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I'm currently studying undergraduate physics at Cambridge UK (and a UK citizen). However, I've found the lack of rigor in the math we use extremely frustrating. I've dabbled in math texts (Spivak's Calculus and Axler's Linear Algebra Done Right) and absolutely loved proof based math, though I appreciate this is at a first-year level.

Does anyone have advice on how to switch to mathematics after a physics degree? I've already seen suggestions to pursue a Masters or Bacc in the US but the issue with that is I couldn't pay for it. Is "catching up" during the first couple of years of a math PhD an option? I've seen even highly competitive programmes such as Yale mention that it is possible to take remedial undergraduate courses during a PhD.

Alternatively, can anyone recommend a route through physics that would allow me to pick up more rigorous math? I've also looked at physics PhD programmes but these tend to require a broad base in physics that I'm slowly losing interest in.

Edit: the UK system means unfortunately switching majors/taking extra years as an undergraduate isn't possible. The closest would be completing the "Part III" (masters) math course but this tends to restrict physics students to courses such as QFT(quantum field theory).

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For doctoral study in Math in the US, switching from a Physics undergraduate is reasonable. You wouldn't need to do another undergraduate degree and the funding for a doctorate is much more likely than that for a masters. Most doctoral students work as TAs with full, though modest, funding.

But, you are a bit light in math at the moment. A course in Analysis, one in Abstract Algebra, and maybe one in Topology would greatly enhance your prospects.

Another option is to switch to math where you are, though it will likely delay you for a bit.

Talk to your advisor about what is possible to increase your theoretical math skills and whether you can do that within your current major or not. Talk to someone in the math department.

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    Hi Buffy, thanks so much for your answer. A major issue is that the UK system doesn't allow "changing majors." The closest I could get would be joining the Cambridge Part III (masters) course in math though I would likely be restricted to taking the more physics-ey courses such as QFT. It might help persuade PhD admissions that I can do serious math but wouldn't fill in knowledge gaps. I'm well aware of how much math I'm missing but am struggling to find a practical way to catch up. Have you heard of anyone doing the full catch-up during a PhD? Eg by being granted an extra year?
    – John T
    Commented Apr 21, 2023 at 14:53
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    Few doctoral programs have much of a time limit and for those that do it is more like seven years, after which questions get asked. But note that the US undergrad degree is more general than a UK degree with lots of general education courses. The three courses I mentioned would also let you know if your "love of theory" is solid. Especially the algebra course, I'd guess.
    – Buffy
    Commented Apr 21, 2023 at 15:06
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    @Buffy your comment about time limits for PhD degrees applies to universities in North America but in UK and Europe PhDs are intended to finish much faster. Questions will be asked far before 7 years and at least at Oxford, the chances of staying in the program beyond 7 years is close to 0.
    – Nik
    Commented Apr 23, 2023 at 7:19
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If you are studying at Cambridge you must be taking Natural Sciences. This course teaches physics, but also chemistry, biology, materials science, plant science, etc., so to fit it all in there may very well be less of an emphasis on rigorous maths, especially in the early stages.

On the positive side, Cambridge lets you specialise more and more during the second, third and fourth years, and should definitely have the opportunity to focus on more mathematical subjects. Perhaps it is possible, for example, to take or audit courses taught by DAMTP (Department of Applied Mathematics & Theoretical Physics), rather than the Cavendish?

When applying for a PhD at Cambridge, many students at DAMTP will be told to first do Part III of the Mathematics tripos - this is an option that I imagine is also open to you.

Alternatively, you could look into Centres for Doctoral Training (CDTs), which are fully-funded Masters (1 year) + PhD (3 years) courses, where the masters can allow you to settle into a new subject area.

Consider talking to your Director of Studies at your college, who is quite literally there to discuss things like this.

Edit: The physics course handbook seems to indicate that it is indeed possible to do the Maths Part III. Be warned: it is notoriously hard!!!

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In the US, you indeed have to pay for Masters. However, if you enroll into a combined Masters/PhD programme, then usually they will pay you a scholarship that will cover both your tuition and living expenses. For that, you will have to work as a Research Assistant or, more probably, as a Teaching Assistant. Grading homeworks and teaching problem-solving sessions for advanced courses will be difficult to you during your first year, but you probably will be offered to teach introductory courses, something at the level of Axler's book.

As you rightly mentioned, many schools in US will enable you to take remedial undergraduate courses during your Masters/PhD studies. I am aware of such a case.

So I see no problem here.

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"I'm currently studying undergraduate physics at Cambridge UK (and a UK citizen). However, I've found the lack of rigor in the math we use extremely frustrating. I've dabbled in math texts (Spivak's Calculus and Axler's Linear Algebra Done Right) and absolutely loved proof based math, though I appreciate this is at a first-year level."

I agree with this answer: You will have more opportunities for exposure to what you call "proof based math" in the more senior years of your undergraduate studies, and if that's not enough then in Part III you can select courses from anywhere among pure mathematics, applied mathematics, statistics and theoretical physics (the latter including highly proof-based courses like quantum information, which is taught more as a math course than a physics course). Plenty of people do a PhD in pure math or applied math after an undergraduate or masters degree in physics. Also, Cambridge students have unusually long break periods between terms, during which you can study more pure mathematics on your own as you have already been doing, while still remaining in the natural sciences program which makes you much more likely to secure a job (whether in academia or elsewhere) after you get your degree.

