how much is one expected to know to go into theory?

 Posts: 198
 Joined: Thu Feb 05, 2009 11:45 pm
how much is one expected to know to go into theory?
Say someone wants to go to grad school for theoretical physics, say string theory or what not, how much knowledge is expected before one entered grad school? What kind of advanced math classes, advanced physics class, undergrad research, etc. etc.? I'm not only talking about getting into grad school part, but also whether one can get maximum benefits from grad school in theoretical physics  so that one does not need to spend the first few years in taking classes to just be prepared for doing anything useful.
More specifically I'm trying to decide what grad physics/math class I should take for next semester...
More specifically I'm trying to decide what grad physics/math class I should take for next semester...
Last edited by axiomofchoice on Sat Feb 28, 2009 11:20 pm, edited 1 time in total.

 Posts: 26
 Joined: Sat Jan 31, 2009 5:59 pm
Re: how much is one expected to know to go into theory?
.
Last edited by Theoretischer on Wed Mar 11, 2009 11:33 pm, edited 1 time in total.
Re: how much is one expected to know to go into theory?
As far as I can tell, first of all you need to ace your GREs. Or you will be screwed regardless of the math/etc. you know.
This is because the tests don't measure your knowledge of things that are important in theory (mathematically). So, even though you might be using some highpowered math to study the solutions to the Einstein equations in your research as an undergrad (this is what I do), unless you know how to answer PGRE problems on the Doppler effect or highpass filters, it doesn't matter. And knowing graduate level geometry or algebra will not refresh your memory for high school polygon nonsense on the general GRE (my mistake  I never reviewed for that test, and ended up forgetting something, apparently, since my score is quite low).
My advice would be to make sure you are very firm on all the fundamentals of physics first. You can probably pick up most of what you need in graduate school (that's why they have graduate school). It is more important to start with a solid background  you can pick up what you need so long as you're not trying to catch up on weaknesses from undergrad physics at the same time. But you should at least strive to get a math minor, and if you want to work on mathematically interesting problems then you will need a firm foundation in linear algebra, group theory, differential geometry and analysis, at the very least. Topology is also important, but I think the first three I listed appear most frequently in physics problems. So if you've already got the fundamentals covered well, then take some math and read like crazy. It is also useful to have a class in "Math Methods", which if taught in a physics department will involve few elegant proofs, but will introduce you to things too "dirty" for pure mathematicians, e.g. actually solving things and not just writing proofs. This can be a nice complementary set of tools.
For my research in GR I use huge amounts of linear & multilinear algebra (tensors, etc.) and differential geometry (and some group theory, come to think of it)  and it's nice to pick up Wald's book or Boothby's book and feel at the right level to read it, though it's a graduate textbook and I'm still an undergrad. So I've found doing research in the field to be very helpful. The problem was, of course, that it took three years before I knew enough math to do any research in GR, so you might not get as much research in if you wait (my fix to this was to do experimentally based research for my first few years, and then when I had the math background I started working with my current research advisor).
Good luck!
PS: Just make sure you don't forget some physics when you're learning all the math. I took a year of just pure math (to finish my math degree on time) the fall/spring before my PGRE and it's amazing how much E&M and mechanics I forgot when I was off doing Galois theory. Don't make my mistake! keep reviewing basics!
This is because the tests don't measure your knowledge of things that are important in theory (mathematically). So, even though you might be using some highpowered math to study the solutions to the Einstein equations in your research as an undergrad (this is what I do), unless you know how to answer PGRE problems on the Doppler effect or highpass filters, it doesn't matter. And knowing graduate level geometry or algebra will not refresh your memory for high school polygon nonsense on the general GRE (my mistake  I never reviewed for that test, and ended up forgetting something, apparently, since my score is quite low).
My advice would be to make sure you are very firm on all the fundamentals of physics first. You can probably pick up most of what you need in graduate school (that's why they have graduate school). It is more important to start with a solid background  you can pick up what you need so long as you're not trying to catch up on weaknesses from undergrad physics at the same time. But you should at least strive to get a math minor, and if you want to work on mathematically interesting problems then you will need a firm foundation in linear algebra, group theory, differential geometry and analysis, at the very least. Topology is also important, but I think the first three I listed appear most frequently in physics problems. So if you've already got the fundamentals covered well, then take some math and read like crazy. It is also useful to have a class in "Math Methods", which if taught in a physics department will involve few elegant proofs, but will introduce you to things too "dirty" for pure mathematicians, e.g. actually solving things and not just writing proofs. This can be a nice complementary set of tools.
