What is the Difference Between a Theoretical Physicist and a Mathematical Physicist?

[mgiddy911]

Just another quick point, What about at some point in your research as a mathematical physicist it comes time for you to publish your research, only at that point you realize that while you spent every second of your college career studying every topic in math and physics, you never learned how to write papers properly. English, History, and Philosophy classes will almost all require you to write multiple essays, which is always beneficial, no matter what career you pursue.

 

[^_^physicist]

To hrc969: I understand that his schedule isn't going to kill him by any means, I take 5+ classes a term, tutor, and teach (as a requirement of one of my classes), and I still have time to do things as well. When I stated reasonable, I had intended it to come acrossed as reasonable in the sense to allow time to take advantage of elective credits outside of the math and physics departments...which most universities and colleges do require to earn a B.S. / B.A. Oh and I still feel that some of the courses on the list are redundent if they have been studied before by the OP.

To the OP:
Particularly redundent Calc are 2 and Linear Algebra II, as it has been my experance that bulk of the material from courses with these names tend to be either covered in other classes, or can be self taught with little difficulty.--------

If Tom is anything like I was at 14, he gets along much better with college and older students than those his own age. (to this day have a harder time relating to people my own age than say my late 20's-40 old classmates).

 

[hrc969]

yes, i'll take my time now. that's why I'm going to finish my undergrad in 4 years. but no electives, i only want to study math and physics.

Very cool. I wish my school would have let me just take math and physics. I would have probably kept my physics major if they had done so.

I don't think you can take only mathematics and physics courses, even if you are a prodigy, however, I could be wrong. Regardless, I would definitely advise you to take at the very least, courses in: philosophy, psychology, biology, english and history, to give you a broad, fundamental understanding of the world.

There are some universities that don't require classes outside the major. A professor that I was doing research with went to such a school. In my opinion more schools should be like that.

Thats all good in theory, however I do not know of many universities that allow you to take just what you want to take and what you deem important to your career, more generally, you take what they require. I think most universities in the US have some form a liberal arts requirement of some sort. So you can tell them all you want that studying history will not help you in your career, but I do not think they will comply to your wants

I guess he hasn't said whether electives outside the major are not required or if he just doesn't want to take them. But the former is definitely possible.

So just curious, Tom1992, do you know for a fact that your school does not require you to take courses outside your major?

 

[Tom1992]

So just curious, Tom1992, do you know for a fact that your school does not require you to take courses outside your major?

i've been told that if your GPA is high enough you can omit some courses and add some of your own.

 

[hrc969]

i've been told that if your GPA is high enough you can omit some courses and add some of your own.

I don´t know if you have done this already but if you haven´t: You should verify this with a counselor or with you math or physics advisor department advisor. You need to make sure you can omit all the courses you want to omit. Otherwise you might have to do them eventually.

I didn´t like that there were courses outside math and physics required, but I just did those in a way that would not mess with the classes that I wanted to take. One of the things I did was take some of those requirements over the summer when the courses I am interested in aren´t offered anyways, just to get them out of the way. That way, I started my 3rd year (this year) knowing that I did not have to worry about taking any non-math classes.

Its annoying to take classes that you don´t think you need. I had to take quite a few of those. If it will be a hassle to try to get them ommitted, then just take them. Since you have studied some of the math subjects before that allows you to take more classes per term and you can probably manage with one of those requirements.
PS:You should try to find out about whether or not you can omit some of those requirements or not as soon as possible. That way you can plan accordingly.

 

[MathematicalPhysicist]

it's not bad to take an elective which isn't from maths or physics.
why not take at the summer a course in a foreign language, it's always good to learn another language.
expecially french and german, it could be handy when you want to read articles on maths or physics in german or french.

 

[complexPHILOSOPHY]

Tom, if you are required to take your GE's, most of them are mind-numbingly boring but all of them are very easy A's so long as you do the work and take the tests.

So, if you are obligated to take things like philosophy (which I recommend to anyone, especially science and mathematics majors), history, english, etc., you can add them into your schedule as if they weren't classes. Just schedule around them as if they weren't there. The only requirement you will have, is to show up and hand the work in.

I do recommend atleast one class on symbolic logic and one class on reality and knowledge, simply as a survey of philosophy. It is important, especially as a practicing mathematical physicist, that you understand a great deal of philosophy. Philosophy consists of abstract formal logic systems, built upon the axioms of deductive, inductive and dialectical reasoning. It is logically rigorous (symbolic logic moreso) and will help construct alternate paradigms of reality to look through and gain distinct insight that would not have been gained without reading philosophy. Reading the works of Hume, Kant, Nietzsche, Popper, Kuhn, Hiegal, Hiedeggar, Russel, etc., gives you amazing insight into the uncertainty of reality.

