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Bachelor In Mathematics and PhD In Physics

  1. Mar 17, 2013 #1
    Bachelors In Mathematics and PhD In Physics

    Hello,

    As of now, I am maneuvering through undergraduate work, with the intention of gaining a bachelors in mathematics and (someday) a PhD in physics. For my mathematics courses I have taken, I have done all three calculi, discrete mathematics, and probability and statistics; and this summer, I shall be taking differential equations. Without respect to any particular school, in general, what is the required amount of mathematics courses to complete a bachelors in mathematics; and of the remaining I need to take, what mathematics classes would be pertain to someone who intends to do research in mathematical/theoretical physics?

    Thank you dearly for your time.
     
    Last edited: Mar 17, 2013
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  3. Mar 17, 2013 #2
    There are usually around 15-18 credit hours (5-6 classes) of "core" mathematics that all math majors are required to take. This includes discrete math, real analysis, abstract algebra, probability, linear algebra, differential equations, and perhaps a research requirement.

    Then you usually get to choose from ~15 hours of mathematics electives, which could include just about anything, for example complex analysis, differential geometry, numerical analysis, mathematics software, more real analysis and algebra, partial differential equations, topology...

    Not sure on the physics side, but my recommendation is that you AT LEAST minor in physics if you plan to attend graduate school in physics. My friend talked to the physics department at Ga Tech and they said they would consider anyone who didn't have at least 5 upper-division physics courses. I assume these would include E&M, Mechanics, Optics, Solid State, Condensed matter, statistical mechanics.

    I urge you to just go ahead and change your major to physics if that is what you want to study. I don't know why you are a math major, but in my case, math was the only STEM degree that my school offers (aside from biology and chemistry), and I didn't want to transfer. It has caused me a LOT of trouble and inconveniences and caused me to postpone my graduation for several semesters while I attend another school part time to get prerequisites for my intended field in graduate school. Save yourself the headache and just major in physics, or double major.
     
  4. Mar 17, 2013 #3

    micromass

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    Take as many analysis classes as you can. Don't stop with one real analysis class. Take complex analysis and functional analysis and more. Be sure to take differential equations classes, such as PDE.
    Also, take whatever linear algebra classes they have. And also take differential geometry.

    I guess you should also take abstract algebra, but that's way less important for physics.

    These are the recommendations I would make for anybody going into mathematical physics. They might not help you get into grad school. But once you got into grad school, you will be happy that you took these classes. The more analysis, linear algebra and differential geometry you took, the better off you will be.
     
  5. Mar 18, 2013 #4
    Micromass, so if I took PDE, ODE, Real Analysis, Complex Analysis, Functional Analysis, Linear Algebra, Differential Geometry, and Abstract Algebra, would that be enough to complete the required courses for the "average" bachelors in mathematics? Too much? Too little?

    Also, is Advanced Calculus simply another name for Real Analysis?
     
  6. Mar 18, 2013 #5
    I realize you were asking Micromass, but I figure more input can't hurt. For the "average" bachelors in math, probably yes, this is fine. However, these types of questions should be directed toward your advisor (that is why they are there). It really depends on the school. For example, at my school, the core required coursework includes Calc 1-3, ODEs, Intro to Abstract Math, Intro Real Analysis, Combinatorics, Prob and Stats 1, Applied Linear Algebra, and 2 programming courses. Then you pick a concentration (not necessarily in math) and choose your electives based on that. Other math departments I've looked at require a year of real analysis, a year of abstract algebra, calc 1-3, ODEs, etc.
     
  7. Mar 18, 2013 #6
    Assuming we're talking about group theory, it comes up a lot in physics. Some examples off the top of my head are: QFT, or in general any field theory (yes, there is more to this, but dont got much time to type), Lie groups used in LQG, and of course they use groups in the standard model! I don't see any reason why this is less important, than say, real analysis...

