solid undergrad background in both math and physics ... passing your first year qualifiers should not be that much of an issue. In your spare time, you can always attend mathematics classes in the areas you are interested in (or may apply to the physics you are doing).

The courses in mathematical physics that are offered in grad school are there for a reason. They expose you to whatever mathematical tools the department feels you'll need to succeed on your way to the PhD.

Most (if not all) schools will not mind a physics PhD candidate sitting in on whatever math courses they want / have time for. You don't need to spend all your time doing proofs and the work required to complete the exams required of mathematicians, but by merely auditing and doing the reading along with lectures, you'd be surprised what you can get from the experience ... plus YOU get to decide what you're really taking away from the course since you get out of it whatever you put into it.

So if you managed to take some algebra, topology, and geometry in undergrad AND have "free time" somehow when you're a graduate physics student, why not audit some grad level algebraic topology or differential geometry or Lie group theory? All of those subjects have a good deal of cross-over into some of the physics you think you like (sorry to say "think you like" like that, but being a sophomore undergrad, you only think you like this stuff now, who knows what will change as your level of sophistication grows in both fields).

Another reason I'd stick away from the masters in math ... unless you find a program that funds masters candidates or offers them TA positions, you'll probably be spending a decent bit of money on the degree (and diverting your eventual PhD in physics by 2-3 years) ... but hey, if you're parents have your education covered or you're already on full scholarship, or you find a fellowship/TA, or you just don't care about money, then maybe the MS in math is still a good idea.

I agree with one of the other guys that posted before: if you end up going into math and pursue it to a PhD ... you're only "kinda" doing physics ... there's a reason people spend 4+ years in graduate level study of physics just to get to the point where you're a "beginning" physicist as far as academics go. Sure, as a mathematician, you may specialize in geometry or PDEs or Bifurcation analysis ... and probably will collaborate with physicists at some point or another ... you're still a mathematician, which is a very different field.

I feel for you with all this stuff, I had a tough time figuring it out myself, and it took me until I was a year into medical school to realize I needed to get out and get back to grad school for this stuff.