Also, I know you can read this stuff on your own and fully comprehend it (I have no problems and I know that most people don't), however, philosophy has no correct solutions. Your lecturer will have received a graduate level education in philosophy and will have been exposed to multi-cultural perspectives which he can express to you. I thought I fully understood everything I read on my own and I thought my critiques were logically sound, until I had my first philosophy class and my teacher found gaping holes in much of my logic. By the end of the semester, my writing had developed exponetially and my ability to construct logically sound arguments really emerged.

Anyways, I am rambling dude. Basically, just try to take a little bit of everything and talk to as many people as you can, so you can always construct a sound model of reality.

 

[complexPHILOSOPHY]

it's not bad to take an elective which isn't from maths or physics.
why not take at the summer a course in a foreign language, it's always good to learn another language.
expecially french and german, it could be handy when you want to read articles on maths or physics in german or french.

I don't think America really cares if we learn a language, so most kids don't. I know some universities require it, although I think if you take it in high school, you are exempt in college.

However, I am taking French again for the hell of it, (I took four years of it in high school) and never really thought I could use it later on when reading articles. That's awesome! I forgot French and German are two huge science languages.

<33

 

[symbolipoint]

I don't think America really cares if we learn a language, so most kids don't. I know some universities require it, although I think if you take it in high school, you are exempt in college.

However, I am taking French again for the hell of it, (I took four years of it in high school) and never really thought I could use it later on when reading articles. That's awesome! I forgot French and German are two huge science languages.

<33

The recommended (or were they "required"?) languages for science students several years ago were German, or French, or Russian. The Russian language is not as commonly taught; this seems to restrict which language a student may choose. If a student sees a particular advantage for studying Russian, and his institution does not offer it, then maybe he is out of luck. I wonder what kind of scientific literature would now be difficult to examine if Russian is a more difficult foreign language choice for todays science students?

Actually, other scientific literature must exist in many other languages; why not also allow the student to choose for his science education, Dutch, Danish, Italian, Norwegian, Korean, Mandarin...? And why not Spanish?
Should not expect to be able to find Spanish scientists publishing reports to journals in their own language?

 

[complexPHILOSOPHY]

I am not sure what your contention was (or if you were just commenting), however, in American universities, I believe they offer many of the courses you mentioned. Spanish is probably the language that most Americans learn, because it has direct applications with immigrants and natural born citizens from Mexico, Puerto Rico, Cuba, etc.

My high school offered German, French, Italian, Spanish and Latin, while I have talked to people from different high schools which only offered Spanish or no language at all. In Europe, there is more exposure to foreign languages, so I believe people start learning them at a young age.

I have read research which indicates that if an individual begins learning languages at a young age, the encoded information for those languages is stored in the same structure of the brain (I can't remember if it was Broci's area or not) as your natural or original language, however, if an individual learns foreign languages later in life, it is stored in a different structure of the brain, distinct from your natural or original language. This might have implications on learning foreign languages for those who are never exposed to it.

As for your question regarding why other languages aren't offered, I would suspect it has something to do with the demand for it? I can't comment on what universities offer but I would imagine among all of the universities available to the student, all of those languages are present in some form. I know they offer Russian at the university I am going to be attending in addition to a lot of other languages.

My community college offers Arabic, Italian, German, French and Spanish, so I am sure the universities offer a broad spectrum.

 

[mgiddy911]

I don't think America really cares if we learn a language, so most kids don't. I know some universities require it, although I think if you take it in high school, you are exempt in college.

I thought it was pretty much a standard requirement of at least two years of foreign language at almost every high school, and a general requirement for most colleges to have some foreign language experience. I am in the College of Arts and Sciences at my school, and it requires at least 2 semesters of foreign language for all of its students unless you take a proficiency test to show that you know a language already.

As to the OP, i know you seem to be very talented and interested in math and physics... but do you think that perhaps you might gt just slightly bored if you are taking 4+ math and physics classes each semester and nothing but that?

 

[complexPHILOSOPHY]

I thought it was pretty much a standard requirement of at least two years of foreign language at almost every high school, and a general requirement for most colleges to have some foreign language experience. I am in the College of Arts and Sciences at my school, and it requires at least 2 semesters of foreign language for all of its students unless you take a proficiency test to show that you know a language already.

As to the OP, i know you seem to be very talented and interested in math and physics... but do you think that perhaps you might gt just slightly bored if you are taking 4+ math and physics classes each semester and nothing but that?

