Torn between Math and Plasma Physics for grad school

[Hercuflea]

I did my undergrad in mathematics, and I really enjoyed studying that subject. After graduating, I've been continuing my study in Real Analysis with one of my undergrad professors, and I really feel like I'm blooming into a decent analysis student. This week I'm going to start learning functional analysis and lebesgue integration as a directed study over Skype, and I'm really looking forward to it.

It's almost as if math is the only "pure" subject out there, i.e. rigorous in the sense that every thing you say must have a precise meaning and there is no tolerance for hand-waviness or ambiguity (at least that's the way I like to do it and some of my best professors did). That is the one thing that I do not like about physics, physicists are too imprecise about what they say, and they introduce variables and formulas without even naming or defining them half the time.

At the same time, I also studied plasma physics and E&M and classical mechanics as an undergrad, and my goal throughout college has been to end up working in the nuclear fusion research field someday. I even did a summer internship in a plasma physics lab in a nuclear engineering department last year, and thoroughly enjoyed it. I was accepted into a M.Sc. in Fusion Energy at the University of York in the UK, and I'm going to go there to do my Master's this year. I also really think I will enjoy that course and it pertains more directly to what I'd like to do as my future career.

So basically, I'm kind of at a crossroads as to whether I should continue to pursue plasma physics as a Ph.D. option, or if I should go into a Math Ph.D. instead. Last year, when I was applying to grad schools, I got into the Uni. of Wisconsin among others for nuclear engineering, and I was offered a spot in a computational/theory group where they do a lot of work in numerical analysis for plasma simulations and also pure theory. It seemed like a really good fit, but I got a fellowship to go to the UK so I'm taking a year to go and do that.

Sometimes I just feel as if I would be "selling out" by not doing a Math Ph.D. I really enjoy analysis and I want to know everything about it that I can learn, but honestly I don't really want to be a professional mathematician. I'm just afraid my knowledge of analysis and other relevant subjects would be less if I didn't do a Math Ph.D. I've always wanted to work in plasma physics and nuclear fusion research, though. My ideal future career would probably be to be a full-time researcher at a national lab or NASA. At Wisconsin, they have an http://www.math.wisc.edu/foundations intended for Ph.D. students in other subjects, which is only course-based. Also, St. Andrews University in Scotland has a Ph.D. in Math, but there is an active group studying solar plasma physics in the math department. I'm just a little confused as to what is the best path for me to take. Right now I'm feeling Wisconsin, but who knows? What if I end up falling in love with analysis/math even more than physics?

edit* The other dilemma is that, if I go into math, I could probably only get into a rank 30-50 school at best with my credentials (i'm guessing, I haven't taken the math GRE yet), but I could get into a top 5 school in Nuc. Eng., since I already have once.

 

[Hercuflea]

Don't get me wrong, I enjoy plasma physics as much as or maybe more than I do mathematics. I'm just pursuing advanced math courses because I believe it will make me a better scientist in the end. An N.E. professor once told me that it's always good to have a "diverse toolbox" as an engineer, for example some results from functional analysis can be used to solve PDEs. I've always been one to want to know the theory cold, and then I can apply it to any situation and that's kind of how I see mathematics.

I just think that physics could be much easier for students to learn if the textbook actually said what it meant and meant what it said. For example, in almost every physics book I've read, they will introduce variables or notation into a formula without even telling the reader what it means. "Einstein notation" for vectors in relativity is horrendous, it's like these people spent more time trying to figure out how to make an incoherent notation for a simple mathematical concept that you learn in linear algebra 1 than if they had just written the summation out properly in the first place. But I digress.

Essentially I really do enjoy both math and plasma physics, and I'd like my career to involve both, but I am not sure which Ph.D. would actually be better for my future. (Can one even get into plasma physics research with a Math Ph.D.? If you play your cards right?)

 

[micromass]

What exactly do you want to do after you do a PhD? Where do you want to work? Both a math PhD and a physics PhD have quite some options, but they tend to be different. Of course, you could continue in academia if that's your wish, but be sure to have a plan B ready since things might develop differently from what you think.

Don't get me wrong, I enjoy plasma physics as much as or maybe more than I do mathematics. I'm just pursuing advanced math courses because I believe it will make me a better scientist in the end. An N.E. professor once told me that it's always good to have a "diverse toolbox" as an engineer, for example some results from functional analysis can be used to solve PDEs. I've always been one to want to know the theory cold, and then I can apply it to any situation and that's kind of how I see mathematics.