    Anyways to OP, if you DO want to stay with math major and then do get a PhD in physics be sure to take PDEs, differential geometry, and a numerical computation course. I dunno of a math program where you wouldn't have to take real analysis, abstract algebra and linear algebra, so of course you're gonna take those.

    Also, see if the math department offers a course on perturbation theory, it's used a lot in physics, and being a math major, you should want to understand the methods and theory behind it.

    Good luck!
     
  8. Mar 18, 2013 #7

    lisab

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  9. Mar 19, 2013 #8
    Thank you everyone for your responses.
     
  10. Mar 20, 2013 #9
    micromass - Take as many analysis classes as you can. Don't stop with one real analysis class.

    - Take complex analysis and functional analysis and more. Be sure to take differential equations classes, such as PDE.

    - Also, take whatever linear algebra classes they have. And also take differential geometry.

    - I guess you should also take abstract algebra, but that's way less important for physics.

    - The more analysis, linear algebra and differential geometry you took, the better off you will be.

    ----

    In a nutshell all a mathematical physics degree is:

    It's just an honours degree in Physics with

    - extra analysis
    - differential geometry

    ---



    [i'll repeat what i said in - who wants to be a mathematician]


    i would think that as an undergrad you'd aim for 70% of this outline... and if you take an extra year for your degree, maybe you dont need to take as much in grad school...

    But the ideal undergrad degree, would be this:

    Mathematical Physics
    -------------------------


    Calculus
    ------------
    Math 151 Calculus I
    Math 152 Calculus II
    Math 251 Calculus III
    Math 252 Vector Calculus I
    Math 313 Vector Calculus II / Differential Geometry
    Math 466 Tensor Analysis [needs Differential Geometry]
    Math 471 Special Relativity [needs Differential Geometry and Butkov] [Butkov needs Diff Eqs and Griffith EM]

    Analysis and Topology
    --------------------------
    Math 242 Intro to Analysis
    Math 320 Theory of Convergence [aka Advanced Calculus of One Variable]
    Math 425 Introduction to Metric Spaces
    Math 426 Introduction to Lebesque Theory
    Math 444 Topology

    Differential Equations
    ------------------------------
    Math 310 Introduction to Ordinary Differential Equations
    Math 314 Boundary Value Problems
    Math 415 Ordinary Differential Equations [needs Complex Analysis]
    Math 418 Partial Differential Equations [needs Differential Geometry]
    Math 419 Linear Analysis [needs Theory of Convergence]
    Math 467 Vibrations [needs Symon]
    Math 470 Variational Calculus [needs Symon and Differential Geometry]

    Complex Analysis
    -------------------------
    Math 322 Complex Analysis
    Math 424 Applications of Complex Analysis

    Linear Algebra
    --------------------
    Math 232 Elementary Linear Algebra
    Math 438 Linear Algebra
    Math 439 Introduction to Algebraic Systems [aka Abstract Algebra]



    minor things

    Fluid Mechanics [fluid motion/air motion/turbulence] - engineering like - turbulent gases and liquids
    ------------------
    Math 362 Fluid Mechanics I [needs Vector Calculus and Symon]
    Math 462 Fluid Mechanics II [needs Boundary Value Problems]

    Continuum Mechanics [aka deformation/stress/elasticity] - engineering like - elastic solids
    --------------------------
    Math 361 Mechanics of Deformable Media [needs Vector Calculus and Engineering Dynamics]
    Math 468 Continuum Mechanics [needs Differential Geometry and Boundary Value Problems]

    Probability and Statistics
    -----------------------------
    Math 272 Introduction to Probability and Statistics
    Math 387 Introduction to Stochastic Processes

    Numerical Analysis
    -----------------------
    Math 316 Numerical Analysis I [needs Fortran or PL/I]
    Math 416 Numerical Analysis II [needs Differential Equations]




    Mechanics - 1
    ------------
    Phys 120 Physics I
    Phys 211 Intermediate Mechanics [Symon and Kleppner-Kolenkow]
    Phys 413 Advanced Mechanics [Goldstein]