It is a standard that students 'take' a foreign language, however, there is no standard for 'proficency'. I finished four years of French in high school, which, if I wanted to, would transfer into my University and fulfill my language requirements and never have to take French again. However, I took French in 6th, 7th, 8th and 9th grade and never spoke or read a word of it since my last year of it. In addition to that, I got mostly B's and C's in French because at the time, I didn't care to learn it, hence the reason I am not transferring it. However, if I wanted to, I could transfer it, claim proficiency in French and continue on, even though I can barely speak it.

Most students I have spoken to, simply transfer their foreign language from high school into university and never have to take a proficiency exam. If you asked them to communicate in that particular language, they have a tough time recalling it.

Granted, I am sure there are plenty of students who did learn the language and retained most of it, so that doesn't apply to them.

 

[tmc]

My story is a bit similar to yours, although I didn't start university quite as early on schedule. I also take no electives, and basically fill my schedule with as many physics and math courses as I can. On average, I end up with 7-8 courses per semester, for about 16 courses per year (one in the summer while doing research too). All of these are useful courses for my future research.

With that said, there are a few problems with the courses you've chosen, so let me go ahead and help you out:

First, unlike most people have said, I'm almost certain that you can handle more than 9 courses per year. You're obviously a very smart kid, and I don't think you'd have trouble handling 12 courses per year, which is 6 per semester. If it's too much, you can always drop one early in the semester anyways. If it's too little, add a few in the later years.

Now to your courses:
First thing that's a glaring omission is the lack of programming courses. That's not a problem if you have some programming background, but if you don't, you will need it. This is one of the most useful tools you'll need as a physicist, no matter how theoretical you want to become. I'd suggest taking the two first comp sci courses, something like intro to comp sci 1 and 2. They'll be easy, generic courses, the idea being that they'll teach you the basics of one programming language, probably Java, and you'll be able to apply that knowledge to very, very quickly pick up new languages (in your case, FORTRAN, C, C++ and Matlab). So add that to second year.

Problem #2: Not a single experimental physics or lab experience. That's very bad, if only because it might go against you when entering grad school. You should at least have some (2-3 years) lab experience. So add those to second and third year.

Problem #3: No mechanics course. Those are useful, but not essential for future work. They're useful mostly because they teach you a lot about using math in physics; it's basically applied calculus for the most part. Another part is getting taught the Lagrangian and Hamiltonian formalisms, which are very useful in more advanced theoretical physics courses, though they can be done without. So no need for this, but if you have room, it's a definite plus.

If possible (ie, if it exists), add another linear algebra class in there, something dealing with stuff like operator algebra and treating dual spaces more thoroughly. That'll definitely be useful in quantum. One of the topology classes can go to make room for this.

Likewise, try to find a Tensor Analysis class, for general relativity. The math that you'll learn in the physics courses will probably be nowhere near formal enough for you, so this is something you would probably regret not taking.

A good Lie Algebra / Lie Theory class would be extremely good for you, if one is offered. Otherwise, grab a textbook on the topic.

You only have one E&M course. I doubt it will be very advanced, since I assume that it will be your first calculus-based E&M. You definitely need another one.

With those recommendations, this is what your schedule would become:

2nd year:
Intro to Comp Sci I
Intro to Comp Sci II
Calculus II
Linear Algebra II
Group Theory
Ordinary Differential Equations
Real Analysis I
Thermal Physics
Oscillations and Waves
Introduction to Quantum Physics
Lab

3rd year:
Partial Differential Equations
Complex Analysis I
Real Analysis II
Point-Set Topology
Differential Geometry I
Classical Mechanics
Electromagnetic Theory
Quantum Mechanics I
Nuclear and Particle Physics
Lab
Ring Theory

4th year:
Complex Analysis II
Differential Geometry II
Differential Topology
Algebraic Topology
Quantum Mechanics II
Relativity I
Relativity II
Introduction to String Theory
Introduction to Quantum Field Theory
Lie Algebra
Tensor Analysis
Electromagnetic Theory II

Again, this is a much harder than your initial plan, but I don't think it's beyond your means. It's definitely doable. Also, get research experience during the summer, you'll need it for grad school. Finally, get an A+ in all of those. That will also help for grad school. If you can't do that, then drop a course or two per year.

Good luck.