Sure, mathematics provides a decent toolbox. But if you go for a PhD in plasma physics, then you would probably get the correct toolbox anyway out of your physics courses. It is only if you're doing something very theoretical that mathematics courses like analysis will help you.

I just think that physics could be much easier for students to learn if the textbook actually said what it meant and meant what it said. For example, in almost every physics book I've read, they will introduce variables or notation into a formula without even telling the reader what it means. "Einstein notation" for vectors in relativity is horrendous, it's like these people spent more time trying to figure out how to make an incoherent notation for a simple mathematical concept that you learn in linear algebra 1 than if they had just written the summation out properly in the first place. But I digress.

Funny because I view Einstein notation as one of the best notations ever invented. The notation is extremely useful and is not just leaving out the summation. Einstein notation can actually tell you which expressions which make no sense and which do, something that the other notation cannot. I recommend reading through Lee's smooth manifold book, there you can sometimes see the power of the notation.

Essentially I really do enjoy both math and plasma physics, and I'd like my career to involve both, but I am not sure which Ph.D. would actually be better for my future. (Can one even get into plasma physics research with a Math Ph.D.? If you play your cards right?)

I don't think you should count on plasma physics research with a Math PhD.

 

[homeomorphic]

"Einstein notation" for vectors in relativity is horrendous, it's like these people spent more time trying to figure out how to make an incoherent notation for a simple mathematical concept that you learn in linear algebra 1 than if they had just written the summation out properly in the first place. But I digress.

It's not really horrendous if you know how to look at it. When I first saw it, coming from more of a math background, it seemed ridiculous, but with a shift of perspective, it's really just as intuitive as the mathematicians notation and better for doing coordinate-based stuff. Roger Penrose talks about this slight shift in attitude, needed to properly comprehend the index notation, in his book, The Road to Reality.

It's almost as if math is the only "pure" subject out there, i.e. rigorous in the sense that every thing you say must have a precise meaning and there is no tolerance for hand-waviness or ambiguity (at least that's the way I like to do it and some of my best professors did). That is the one thing that I do not like about physics, physicists are too imprecise about what they say, and they introduce variables and formulas without even naming or defining them half the time.

Sounds like you are falling in love with upper level UNDERGRADUATE math. You might be in for a nasty surprise if you go further:

http://terrytao.wordpress.com/career-advice/there%E2%80%99s-more-to-mathematics-than-rigour-and-proofs/


Personally, although I went through my phase where I really enjoyed rigor, I would have been happy to let go of that to the extent that Tao is talking about there. What was more troubling to me was how complicated things became. There are results from the 1960s that no one really understands thoroughly anymore, but they use them all the time. It's really awful if you are someone who likes things to be like a thorough undergraduate math textbook where you just write down your axioms and happily proceed to prove everything from first principles (this sentence is not just me by the way--I'm basically just passing on and agreeing with the sentiment of my friend who studied PDE, here). It's really a very misleading picture of what being a mathematician has now become like.

I'll give you my favorite two links that illustrate this problem.

http://mathoverflow.net/questions/99506/blackbox-theorems

http://blogs.scientificamerican.com/guest-blog/2013/10/01/voevodskys-mathematical-revolution/

In the 2nd one, the Fields medalist Voevodsky points out how his computer-aided proof ideas will improve collaboration because, at the moment, it requires too much trust in your collaborators results, since there isn't enough time to check them yourself. Of course, I'm sure this doesn't apply to ALL research, but it is never the less pretty terrifying to someone like myself, who doesn't really consider any of his knowledge to be genuine unless he fully understands it for himself.

Upon leaving math, I noticed how the same trend was apparent more generally, and then found this article, which struck a bit of a chord, and pretty well sums up my dispair in not being able to understand enough of the world we have created:

http://aeon.co/magazine/technology/is-technology-making-the-world-too-complex/

Sometimes I just feel as if I would be "selling out" by not doing a Math Ph.D.

Doing physics instead would hardly be selling out. I think of math and physics PhDs as people who are willing to make the ultimate sacrifice of their careers in order to pursue knowledge. It takes many years of being a grad student and not making money and being too busy to have much of a life. Yet, for most people, they won't get much out of all this sacrifice and will just end up having to find something else to do when they fail to get a tenure-track position. If they do get one, woo-hoo, they just won the right to be constantly inundated with mountains of work and pressure to publish, or possibly spend an inordinate amount of time being a grant-beggar.

