Admissions Want a PhD in math, but stuck in a physics degree. What to do?

[BamboozledBoi]

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I am a 2nd year BSc Physics student in India. But due to a change of interests, I now want to become a mathematician. I wish to do my PhD in the TOP programs in the world. (I want an inspiring environment full of people more capable than me.)
My uni doesn't allow a major switching, and I can't take pure math courses apart from intro real analysis. I am self-learning undergrad math, but I have no credits to show for it.
I have some doubts ( categorized for ease of answering):

1. Given this condition, what steps must I take to land a top PhD program? ( Note: I'll do a master's in math before entering a PhD program.)
2. I will do research during my master's degree anyway. But how much will Undergrad research help me in PhD admissions? How do I get professors to take me in for a pure math project, when I have no math credits to prove my knowledge and passion?
3. I am currently about to start a year-long neural networks research project ( supported by a prestigious program). I am interested in the topic too. Will this count during a math PhD admission? Should I find something in pure math instead of this? (some low hanging fruit)
4. USA has PhD programs that you can enter straight after undergrad. Do I, with a 3-year physics degree, have a shot at this? What must I do if I want to land such programs? (I have no chance in top programs; here,I am talking about mid and low-tier. I would exit with an MS if I make it to such progr

 

[WWGD]

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I am a 2nd year BSc Physics student in India. But due to a change of interests, I now want to become a mathematician. I wish to do my PhD in the TOP programs in the world. (I want an inspiring environment full of people more capable than me.)
My uni doesn't allow a major switching, and I can't take pure math courses apart from intro real analysis. I am self-learning undergrad math, but I have no credits to show for it.
I have some doubts ( categorized for ease of answering):

1. Given this condition, what steps must I take to land a top PhD program? ( Note: I'll do a master's in math before entering a PhD program.)
2. I will do research during my master's degree anyway. But how much will Undergrad research help me in PhD admissions? How do I get professors to take me in for a pure math project, when I have no math credits to prove my knowledge and passion?
3. I am currently about to start a year-long neural networks research project ( supported by a prestigious program). I am interested in the topic too. Will this count during a math PhD admission? Should I find something in pure math instead of this? (some low hanging fruit)
4. USA has PhD programs that you can enter straight after undergrad. Do I, with a 3-year physics degree, have a shot at this? What must I do if I want to land such programs? (I have no chance in top programs; here,I am talking about mid and low-tier. I would exit with an MS if I make it to such progr

Welcome to PF! A good score on the Math GRE could help you. If you have a specific topic you're interested in, or, better, a specific problem, it would help too.

 

[jedishrfu]

Math is a diverse discipline. There are many physics courses that you can take with a lot of applied math. General Relativity with differential geometry, or Quantum Mechanics with Bessel, Legendre, and Laguerre differential equations, or Stat Thermo with statistical analysis, or EM Theory with Maxwell's equations and boundary conditions.

I don't know how things work in India, but a US-based graduate program would probably accept good grades from a BS Physics background.

Check the graduate program requirements at the colleges you are looking at. They should tell you whether they will accept your BS in Physics.

In general, you lose a lot of momentum from switching majors. In the US, students may spend an extra year recovering from the switch, getting enough credits to graduate in their field.

 

[CrysPhys]

My uni doesn't allow a major switching, and I can't take pure math courses apart from intro real analysis. I am self-learning undergrad math,

You keep emphasizing pure math; yet, by your own admission, your exposure to pure math has been mainly limited to self-learning. So what have you covered so far? Is it enough to persuade you that your future lies in pure math? What about applied math? That might be a more viable transition.

 

[Choppy]

There appears to be something of a sunken cost effect at play here.

You want to switch majors, but your university does not allow one to do that. The solution, it would seem, would be to switch universities.

I know that's not always easy.