 

[Tom1992]

thanks a lot tcm. I've considered your recommendation but i had to chop off a bit. also electromagnetic theory II i moved to third year due to the additional 4th year courses. I'm not going to do computational research so i'll forgo computer science courses for now and crash study it later if it turns out i need to. lagrangian and hamiltonian dynamics is covered in the third year classical mechanics course. tensor analysis is already covered in differential geometry II. my first year physics course is already pretty heavy on the labs. this is what i have so far:

2nd year:
Calculus II
Linear Algebra II
Group Theory
Ordinary Differential Equations
Real Analysis I
Ring Theory
Thermal Physics
Oscillations and Waves
Introduction to Quantum Physics
Lab

3rd year:
Partial Differential Equations
Complex Analysis I
Real Analysis II
Point-Set Topology
Classical Mechanics
Electromagnetic Theory
Quantum Mechanics I
Nuclear and Particle Physics
Differential Geometry I
Electromagnetic Theory II

4th year:
Complex Analysis II
Differential Geometry II
Differential Topology
Algebraic Topology
Quantum Mechanics II
Relativity I
Relativity II
Introduction to String Theory
Introduction to Quantum Field Theory
Lie Algebra


tcm, you might be the closest match to my programme. could you tell us the courses in your programme, please?

 

[tmc]

First: you seem to be going towards string theory. You will do computational research. I understand if your schedule is full and you don't have room for them and prefer taking math courses (which you will also use), but you will need to learn programming at some point. Just keep that in mind.

My program is nothing special really; I take a lot of extra classes as electives to compensate. Here are the courses I took:

2nd year, fall:
Calculus III
Honours Linear Algebra I (2nd lin algebra course)
Introduction to Probability
Applied Optics
Mecanics I
Physics Lab

2nd year, Winter
Electricity and Magnetism
Mecanics II
Modern Physics
Group Theory and Applications
Ordinary Diff. Eq. and Numerical Methods
Introduction to Topology
Lab
Mathematical Analysis I

Summer, while doing research: applied algebra

3rd year, Fall
Analysis III
Theoretical Physics
Intro Quantum Mechanics
Thermodynamics
Physics Lab
Subatomic Physics I
Intro to Microeconomics

I'm currently in winter of 3rd year, taking a semester off (well, actually 3, but this is the first) to do full-time research.

A few things to note: I'm not sure if you've already chosen a university, but most will force you to take a number of electives which have nothing to do with your major, so consider that. Also, most schools won't have all the courses you seek offered, or at least, not necessarily offered on the year you want to take them (many universities alternate which courses are offered). Most of your courses should be easy to take, but things like intro to String, intro to QFT, lie algebra, relativity II, diff geometry II, etc. might be hard to take. These are the kinds of courses that few students would take, especially at the undergrad level, and as such, are bound to be only rarely offered except at few select schools.

 

[tmc]

don't know yet. I'm in a fairly small department, so what I take strongly depends on what happens to be offered (which is why I take many courses - they might not be offered the following year, or not fit in my schedule). But I'd like

Lie Theory
Lie Algebra
Linear Algebra II
Linear Algebra III
Tensor Analysis
Complex Analysis II
General Relavitity
Quantum Mechanics II
E&M 2
Statistical Mechanics
Subatomic Physics II
Nuclear Physics

 

[Tom1992]

i think you will need riemannian geometry, differential topology, and algebraic topology math courses to accompany your general relativity and particle physics. at any rate, looks like you will be more knowledgeable than me when you finish your undergraduate degree, because you are cramming in more than me. good luck.

 

[imastud]

wow you guys are nuts, i guess you won't have time for DR 210 Introduction to Beer Pong, that was my favorite class, i thought everyone took that

 

[tmc]

i think you will need riemannian geometry, differential topology, and algebraic topology math courses to accompany your general relativity and particle physics. at any rate, looks like you will be more knowledgeable than me when you finish your undergraduate degree, because you are cramming in more than me.

I'd love to fit those in, but as I said earlier, those courses are fairly rarely offered (though a good deal of differential geometry is done in tensor analysis, which is offered more often). I guess I have to leave some for grad school.

There are also other things which can be useful. For example, TAing labs or a class to develop your teaching skills, as well as doing research. My plan, when I started my undergrad, was to become one of the best grad school applicants in the world. If that's what you seek, then follow what I did: take as many courses as you can while keeping a near-perfect average, do research and publish as early and as often as possible (my goal is to have 4-5 publications by the end of my undergrad), and TA for a semester or two.

I'm planning on going in very theoretical physics, probably QFT or LQG, and will probably do a masters in mathematics next, before a phd back in physics.

 

[tmc]

wow you guys are nuts, i guess you won't have time for DR 210 Introduction to Beer Pong, that was my favorite class, i thought everyone took that

Beer Pong is a freshman class, whereas you will notice that the classes we're talking about are for years 2-4.

Indeed, first year is mostly dedicated to those killer beer pong assignments. They make for some long, long nights.

 

[imastud]

Beer Pong is a freshman class, whereas you will notice that the classes we're talking about are for years 2-4.

whoops my mistake wasn't paying attention

Indeed, first year is mostly dedicated to those killer beer pong assignments. They make for some long, long nights.

yeah i spent hours and hours on my beer pong class my freshman year to the deteriment of my other classes. its a tough weed out class. I've been able to balance things better since.