    Electricity and Magnetism - 2
    ------------------------------
    Phys 121 Physics II [Halliday-Resnick/Purcell/Kip/Griffith/Lorrian/Stump-Pollack]
    Phys 221 Intermediate Electricity and Magnetism
    Phys 325 Relativity and Electromagnetism
    Phys 326 Electronics and Instrumentation [tube/transistor/ic]
    Phys 425 Electromagnetic Theory [with Griffiths and Pollack you can do Jackson]

    Waves and Optics - 3
    ---------------------
    Phys 355 Optics

    Quantum Mechanics - 4
    ------------------------
    Phys 385 Quantum Physics [Griffiths]
    Phys 415 Quantum Mechanics
    Phys 465 Solid State Physics - [should be separate but basic QM is needed for these branches]
    Phys/Nusc 485 Particle Physics - [should be separate but basic QM is needed for these branches]

    Thermodynamics and Statistical Mechanics - 5
    --------------------------------------------------
    Phys 344 Thermal Physics [Baerlein/Schroeder]
    Phys 345 Statistical Mechanics [Reif]

    Mathematical Physics
    -------------------------
    Phys 384 Methods of Theoretical Physics I [Butkov]
    Phys 484 Methods of Theoretical Physics II [Butkov]

    Plasma Physics
    -----------------
    Phys 477 Plasma Physics [Chen]
     
  11. Mar 20, 2013 #10

    micromass

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    I know that groups are used very frequently in physics. But the point is that I *think* that an abstract algebra course will not be very useful. I say this because I think that an abstract algebra course doesn't really deal with those parts of group theory that are useful in physics.

    A group theory course in the math department is usually about finite groups. Things like Lie groups are not treated at all and I don't see how the theory of finite groups is going to be useful in Lie group theory (apart from usual definitions such as subgroups or quotient groups).

    Very useful in physics would be representation theory. But the representation theory course I was taught was for finite groups. And again, I don't think that this would be very useful to physicists.

    The OP should absolutely do an abstract algebra course though, if only to see what it is about. But I think it might be better to just self-study the group theory you need for physics. I've seen here a lot that it is possible to self-study those things in a very short time period.
     
  12. Mar 20, 2013 #11
    Although I agree with most of micromass's advice but I would take it with a grain of salt.. since he or she is still in high school.

    Hercuflea has the right idea. Your extra classes should be at least some physics courses or you'll have to make them up if you intend on pursuing a PhD in physics. Now if you go for a PhD in math with an adviser that does work in mathematical physics this won't be a problem and you could probably get by with no physics courses.. but it would be program dependent.

    Physics classes are a different animal than math classes, most of them will be grinding out solutions using approximation techniques or learning about assumptions to the model at hand. Unless it's a "____ for mathematicians" class the solutions and models will be most important not the theory. If this isn't your intention then I highly suggest looking into a math PhD with an adviser that specializes in mathematical physics. They're aren't many of these but I found a couple and almost considered that route too.

    If you go the math PhD route then take what others have said and take a good course(s) in probability you'd be surprised how much probability is used in physics and unfortunately many students don't recognize this.
     
  13. Mar 20, 2013 #12

    micromass

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    Ad hominem??

    Where exactly is my advice wrong??
    Either my advice was wrong and then you should point out what was wrong about it. Or my advice was correct. There is no need about snide remarks about my education level.
     
    Last edited: Mar 20, 2013
  14. Mar 20, 2013 #13
    - I say this because I think that an abstract algebra course doesn't really deal with those parts of group theory that are useful in physics.

    don't most people learn it through some of their higher up quantum mechanics courses though?

    Lie Algebra and Representation Theory wouldnt be touched by most undergrads, and same with group theory too.