"Does anyone have advice on how to switch to mathematics after a physics degree?"

Take mathematical courses that are available to you during your undergraduate physics degree, continue reading mathematics books like the ones you mentioned, and ask people working in mathematical areas if they are working on any projects in which an undergraduate research assistant (you!) can participate. Be careful that you're really doing the right thing for you (it seems that one year ago you thought that physics was the wiser choice, and now you've changed your mind, so keep in mind that it can change again, and continue talking to as many people as you can in both fields: peers, students more senior to you, professionals, professors, advisors, etc.).

"I've already seen suggestions to pursue a Masters or Bacc in the US but the issue with that is I couldn't pay for it."

The US isn't the only place that offers Masters degrees. Your own university has Part III of the Mathematical Tripos, and plenty of universities and institutes in Europe, Canada, etc. have Masters degrees with scholarships or lower costs than universities in USA. In Canada you usually get paid to do Masters degrees, but at your stage it might be better to aim to get into a PhD program, since you can switch from a PhD program into a Masters one if you want to (or have to) exit early.

"Is "catching up" during the first couple of years of a math PhD an option? I've seen even highly competitive programmes such as Yale mention that it is possible to take remedial undergraduate courses during a PhD."

Yes. Catching up is what almost all PhD students do at the start of their PhD program, and taking undergraduate level courses as a PhD student is pretty much universally accepted.

"Alternatively, can anyone recommend a route through physics that would allow me to pick up more rigorous math? I've also looked at physics PhD programmes but these tend to require a broad base in physics that I'm slowly losing interest in."

You can take mathematical courses whenever you have the option, continue studying rigorous mathematics during the unusually long Cambridge break periods, and you can try to work as a research assistant with mathematicians (I would be surprised if your college didn't send you information about the research assistantship program that we sent towards the end of Lent term 2023 by the way, since more than 50 Oxbridge students responded).

"Edit: the UK system means unfortunately switching majors/taking extra years as an undergraduate isn't possible. The closest would be completing the "Part III" (masters) math course but this tends to restrict physics students to courses such as QFT."

No, in Part III courses are offered in:

  • Algebra,
  • Algebraic Geometry,
  • Analysis and PDEs,
  • Combinatorics,
  • Differential Geometry and Topology,
  • "Foundations",
  • Number Theory,
  • Information and Finance,
  • Probability, Statistics,
  • Quantum Computation and Information and Foundations,
  • Particle Physics and QFT,
  • Reltivity and Cosmology,
  • Applied and Computational Analysis,
  • Astrophysics,
  • Soft Matter and Biological Physics, and
  • Continuum Mechanics

"Students may select courses freely from those available, subject to the constraints of the lecture timetable, regardless of which department they registered with or which application stream they applied through." so you are not restricted to taking QFT (Quantum Field Theory) and similar courses.

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  • It seems to me that you are theorizing about how things like Part III work based on reading about things you have not experienced. My own experience of the Maths Tripos + Part III is somewhat out of date but I think your final quote is a misunderstanding of what is implied. There might not be a rule saying that a student who has done mostly Physics courses at UG cannot take Part III algebra, but I think it is very likely they will not have sufficient preparation in UG algebra
    – Yemon Choi
    Commented Apr 22, 2023 at 23:42
  • "You will have more opportunities for exposure to what you call "proof based math" in the more senior years of your undergraduate studies" - on what do you base this claim? The pure maths stream at Cambridge tends to be rather cumulative in nature, and I suspect that lecturers of Y3 pure maths courses will assume that the students in the class have absorbed basic techniques of "proof based maths" from previous years.
    – Yemon Choi
    Commented Apr 23, 2023 at 0:00
  • @YemonChoi when did I say that a student that has done mostly physics courses (and nothing else) would be sufficiently prepared for a Part III course on algebra? I replied to OP's incorrect claim that they restrict physics students to "courses like QFT". Also, pure math courses aren't the only courses that involve proofs. My "claim" was partially based on that.
    – Nik
    Commented Apr 23, 2023 at 2:31
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I did NatSci at Cambridge and I have a friend who did exactly what you describe, i.e. switching from physics to maths though that was in the US. My answer is based on this experience.

The answer depends on how much maths you did. I was mostly interested in experimental physics and I couldn't even understand what my maths friends were talking about let alone do a PhD in maths. If your experience is similar then I think your only option would be to start afresh with a maths degree. I'm sure you'll find a university happy to take your money if you want to do this - Cambridge may even be willing to do it. Discuss it with your tutor.

My friend did theoretical physics, and pretty advanced theoretical physics at that. He learned quantum field theory with irritating ease and even got started in string theory. This meant he learned vastly more maths than I did, though still far less than a student doing the Maths Tripos at Cambridge would learn. He then went on to do a maths PhD in the areas of maths he had learned as an undergrad. I can't give any more detail since I understand next to nothing of his research, and he was studying in the US not Cambridge. I guess when you talk about restricting courses to QFT you mean you could only do a PhD in areas of maths you had already studied, and that would be physics related areas like the maths used in QFT.

You don't say how much maths you have learned, but from the tone of your question I'd guess it's nowhere near my friend's experience (so far). NatSci does allow you to take advanced theoretical physics courses and learn the associated maths, but as you say this would leave you lacking in many areas that maths students study and hence limit the areas you could do research in. But my friend managed it and last time I spoke was thoroughly enjoying his work.

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