 

[Stingray]

i think you will need riemannian geometry, differential topology, and algebraic topology math courses to accompany your general relativity and particle physics.

Ok, this statement has convinced me to get back into this thread. Those subjects are most certainly not required to take relativity or particle physics. In fact, almost no upper-level math course will ever be directly applicable in any physics course. The math you learn in those courses is not used or even known by the vast majority of physicists (yes, I mean theoretical physicists). The bits which are used will be presented in a completely different ways in physics and math classes. Physics books will never assume you've taken extra math anyway. Everything will be explained in the relevant courses. You can take lots of math if you like, but unless you end up in a very unusual research field, don't expect it to be useful in physics.

Some other points in this thread need reality checks too. I still don't understand how you can get out of general ed courses by having a good GPA. I've never heard of such a thing, and I'd be very skeptical of any university which allowed that.

Also, labs are good things (at least if they're organized properly). I'm finishing my Ph.D. in relativity right now, so I'm pretty far removed from experiments. Yet I still consider my undergrad labs some of the most important courses I ever took. You can't get that experience from a book. I should also say that I've never seen freshman labs that had any real substance. You should take real lab courses. If you're actually getting a physics degree, any respectable university should require many of them.

Why do you have QM II, QFT, and string theory all in one year? QFT requires a thorough knowledge of QM, and I doubt the course starts in the middle of the year. You also need a solid knowledge of QFT to learn any meaningful string theory, so it wouldn't be reasonable to take those concurrently.

 

[imastud]

you might want to take an advanced stats/probability course or two

 

[Tom1992]

Ok, this statement has convinced me to get back into this thread. Those subjects are most certainly not required to take relativity or particle physics. In fact, almost no upper-level math course will ever be directly applicable in any physics course. The math you learn in those courses is not used or even known by the vast majority of physicists (yes, I mean theoretical physicists). The bits which are used will be presented in a completely different ways in physics and math classes. Physics books will never assume you've taken extra math anyway. Everything will be explained in the relevant courses. You can take lots of math if you like, but unless you end up in a very unusual research field, don't expect it to be useful in physics.

tmc and i want to specialize in BOTH physics and mathematics (as the title of this thread indicates). tmc stated that he wants to get a masters in math before doing his phd, so we want to be very versed in both mathematical knowledge and rigour and then use that extra skill to our advantage in researching theoretical/mathematical physics.

besides, I've been looking at the book "mathematical perspectives on theoretical physics" by prakash
and in it summarizes in chapter ZERO all "mathematical preliminaries" up to, and including, riemannian geometry, differential topology, and algebraic topology. chapters 1 to 5 in that book then proceeds to MORE ADVANCED mathematics. look:
https://www.amazon.com/gp/product/1860943659/?tag=pfamazon01-20
and then chapters 6 onwards it finally begins discussing relativity, quantum field theory and superstrings. so I've gotten the impression that the math courses I've listed for my undergrad program just touches on what math i eventually need to know.

Some other points in this thread need reality checks too. I still don't understand how you can get out of general ed courses by having a good GPA. I've never heard of such a thing, and I'd be very skeptical of any university which allowed that.

i've been told by one of my TA's that his high GPA and experience in working with his programme coordinator that he has been allowed by his programme coordinator to customize his undergrad programme.

Why do you have QM II, QFT, and string theory all in one year? QFT requires a thorough knowledge of QM, and I doubt the course starts in the middle of the year. You also need a solid knowledge of QFT to learn any meaningful string theory, so it wouldn't be reasonable to take those concurrently.

i did not know that one has to master qft before string theory, i thought they could be studied concurrently. but i know that there is an undergraduate textbook for string theory already called "a first course in string theory" by zwiebach:
https://www.amazon.com/dp/0521831431/?tag=pfamazon01-20

 

[Stingray]

tmc and i want to specialize in BOTH physics and mathematics (as the title of this thread indicates). tmc stated that he wants to get a masters in math before doing his phd, so we want to be very versed in both mathematical knowledge and rigour and then use that extra skill to our advantage in researching theoretical/mathematical physics.

That's fine. I was just saying that all of those math courses may not be as applicable to physics as you're expecting.

The math described in chapter 0 of Prakash is really pretty basic. Almost all of it would already be included in the relevant physics courses anyway. Math courses would probably focus on completely different topics, and may even ignore the material relevant to physics anyway. In most cases, the notation is also completely different.