    I'm under the impression that like 5% of physics degree people will take a complex variables class [usually the honours degrees in physics take it, but not the main stream]

    and Abstract Algebra or a higher up class in Linear Algebra is way rarer. Some that focus a lot in QM might take a math elective in that, but it depends if the math courses demand 1-2 classes in Analysis for a rather abstract and pure treatment of it. [Some Abstract Algebra or Linear Courses toss in that hurdle, some don't]

    but yeah if you're going for a Physics and Math double honours/Mathematical Physics option...

    the Abstract Algebra, Advanced Linear Algebra, and Analysis courses will be optional electives, a few required...

    and as you need it in physics, people will be picking up Tinkham's Group theory book, or Lie Algebra for Physics, but i'd think 96% will not touch that till 4th year or more likely grad school, or reading on their own

    I'd think people would just take QM I II III
    and getting a chapter in those things, will seek out the math classes for them later or the recommended readings in their texts.


    but i think Differential Geometry would be the course most often done... [though some places will want some analysis or place it way into 4th year math classes rather than something taken right after Calculus IV/Vector calculus]

    taking the Differential Geometry and Analysis courses, basically opens the doors to the most textbooks and prerequisites...

    just like vector calculus and one/two courses in Differential Equations is about 99% of all you need as an undergrad. [aka normal physics degree with a Rudin/Boundary Values Text or the second half of your Diff Eqns book/Complex variables book]

    and most honours physics people will take the the Boundary Values and Complex both anyways, but not the Analysis.

    but my guess is 99% of souls would be taking a 300 level Quantum Course after a Modern Physics course, and after that, they say, do i need Abstract Algebra class or text, and one more Linear Textbook...


    I think one year of Analysis, a course in Diff Geometry right after Vector Calculus, and Boundary Values right after Differential Equations, will make any scary physics or math prerequisites disappear for people early on...

    and the sooner someone has read Symon [Intermediate Mechanics] and Butkov [Mathematical Physics] the smoother it might be, for choosing most any course later on
     
  15. Mar 20, 2013 #14
    Hmm, yeah that's true. I took abstract algebra from the math department last semester and it was just finite groups/rings. It was undergraduate though (book was Fraleigh, so pretty easy :x).

    For Lie groups (I know that micromass knows this, but for others), it's just a differentiable manifold s.t the differentiable structure can work with the group structure (I.E C X C -> C by (a,b)-> ab^-1 would be a diff. mapping). Without knowing group notation/axioms this makes no sense! Could you self study this, I guess so. But, can't I say that about analysis too?

    Ok, i think it's about time I understand why real analysis is important for physics! From what I know, analysis is pretty much another name for calculus? Thus, complex analysis is important because it offers more integration techniques for physics students. But why would real analysis be important? Does a physics student need to know proofs for calculus on the reals? How would constructing the reals be any different than say, proving H is a subgroup of G using axioms and group notation?

    Maybe I should make a new thread/pm for this.
     
  16. Mar 20, 2013 #15

    micromass

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    I don't think real analysis would be useful for all physicists. But the OP specifcally said "mathematical physics", so my responses were tailored to that.

    In my opinion, to decently grasp the mathematics behind quantum mechanics, you absolutely need to know real analysis, functional analysis and measure theory. Of course, one can get a very decent understand of QM without all these analysis things, but you'll need them you're into mathematical physics.

    Of course, an abstract algebra course will be useful to understand Lie groups and stuff. But only a very small part of the abstract algebra course. I'm no physicist, but I doubt that things like Sylow's theorems or even Lagrange's theorems are used very often in physics. People are of course welcome to show me wrong!

    On the other hand, most of real analysis will actually be useful somewhere. But I'm not talking about a first analysis course. A first analysis course will just be "calculus made rigorous" and won't be very interesting. Constructing the reals is certainly not interesting for physics (or even math) students. But later analysis courses will be extremely useful!!

    Also, I consider real analysis to be more difficult (but also more fun) than abstract algebra. So I think it is very possible to self-study all the abstract algebra you need. But it would be more difficult with analysis. But that is personal.
     
  17. Mar 20, 2013 #16
    OP, what are your goals?

    Why are you doing a degree in mathematics for a PhD in physics?