For my research in GR I use huge amounts of linear & multilinear algebra (tensors, etc.) and differential geometry (and some group theory, come to think of it)  and it's nice to pick up Wald's book or Boothby's book and feel at the right level to read it, though it's a graduate textbook and I'm still an undergrad. So I've found doing research in the field to be very helpful. The problem was, of course, that it took three years before I knew enough math to do any research in GR, so you might not get as much research in if you wait (my fix to this was to do experimentally based research for my first few years, and then when I had the math background I started working with my current research advisor).
Good luck!
PS: Just make sure you don't forget some physics when you're learning all the math. I took a year of just pure math (to finish my math degree on time) the fall/spring before my PGRE and it's amazing how much E&M and mechanics I forgot when I was off doing Galois theory. Don't make my mistake! keep reviewing basics!
Re: how much is one expected to know to go into theory?
If you knew everything already then what's the point of going to grad school?
What you need, as mhazelm said, is to master all you are supposed to know as undergrad, i.e. Advanced Mechanics, E&M, Intermediate QM, Stat Mech, Optics, and maybe Intro to Particle Physics and Solid State. Anything else in addition will be a plus, but you don't have to have it.
As for what math classes to take, I would say try to take as much as you can from the following:
Linear Algebra and Calculus (do I even need to mention them)
Ordinary Differential Equations
Partial Differential Equations
Real Analysis
Complex Analysis
Numerical Analysis
At least one more proofbased class (I'd say Abstract Algebra)
Those are pretty much the exact courses I have taken, and I've been accepted into several of the most prestigious theoretical physics programs.
What you need, as mhazelm said, is to master all you are supposed to know as undergrad, i.e. Advanced Mechanics, E&M, Intermediate QM, Stat Mech, Optics, and maybe Intro to Particle Physics and Solid State. Anything else in addition will be a plus, but you don't have to have it.
As for what math classes to take, I would say try to take as much as you can from the following:
Linear Algebra and Calculus (do I even need to mention them)
Ordinary Differential Equations
Partial Differential Equations
Real Analysis
Complex Analysis
Numerical Analysis
At least one more proofbased class (I'd say Abstract Algebra)
Those are pretty much the exact courses I have taken, and I've been accepted into several of the most prestigious theoretical physics programs.
Re: how much is one expected to know to go into theory?
If you're interested in string theory, you eventually need to be familiar with this kind of math (I'm taking this list from a response from a family friend who is now a string theory prof at a top5 university, in response to a very similar question I asked him a few years ago):
Complex analysis (at the level of Chapter Zero of Griffiths and Harris)
Differential geometry (Boothby is a good intro text; I also like "Differential Geometry and Lie Groups for Physicists" by Fecko)
Algebra (Cox, Little and O'Shea)
Group theory/Lie theory/representation theory (Georgi is more physicsoriented, while Fulton and Harris is more mathoriented)
Algebraic topology (Bott and Tu is the bible)
Algebraic geometry (Cox Little and O'Shea is a good introduction)
Pretty much everything on this list is touched on in Nakahara, "Geometry, Topology, and Physics"; it's not a great book to learn from, but it's an excellent compendium of all the math you'll eventually need to know.
The standard wisdom is that you're supposed to take physics classes and pick up the required math on the side, but I've found that taking these kinds of classes in the math department helps solidify my understanding more than trying to teach myself. Also, the more of this math you know, the more compelling your grad school apps will be, and the quicker you'll be able to start real research.
Complex analysis (at the level of Chapter Zero of Griffiths and Harris)
Differential geometry (Boothby is a good intro text; I also like "Differential Geometry and Lie Groups for Physicists" by Fecko)
Algebra (Cox, Little and O'Shea)
Group theory/Lie theory/representation theory (Georgi is more physicsoriented, while Fulton and Harris is more mathoriented)
Algebraic topology (Bott and Tu is the bible)
Algebraic geometry (Cox Little and O'Shea is a good introduction)
Pretty much everything on this list is touched on in Nakahara, "Geometry, Topology, and Physics"; it's not a great book to learn from, but it's an excellent compendium of all the math you'll eventually need to know.
The standard wisdom is that you're supposed to take physics classes and pick up the required math on the side, but I've found that taking these kinds of classes in the math department helps solidify my understanding more than trying to teach myself. Also, the more of this math you know, the more compelling your grad school apps will be, and the quicker you'll be able to start real research.