Anyway, I can't see the other chapters of that book very efficiently online. I'll just say that the GR chapter looks pretty basic. Most serious textbooks on the subject actually require much more math. But they also spend a lot of time developing it. A lot of that comes from Riemannian geometry, but I don't know how useful a mathematician's course in that subject would be. Most of relativity uses pseudo-Riemannian geometry anyway, and from what I understand, many of the theorems don't carry over. I can't recall ever seeing differential topology or algebraic topology in GR or (standard textbook) QFT. If they were there, I didn't recognize them as such.

I also thought I needed to learn tons of math to understand GR when I was a freshman. I was taking too many cues from popular books going on about how esoteric these subjects were. That impression turned out to be completely wrong.

i've been told by one of my TA's that his high GPA and experience in working with his programme coordinator that he has been allowed by his programme coordinator to customize his undergrad programme.

That sounds more like he was able to pass out of certain classes because of his experience. That's different than getting out of courses you don't know anything about.

i did not know that one has to master qft before string theory, i thought they could be studied concurrently. but i know that there is an undergraduate textbook for string theory already called "a first course in string theory" by zwiebach:
https://www.amazon.com/First-Course-S...e=UTF8&s=books&tag=pfamazon01-20

From what I've heard, that book introduces string theory in a very roundabout and handwavy way. Lots of things are just presented out of nowhere. Given the rigorous background that you seem to want, I think it's better to wait until you can learn the subject properly.

 

[^_^physicist]

Posted by Stingray:
Most of relativity uses pseudo-Riemannian geometry anyway, and from what I understand, many of the theorems don't carry over. I can't recall ever seeing differential topology or algebraic topology in GR or (standard textbook) QFT. If they were there, I didn't recognize them as such.

From what I can gather from my diff. geo. texts; this is true. However, I would like to add, however, that taking a course in Diff. Geometry is not going to hurt. My math professor for differential geometry was once a mathematical physicists, and still does research as a mathematician into mechanics...so Differential geometry can still be useful (there are entire books deticated to geometric mechanics such as: Geometric Mechanic by: Richard Talman). I do believe, however, Stingray, that you are correct, from a purely physics point of view you can/will pick up the mathematics to preform your subject while within the subject.

Though, I would argue having an excess of mathematics isn't necessarily going to be a bad thing, it is often harder to pick up the math on the fly than the lab procedures (assuming a minimual background is in place).Still the Diff. Topology and Algebraic Topology hasn't cross my radar in terms of needing it for any of the subjects listed. I do know that either one will be of use, however, if the OP plans on studying Quantum Topology (which is what a professor at my university does...still not sure what exactly that encomposses).

 

[Tom1992]

again i am a mathematics specialist who now wants to expand into theoretical physics so i have only some exposure to general relativity and quantum mechanics. having no background in quantum field theory or string theory, i am relying on the book "mathematical perspectives on theoretical physics" by prakash
https://www.amazon.com/gp/product/1860943659/?tag=pfamazon01-20
to determine ahead of time what math courses i need to take. in chapter zero, it clearly states a need to know:

homotopy, category and functors, de rham cohomology, mayer-vietoris sequence.

this to me suggests that i need to study algebraic topology and category theory (and i haven't studied those yet).

and then in chapter one, it goes on to talk about elliptic curves, riemman surfaces, complex manifolds, kahler manifolds, etc... topics beyond the riemannian geometry and differential topology topics i spoke of. so i feel that the mathematics courses i have listed are not even sufficient to cover the math i need to know--contrary to your claim that these math courses are excessive.

 

[complexPHILOSOPHY]

however, the OP plans on studying Quantum Topology (which is what a professor at my university does...still not sure what exactly that encomposses).

Isn't that like a lot of Kahler Geometry? When I consider 'quantum topology,' I think of Calabi-Yau spaces, which I guess, are essentially Ricci-flat complex manifolds that admit a closed Kahler form and have a vanishing first Chern class, (not that I fully understand what that means, I need to learn more maths first).

Although, I think this only has applications in String Theory but I am not aware of other widely used 'quantum topological spaces' used elsewhere in QM. So, I wonder, is it mostly Kahler Geometry, Poincare Groups, SU(5)?

It sounds like a tight research field, which is why I am trying to figure out what exactly it is.

Then again, I am not really aware of a whole lot at all, so I probably am just retarded.

 

[Tom1992]

ok, here is the official list of the mathematics one needs to know for string theory (i'm skipping the simple 1st and 2nd year math courses).
http://superstringtheory.com/math/index.html

Real analysis
In real analysis, students learn abstract properties of real functions as mappings, isomorphism, fixed points, and basic topology such as sets, neighborhoods, invariants and homeomorphisms.