I really enjoy analysis and I want to know everything about it that I can learn, but honestly I don't really want to be a professional mathematician.

Good. If you don't really want to, then don't even consider it. Even people who really want to do it may be in for a nasty surprise, as I found out the hard way.

I'm just afraid my knowledge of analysis and other relevant subjects would be less if I didn't do a Math Ph.D.

Not sure you'd really be missing that much.

 

[caldweab]

I did my undergrad in mathematics, and I really enjoyed studying that subject. After graduating, I've been continuing my study in Real Analysis with one of my undergrad professors, and I really feel like I'm blooming into a decent analysis student. This week I'm going to start learning functional analysis and lebesgue integration as a directed study over Skype, and I'm really looking forward to it.

It's almost as if math is the only "pure" subject out there, i.e. rigorous in the sense that every thing you say must have a precise meaning and there is no tolerance for hand-waviness or ambiguity (at least that's the way I like to do it and some of my best professors did). That is the one thing that I do not like about physics, physicists are too imprecise about what they say, and they introduce variables and formulas without even naming or defining them half the time.

At the same time, I also studied plasma physics and E&M and classical mechanics as an undergrad, and my goal throughout college has been to end up working in the nuclear fusion research field someday. I even did a summer internship in a plasma physics lab in a nuclear engineering department last year, and thoroughly enjoyed it. I was accepted into a M.Sc. in Fusion Energy at the University of York in the UK, and I'm going to go there to do my Master's this year. I also really think I will enjoy that course and it pertains more directly to what I'd like to do as my future career.

So basically, I'm kind of at a crossroads as to whether I should continue to pursue plasma physics as a Ph.D. option, or if I should go into a Math Ph.D. instead. Last year, when I was applying to grad schools, I got into the Uni. of Wisconsin among others for nuclear engineering, and I was offered a spot in a computational/theory group where they do a lot of work in numerical analysis for plasma simulations and also pure theory. It seemed like a really good fit, but I got a fellowship to go to the UK so I'm taking a year to go and do that.

Sometimes I just feel as if I would be "selling out" by not doing a Math Ph.D. I really enjoy analysis and I want to know everything about it that I can learn, but honestly I don't really want to be a professional mathematician. I'm just afraid my knowledge of analysis and other relevant subjects would be less if I didn't do a Math Ph.D. I've always wanted to work in plasma physics and nuclear fusion research, though. My ideal future career would probably be to be a full-time researcher at a national lab or NASA. At Wisconsin, they have an http://www.math.wisc.edu/foundations intended for Ph.D. students in other subjects, which is only course-based. Also, St. Andrews University in Scotland has a Ph.D. in Math, but there is an active group studying solar plasma physics in the math department. I'm just a little confused as to what is the best path for me to take. Right now I'm feeling Wisconsin, but who knows? What if I end up falling in love with analysis/math even more than physics?

edit* The other dilemma is that, if I go into math, I could probably only get into a rank 30-50 school at best with my credentials (i'm guessing, I haven't taken the math GRE yet), but I could get into a top 5 school in Nuc. Eng., since I already have once.

Sounds like you answered your own question. You should be doing a phd in plasma physics or nuclear engineering at school that does fusion research, University of Wisconsin is well known for that actually. If you can get into the computational/theory group there will be plenty of mathematics to keep you happy. I'm a nuclear engineering student and there is a lot of interesting math problems in the field. If you do a phd in mathematics, how is that going to help you in your dreams to do fusion research? Granted there are some mathematicians that work on fusion research but you also said you don't want to be a professional mathematician.

 

[Hercuflea]

Thank you all for the very helpful responses.

What exactly do you want to do after you do a PhD? Where do you want to work? Both a math PhD and a physics PhD have quite some options, but they tend to be different. Of course, you could continue in academia if that's your wish, but be sure to have a plan B ready since things might develop differently from what you think.

I don't think you should count on plasma physics research with a Math PhD.

I am sure I want to work in the field of plasma physics after my PhD. Whether it's as a post-doc, national lab researcher, NASA, or overseas I have always wanted to go where the plasma jobs are. I used to think I wanted to be an experimentalist but I find myself being drawn more towards the theory/parallel computing areas.