 Posts: 198
 Joined: Thu Feb 05, 2009 11:45 pm
Re: how much is one expected to know to go into theory?
Thanks guys, that is useful! I'll be finishing my math major requirements soon (so the standard basics + analysis + algebra + topology), and I'm trying to decide between a few grad math classes. I figure my "hole" in mathematical understanding for uses in physics is differential geometry, so I guess I'll take something on that line. But between algebraic topology and algebra (grad level) I can't decide; would algebraic topology be more useful (since I already have a year of undergrad algebra)?
Physics grad course wise, among the "core" first year grad courses like quantum mechanics, e&m, classical mechanics, what would you recommend taking if I can only take one (or at most two)?
Physics grad course wise, among the "core" first year grad courses like quantum mechanics, e&m, classical mechanics, what would you recommend taking if I can only take one (or at most two)?
Re: how much is one expected to know to go into theory?
Differential geometry is an excellent choice. It shows up in physics in more ways than you can imagine. Algebraic topology would definitely be more useful (read: more readily applicable) than pure abstract algebra. If you can take the diff geo and alg. top. courses concurrently, that would give you an introduction to differential forms from two very different angles, and the courses would complement each other nicely. But neither is really a prerequisite for the other.
Physicswise, E+M and classical mech. are most useful for passing your quals  wait until grad school to take those. You will learn a LOT in grad. quantum, though, especially if you want to go into highenergy theory.
Physicswise, E+M and classical mech. are most useful for passing your quals  wait until grad school to take those. You will learn a LOT in grad. quantum, though, especially if you want to go into highenergy theory.
Re: how much is one expected to know to go into theory?
Axiom...
Refer to this link by T'hooft at Utrecht on How to become a good theoretical Physicist:
http://www.phys.uu.nl/~thooft/theorist.html
Best wishes.
Refer to this link by T'hooft at Utrecht on How to become a good theoretical Physicist:
http://www.phys.uu.nl/~thooft/theorist.html
Best wishes.
Re: how much is one expected to know to go into theory?
Commutative algebra is essential if you're serious about Algebraic geometry. Also, if you're interested in String theory, Complex manifolds may also be useful (book by Huybrechts is a good intro). Algebraic topology is also essential (book by May is good).
Re: how much is one expected to know to go into theory?
Quantum Field Theory at the level of the textbooks Ramond, Field theory: A Modern Primer and
Itzykson &Zuber, Quantum Field Theory
General Relativity at the level of Hawking and Ellis, The large scale structure of spacetime
Differential Geometry at the level of Kobayashi and Nomizu, Foundations of Differential Geometry
Algebraic Topology and fiber bundles at the level of Spanier, Algebraic Topology;
Milnor and Stasheff, Characteristic classes; Steenrod, The Topology of Fiber Bundles
Riemann Surfaces at the level of Farkas and Kra, Riemann Surfaces
Commutative Algebra at the level of D. Eisenbud, Commutative Algebra with a View Toward Algebraic Geometry
and Matsumura, Commutative ring theory
Complex Geometry at the level of Griffiths and Harris, Principles of Algebraic Geometry
Algebraic geometry at the level of Hartshorne, Algebraic Geometry
Homological algebra at the level of Stammbach & Hilton, A course in homological algebra and I. Manin, Methods of homological algebra
Toric Geometry at the level of T. Oda, Convex bodies and algebraic geometry.
source: http://www.maths.tcd.ie/~string/phd_program.html
Itzykson &Zuber, Quantum Field Theory
General Relativity at the level of Hawking and Ellis, The large scale structure of spacetime
Differential Geometry at the level of Kobayashi and Nomizu, Foundations of Differential Geometry
Algebraic Topology and fiber bundles at the level of Spanier, Algebraic Topology;
Milnor and Stasheff, Characteristic classes; Steenrod, The Topology of Fiber Bundles
Riemann Surfaces at the level of Farkas and Kra, Riemann Surfaces
Commutative Algebra at the level of D. Eisenbud, Commutative Algebra with a View Toward Algebraic Geometry
and Matsumura, Commutative ring theory
Complex Geometry at the level of Griffiths and Harris, Principles of Algebraic Geometry
Algebraic geometry at the level of Hartshorne, Algebraic Geometry
Homological algebra at the level of Stammbach & Hilton, A course in homological algebra and I. Manin, Methods of homological algebra
Toric Geometry at the level of T. Oda, Convex bodies and algebraic geometry.
source: http://www.maths.tcd.ie/~string/phd_program.html