    Algebra is fairly useful but not as broadly as analysis courses and linear algebra are.
     
  18. Mar 20, 2013 #17
    romsofia - i think it's about time I understand why real analysis is important for physics! ... But why would real analysis be important? Does a physics student need to know proofs for calculus on the reals?

    micromass - I don't think real analysis would be useful for all physicists. But the OP specifcally said "mathematical physics"

    ---

    I think if you're going into mathematical physics, topology would be a cool option [of many options], and there you'd need one textbook [or two small classes] of analysis.

    One could argue that for a degree in physics, you dont really need a class in complex variables, but an honours physics person would probably be forced into taking it.

    When you get hit with heat and wave equations and other pdf's and you're dealing with courant-hilbert [is that applied analysis after complex analysis, or is it just mathematical physics]

    But if you were doing a mathematical physics option, you'll probably be taking Complex Variables and Differnetial Geometry

    and weirder crap like fluid mechanics, or topology...


    Mathematical Physics is just a good way of getting a physics and math degree in one for an extra year's suffering, without getting too crazy into the pure math, but then again you did take analysis and might get tricked into topology...

    so there is some truth that maybe a physics degree in some cases needs more math than a 'math degree'

    ---

    for my money, a mathematical physics degree syllabus is a nice way of showing what math is most useful for future physics study.

    it also shows you how a lot of doors can open with a course in analysis.

    ----

    again

    a. get your vector calculus fast and try differential geometry [aka vector calc II]
    b. take your Differential Equations [and finish your book! or get Powers Boundary Value Problems book]

    [now you got your physics hurdles solved with a. and b.]

    c. take your analysis
    d. take your Symon/Butkov/Griffith's EM early!

    most any courses will now have almost no hurdles with prerequisites



    if you want physics with differential geometry and topology, congrats someone tricked you into being a wannabe mathematical physicist!
     
  19. Mar 20, 2013 #18
    Someone mentioned that I might be in High school--that would be incorrect. I am not working exclusively towards a degree in mathematics; I am doing a double major in mathematics and physics.
     
  20. Mar 22, 2013 #19
    Well, yes, it is ad hominem. It's not a snide remark either, it's a fact about you which is stated in your profile. Since you are in high school, I'm assuming that you've never taken a graduate class nor gone through the graduate admissions process. I just find it strange, if that previous statement is true, that you talk about graduate school. I'm not trying to be offensive and I'm sorry if that's how you took it. If you have taken a graduate course then I would be assuming that you aren't in high school and I would be under the impression that you're impersonating a minor which I'm not really sure how to react to that..

    Anyway, to the OP and what I said earlier, I believe the clear distinction should be made of which PhD you intend to pursue. If it's math then take more math than physics courses to prepare you for graduate level math courses, aka lots of proofs, theorems, etc. If it's physics then you should focus more on physics classes. I did my undergrad as a double major in math and physics as well and there is a huge difference between grad and undergrad level classes. I have also taken both math and physics graduate classes. Since I'm doing a physics PhD the only courses that were absolutely required were physics and a couple of math methods classes in the physics department. I've taken a handful of pure math courses because I really like it but by no means were they required.

    I'm guessing that since you said "mathematical physics" you'll be better off in the math department for a PhD. Those people are the ones that add rigor to physics. They might spend their time proving results that might have be taken on faith by physicists. If you do "theory" in the physics department you will be trying to make models of physical phenomena. Some number crunching, programming, maybe even experimental data analysis. Most of all you will learn how to approximate results and come up with a theoretical model. I stress these differences because you probably won't see them in an undergrad environment. Ask you professors that work in the math department that do research in physics vs. professors that are theoretical physicists. There is a notable difference in how they approach the same problems. When you answer which one you enjoy more then you will know side you fall on.
     
  21. Mar 22, 2013 #20

    micromass

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    Again, what was wrong about my advice? If you warn the OP to take my advice with a grain of salt, then that means that something was wrong. I'd like to know what it was.
     
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