I met a professor at Michigan who did all of his education through PhD in Mathematics, and he is a full professor of Nuclear Engineering there at UM. He works in theoretical/computational radiation transport problems. He told me he ended up in NE because of some connection he had to Los Alamos after his PhD was finished. So I guess it's possible to go into another field with a Math PhD, but I probably shouldn't count on it.

It's not really horrendous if you know how to look at it. When I first saw it, coming from more of a math background, it seemed ridiculous, but with a shift of perspective, it's really just as intuitive as the mathematicians notation and better for doing coordinate-based stuff. Roger Penrose talks about this slight shift in attitude, needed to properly comprehend the index notation, in his book, The Road to Reality.

Sounds like you are falling in love with upper level UNDERGRADUATE math. You might be in for a nasty surprise if you go further:

http://terrytao.wordpress.com/career-advice/there%E2%80%99s-more-to-mathematics-than-rigour-and-proofs/


Personally, although I went through my phase where I really enjoyed rigor, I would have been happy to let go of that to the extent that Tao is talking about there. What was more troubling to me was how complicated things became. There are results from the 1960s that no one really understands thoroughly anymore, but they use them all the time. It's really awful if you are someone who likes things to be like a thorough undergraduate math textbook where you just write down your axioms and happily proceed to prove everything from first principles (this sentence is not just me by the way--I'm basically just passing on and agreeing with the sentiment of my friend who studied PDE, here). It's really a very misleading picture of what being a mathematician has now become like.

I'll give you my favorite two links that illustrate this problem.

http://mathoverflow.net/questions/99506/blackbox-theorems

http://blogs.scientificamerican.com/guest-blog/2013/10/01/voevodskys-mathematical-revolution/

In the 2nd one, the Fields medalist Voevodsky points out how his computer-aided proof ideas will improve collaboration because, at the moment, it requires too much trust in your collaborators results, since there isn't enough time to check them yourself. Of course, I'm sure this doesn't apply to ALL research, but it is never the less pretty terrifying to someone like myself, who doesn't really consider any of his knowledge to be genuine unless he fully understands it for himself.

Upon leaving math, I noticed how the same trend was apparent more generally, and then found this article, which struck a bit of a chord, and pretty well sums up my dispair in not being able to understand enough of the world we have created:

http://aeon.co/magazine/technology/is-technology-making-the-world-too-complex/




Doing physics instead would hardly be selling out. I think of math and physics PhDs as people who are willing to make the ultimate sacrifice of their careers in order to pursue knowledge. It takes many years of being a grad student and not making money and being too busy to have much of a life. Yet, for most people, they won't get much out of all this sacrifice and will just end up having to find something else to do when they fail to get a tenure-track position. If they do get one, woo-hoo, they just won the right to be constantly inundated with mountains of work and pressure to publish, or possibly spend an inordinate amount of time being a grant-beggar.

Not sure you'd really be missing that much.

Wow, you posted some really interesting stuff...I will definitely look into it. It's just that I've always heard that functional analysis and operator theory provides a lot of useful results in physics and different ways of solving PDEs. Problem is you have to take years of math courses to actually get to the level of understanding it, and by then you might as well have done an MS in math or something.

Sounds like you answered your own question. You should be doing a phd in plasma physics or nuclear engineering at school that does fusion research, University of Wisconsin is well known for that actually. If you can get into the computational/theory group there will be plenty of mathematics to keep you happy. I'm a nuclear engineering student and there is a lot of interesting math problems in the field. If you do a phd in mathematics, how is that going to help you in your dreams to do fusion research? Granted there are some mathematicians that work on fusion research but you also said you don't want to be a professional mathematician.

Yeah...I've been pretty committed to reapplying to Wisconsin this year, hopefully getting in again and going there. I would still like to do the optional MA in Math though, because you can pick which courses you want to take, and I'd probably pick all the ODE, PDE, and analysis courses.

 

[Arsenic&Lace]

Thank you all for the very helpful responses.

It's just that I've always heard that functional analysis and operator theory provides a lot of useful results in physics and different ways of solving PDEs. Problem is you have to take years of math courses to actually get to the level of understanding it, and by then you might as well have done an MS in math or something.

Use Web of Science to get a sense of how deeply functional analysis has pentrated whatever fields of physics you find interesting. Look for papers published by mathematicians regarding those fields with keywords from operator theory and look at citation maps; see where they go. Look at papers published by physicists and sort them by various keywords; an interesting question would be if physicists who make use of very formal mathematical arguments produce results which spread through the literature. This will provide you an indication which is quantitative of how powerful these methods may be.