Complex analysis
Complex analysis is an important foundation for learning string theory. Functions of a complex variable, complex manifolds, holomorphic functions, harmonic forms, Kähler manifolds, Riemann surfaces and Teichmuller spaces are topics one needs to become familiar with in order to study string theory.

Group theory
Modern particle physics could not have progressed without an understanding of symmetries and group transformations. Group theory usually begins with the group of permutations on N objects, and other finite groups. Concepts such as representations, irreducibility, classes and characters.

Differential geometry
Einstein's General Theory of Relativity turned non-Euclidean geometry from a controversial advance in mathematics into a component of graduate physics education. Differential geometry begins with the study of differentiable manifolds, coordinate systems, vectors and tensors. Students should learn about metrics and covariant derivatives, and how to calculate curvature in coordinate and non-coordinate bases.

Lie groups
A Lie group is a group defined as a set of mappings on a differentiable manifold. Lie groups have been especially important in modern physics. The study of Lie groups combines techniques from group theory and basic differential geometry to develop the concepts of Lie derivatives, Killing vectors, Lie algebras and matrix representations.

Differential forms
The mathematics of differential forms, developed by Elie Cartan at the beginning of the 20th century, has been powerful technology for understanding Hamiltonian dynamics, relativity and gauge field theory. Students begin with antisymmetric tensors, then develop the concepts of exterior product, exterior derivative, orientability, volume elements, and integrability conditions.

Homology
Homology concerns regions and boundaries of spaces. For example, the boundary of a two-dimensional circular disk is a one-dimensional circle. But a one-dimensional circle has no edges, and hence no boundary. In homology this case is generalized to "The boundary of a boundary is zero." Students learn about simplexes, complexes, chains, and homology groups.

Cohomology
Cohomology and homology are related, as one might suspect from the names. Cohomology is the study of the relationship between closed and exact differential forms defined on some manifold M. Students explore the generalization of Stokes' theorem, de Rham cohomology, the de Rahm complex, de Rahm's theorem and cohomology groups.

Homotopy
Lightly speaking, homotopy is the study of the hole in the donut. Homotopy is important in string theory because closed strings can wind around donut holes and get stuck, with physical consequences. Students learn about paths and loops, homotopic maps of loops, contractibility, the fundamental group, higher homotopy groups, and the Bott periodicity theorem.

Fiber bundles
Fiber bundles comprise an area of mathematics that studies spaces defined on other spaces through the use of a projection map of some kind. For example, in electromagnetism there is a U(1) vector potential associated with every point of the spacetime manifold. Therefore one could study electromagnetism abstractly as a U(1) fiber bundle over some spacetime manifold M. Concepts developed include tangent bundles, principal bundles, Hopf maps, covariant derivatives, curvature, and the connection to gauge field theories in physics.

Characteristic classes
The subject of characteristic classes applies cohomology to fiber bundles to understand the barriers to untwisting a fiber bundle into what is known as a trivial bundle. This is useful because it can reduce complex physical problems to math problems that are already solved. The Chern class is particularly relevant to string theory.

Index theorems
In physics we are often interested in knowing about the space of zero eigenvalues of a differential operator. The index of such an operator is related to the dimension of that space of zero eigenvalues. The subject of index theorems and characteristic classes is concerned with

Supersymmetry and supergravity
The mathematics behind supersymmetry starts with two concepts: graded Lie algebras, and Grassmann numbers. A graded algebra is one that uses both commutation and anti-commutation relations. Grassmann numbers are anti-commuting numbers, so that x times y = –y times x. The mathematical technology needed to work in supersymmetry includes an understanding of graded Lie algebras, spinors in arbitrary spacetime dimensions, covariant derivatives of spinors, torsion, Killing spinors, and Grassmann multiplication, derivation and integration, and Kähler potentials.

K-theory
Cohomology is a powerful mathematical technology for classifying differential forms. In the 1960s, work by Sir Michael Atiyah, Isadore Singer, Alexandre Grothendieck, and Friedrich Hirzebruch generalized coholomogy from differential forms to vector bundles, a subject that is now known as K-theory.
Witten has argued that K-theory is relevant to string theory for classifying D-brane charges. D-brane objects in string theory carry a type of charge called Ramond-Ramond charge. Ramond-Ramond fields are differential forms, and their charges should be classifed by ordinary cohomology. But gauge fields propagate on D-branes, and gauge fields give rise to vector bundles. This suggests that D-brane charge classification requires a generalization of cohomology to vector bundles -- hence K-theory.