Of course given the complexity of the data, you need to be extremely careful, as it is very possible to produce results which are misleading. For instance, it could be that there is a cultural barrier between physics and math departments where low levels of citation involving ideas from functional analysis are merely caused by a distaste for more theoretical mathematics, and that the ideas are still powerful tools which could produce useful research results, meaning that, ultimately, you will need to thoroughly investigate the topics involved yourself if you want to decide with any certainty what is useful and what isn't.

 

[the_wolfman]

I think we have talk a few times before.

It's just that I've always heard that functional analysis and operator theory provides a lot of useful results in physics and different ways of solving PDEs. Problem is you have to take years of math courses to actually get to the level of understanding it, and by then you might as well have done an MS in math or something.

We use a lot of functional analysis and operator theory in plasma physics. Ultimately the spectrum of our operators determines the stability of the plasma. In the worst cases, instabilities can be disruptive leading to the termination of a discharge. Or they can be "weaker" and degrade confinement. Understanding the spectrum of instabilities is a necessary part of optimising our experiments allowing us to go to high pressure.

The operators we work with in plasma physics are ugly beasts. They often have accumulation points, continua, and discrete eigenvalues. Not to mention that they are often complex and non-hermitian.

Also, we have to pay careful attention to the spectrum of the operators in numerical simulations. Its not enough to know that our numerical approximations converge to the physical model. We have to know how it converges. Often the numerical approximation will under/over-predict stability. Knowing which, will help us interrupt our simulation results.

I guess my point is that an interest in functional analysis/operator theory should not dissuade you from plasma physics. It is an important part of the work we do.

Yeah...I've been pretty committed to reapplying to Wisconsin this year, hopefully getting in again and going there. I would still like to do the optional MA in Math though, because you can pick which courses you want to take, and I'd probably pick all the ODE, PDE, and analysis courses.

Plasma/Fusion PhD students and the UW take a lot courses! My course load was almost twice that of other non-plasma PhD students in my program. The UW also requires all PhD students to have a secondary (minor) focus. The problem with the Math masters is that it will limit what courses you can take. If you get the math masters, most of your math courses will have to be taught by the math department.

For example, my secondary focus was computational math. I was able to take any math course offered by the math department, but I was also free to count "math" courses taught be other departments towards my minor focus. I took CFD courses offered by the ME department (a great course) and FE courses offered by EMA. These courses focused on the math (not how to use a specific code), yet they would not have counted towards a math masters. In a weird way, forgoing the math masters gave me more flexibility to study math.

 

[Hercuflea]

I think we have talk a few times before.



We use a lot of functional analysis and operator theory in plasma physics. Ultimately the spectrum of our operators determines the stability of the plasma. In the worst cases, instabilities can be disruptive leading to the termination of a discharge. Or they can be "weaker" and degrade confinement. Understanding the spectrum of instabilities is a necessary part of optimising our experiments allowing us to go to high pressure.

The operators we work with in plasma physics are ugly beasts. They often have accumulation points, continua, and discrete eigenvalues. Not to mention that they are often complex and non-hermitian.

Also, we have to pay careful attention to the spectrum of the operators in numerical simulations. Its not enough to know that our numerical approximations converge to the physical model. We have to know how it converges. Often the numerical approximation will under/over-predict stability. Knowing which, will help us interrupt our simulation results.

I guess my point is that an interest in functional analysis/operator theory should not dissuade you from plasma physics. It is an important part of the work we do.



Plasma/Fusion PhD students and the UW take a lot courses! My course load was almost twice that of other non-plasma PhD students in my program. The UW also requires all PhD students to have a secondary (minor) focus. The problem with the Math masters is that it will limit what courses you can take. If you get the math masters, most of your math courses will have to be taught by the math department.

For example, my secondary focus was computational math. I was able to take any math course offered by the math department, but I was also free to count "math" courses taught be other departments towards my minor focus. I took CFD courses offered by the ME department (a great course) and FE courses offered by EMA. These courses focused on the math (not how to use a specific code), yet they would not have counted towards a math masters. In a weird way, forgoing the math masters gave me more flexibility to study math.

Thanks wolfman. It's good to know that I can expect to work with results from functional analysis in grad school for plasma simulations. That was one of the subjects I found most interesting as an undergrad, although I've only touched the surface in learning about it.