Noncommutative geometry (NCG for short)
Geometry was originally developed to describe physical space that we can see and measure. After modern mathematics was freed from Euclid's Fifth Axiom by Gauss and Bolyai, Riemann added to modern geometry the abstract notion of a manifold M with points that are labeled by local coordinates that are real numbers, with some metric tensor that determines an extremal length between two points on the manifold.
Much of the progress in 20th century physics was in applying this modern notion of geometry to spacetime, or to quantum gauge field theory.
In the quest to develop a notion of quantum geometry, as far back as 1947, people were trying to quantize spacetime so that the coordinates would not be ordinary real numbers, but somehow elevated to quantum operators obeying some nontrivial quantum commutation relations. Hence the term "noncommutative geometry," or NCG for short.
The current interest in NCG among physicists of the 21st century has been stimulated by work by French mathematician Alain Connes.


and i think this list of math topics is still incomplete because it does not mention kahler manifolds and calabi-yau manifolds that prakash claims is essential

the undergrad math courses i listed for myself, including riemannian geometry, differential topology, and algebraic topology, does not even cover HALF of this list. so anybody still believes that my list of math courses is too excessive for physics?

 

[complexPHILOSOPHY]

Tom,

Are you aware that String Theory has yet to produce a single empirical prediction (other than the general predictions of quantum mechanics regarding supersymmetry, extra dimensions, etc. which CERN is hopefully going to experiment with)? If it has made predictions, I have never read any research indicating that we have empirically verified them.

I am sure you are a genius, so this field of maths and physics looks like it might one of the few things challenging for you, however, I would be weary that even the brightest physicists can't rely on String Theory after graduate school.

I also don't understand the appeal of working in such a complex and comprehensive field that has no empirical support. In all of it's elegance and mathematical beauty, it still doesn't reflect any aspect of reality that we have empirically verified.

I am not discouraging you from Quantum Mechanics or Quantum Theory or whatever it's colloquially known as, however, I see a very limited future for employment in that field, even academia. I would imagine it's hard to get funding for a theory that has no direct applications, unless your name is Witten, Greene, Polchinski, et Al.

Perhaps, however, my perception is distorted and String Theory has far greater implications than I am aware of. I might be completely wrong my friend.

 

[imastud]

yeah its kinda sad how the allure of string theory is sucking the greatest minds of the past few generations away from doing stuff that's actually beneficial to the world

 

[CPL.Luke]

thats assuming that string theory is physics and not math ;)

quips aside if you enjoy math for math than you should definatly take as many math courses as you can, similarly if you enjoy physics for physics than you should take as many physics courses as you can.

the math can only help you, although if you decide that you don't like math as much as you thought you did (which may occur at some point) than don't think that it will destroy your ability to destroy your ability to do advanced physics. Also keep in mind that there are multiple ways of doing theoretical physics, one is to be mathematically rigorous and the other is to be intuitive. Both have their uses.

Personally I like math, but find that the average mathematicians approach loses the beauty and the applications. For instance today in my techniques class the professor was lecturing on how to find the components of a vector, his approach was incredibly useful once I was able to decode the significance of it, however he spent the entire class just showing the approach and "proving" that it worked without elaborating on the situations you might need it for or even talking about what a vector represented without just listing the axioms.

EDIT: wrote this after tom's last post, didn't see the other posters

 

[mgiddy911]

the posts about beer pong may be some of the funniest that I have seen in a long time, being a freshmen myself, they hit home nice and well

 

[Tom1992]

well, even if i later decide that studying string theory is a dead-end street, i still believe that all the "extra maths" i will have studied will be useful for whatever other branch of physics i decide to switch to. i believe knowing more math than necessarily actually improves your physics understanding.

for example, I'm taking 1st year physics right now. the math used is rudimentary. work is taught in one dimension with just a hint of integration. my knowledge of n-dimensional calculus has allowed me to see work more broadly than how work is taught in class. the lorentz transformation in special relativity makes more sense to me now that i have group theory under my belt. kepler's laws is also taught with minimal math in 1st year physics, but my knowledge of geodesics allows me to understand the planetary motion more profoundly than I would otherwise see it if I did not know any riemannian geometry.

so the same sort of deeper appreciation for more advanced physics can be had with the extra knowledge of math that i will gain by attempting to study string theory whether or not i maintain interest in it. I'm going ahead to study the mountains of math, not only because i find it intriguing, but because i believe that studying physics while being already very versed in the math will allow me to learn physics more efficiently and thoroughly than a physics student who only learns the bare minimum math required for his physics courses.

 

[imastud]

well, even if i later decide that studying string theory is a dead-end street, i still believe that all the "extra maths" i will have studied will be useful for whatever other branch of physics i decide to switch to. i believe knowing more math than necessarily actually improves your physics understanding.

thats definitley true. especially once you start research you never really know what sort of math you're going to need because you're doing stuff no one's ever done before so the bigger your "mathematical toolbox" is the more potential you have to be successful with your theories.