Yeah, I don't understand all of these people saying to take as few courses as possible in grad school...I mean I get that PhD's are all about your research, but if a knowing the theory cold from a course will help you to understand your field better and to be more efficient at your research, I do not see why you wouldn't opt to take it. For me, it's better to spend a few semesters at the beginning learning the material in a proper classroom with someone who knows what they're doing teaching you, than to spend years trying to teach yourself a subject and going slowly at your research because you don't fully comprehend it.

 

[Arsenic&Lace]

Thanks wolfman. It's good to know that I can expect to work with results from functional analysis in grad school for plasma simulations. That was one of the subjects I found most interesting as an undergrad, although I've only touched the surface in learning about it.

Depends upon the group you get into. Several of my friends have been at or are currently at Princeton in its plasma physics group, and what I was told is that they were moving away from more theoretical approaches towards (if I recall correctly) approaches more along the lines of machine learning or something (basically an approach that was more along the lines of heuristics utilizing powerful computing resources). At the end of the day it seems as if you can still do what you want though. I'd need to talk to them again to make sure I understood them correctly however.

Yeah, I don't understand all of these people saying to take as few courses as possible in grad school...I mean I get that PhD's are all about your research, but if a knowing the theory cold from a course will help you to understand your field better and to be more efficient at your research, I do not see why you wouldn't opt to take it. For me, it's better to spend a few semesters at the beginning learning the material in a proper classroom with someone who knows what they're doing teaching you, than to spend years trying to teach yourself a subject and going slowly at your research because you don't fully comprehend it.

Really depends upon how smart you are, which I would measure by useful results/hour, which is usually a number much less than one. My experience with research is that having knowledge that does not at first glance seem relevant can be very powerful, since paradigm shifts or novel methods are novel because nobody anticipated them, and nobody anticipated them because they did not sit in the conventional tool set. So I'm applying currently with (so far) good results a method which has never been applied before to the problem I am working on, and I wouldn't have thought of it had I not explored completely "irrelevant" fields.

Given finite spare time, though, I can either invest it in a). being committed to a course, b). learning snippets of outside knowledge on my own time, or c). pursuing alternative routes or riskier projects my advisor is less keen on. So far c has paid dividends, without detracting from my main output because I only pursue it in my spare time, although it does result in 80+ hour weeks of labor some times. If you think a). is the best use of your time then by all means pursue it, but I think c). is likely more useful since it is very unlikely that you will stand out as a researcher unless you take a stab at pioneering something risky.

I am taking a graduate course this semester taught by one of the more accomplished professors in the department, and his opinion (just an opinion!) was that additional courses are a waste of time. But that might be a bit harsh.

 

[the_wolfman]

Depends upon the group you get into. Several of my friends have been at or are currently at Princeton in its plasma physics group, and what I was told is that they were moving away from more theoretical approaches towards (if I recall correctly) approaches more along the lines of machine learning or something (basically an approach that was more along the lines of heuristics utilizing powerful computing resources). At the end of the day it seems as if you can still do what you want though. I'd need to talk to them again to make sure I understood them correctly however.

FYI I happen to know the group at Wisconsin that the OP is talking about. I mentioned functional analysis and operator theory because the OP expressed interest in these subject areas of math. But the truth is that plasma physics is a hard problem, and we uses many advance areas of math. Machine learning is nice, because you can get semi-accurate solutions quickly. This is important for experimentalists who often have a few minutes between discharges. And they need to analyse a lot of data quickly so that the can use the previous experiment to guide the next one. However, functional analysis is also a useful tool in generating approximate solutions quickly.

Yeah, I don't understand all of these people saying to take as few courses as possible in grad school...I mean I get that PhD's are all about your research, but if a knowing the theory cold from a course will help you to understand your field better and to be more efficient at your research, I do not see why you wouldn't opt to take it. For me, it's better to spend a few semesters at the beginning learning the material in a proper classroom with someone who knows what they're doing teaching you, than to spend years trying to teach yourself a subject and going slowly at your research because you don't fully comprehend it.

Different people have different learning styles. Some people learn best in class, others learn best in the "lab". To each their own. But there does come a point of dimensioning returns. And one of the goals of getting a PhD is learning how to teach yourself new subject matter. That being said, you'll be encouraged to take a lot of advanced courses if you go to Wisconsin.