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I want to do a PhD in mathematics (Specializing in algebraic topology or algebraic geometry ) or mathematical physics (String theory). I have a bachelor degree in a totally different field (medicine). Here's my situation :

I have independently studied mathematics and physics. I have spent a lot of time with hatcher's textbook in algebraic topology and lang's complex analysis among with other topics in math and physics. These are proof-heavy textbooks and I usually do the exercises.

Now , after having studied analysis ,differential topology ,geometry, abstract algebra , quantum mechanics and QFT .

Is this a good preparation for a PhD in math/physics ?

Please note that I have studied from standard textbooks and solved a lot of exercises.

How do I convince PhD committee to accept my application ?

If I apply now , Apart from formal qualifications,my application will be missing one crucial component which is the letters of recommendations. I'm not sure where do I get them ? I don't know any physics or math professors who can evaluate my knowledge.

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    "I have studied from standard textbooks and solved a lot of exercises" Did you ever ask someone to look at/review your solutions/proofs to those exercises? If not, how do you know you are qualified to do a PhD in mathematics? If yes, that person can write recommendation letter for you.
    – scaaahu
    Apr 8 '15 at 8:34
  • No , I usually solve the exercises on my own. If I get stuck , I google. I try to get the correct answer or prove the required statements , If I can prove them and the same strategy I use work correctly for other kinds of problems. , I think the steps of my calculations/proof are very likely correct.
    – nabil
    Apr 8 '15 at 8:37
  • That "very likely correct" is your problem. If you take your proofs/solution to someone (preferably a Math prof)) to ask him to check, then not only you would have more confidence, but also, you would have someone to recommend you. I'll urge you to do so, now!
    – scaaahu
    Apr 8 '15 at 8:43
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    An alternative would be to do it on our sister site Math SE. There are plenty of people would like to check the correctness of exercise solutions. Then at least, you can have some evidence to say in your cover letter that you do have math knowledge. Whether this is enough is up to your luck. Also, you can take GRE Math Subject test to prove your Math capability. You need to take it anyway.
    – scaaahu
    Apr 8 '15 at 8:47
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    You probably need a master's degree in math or in physics. In some countries (e.g. in some European countries) you might be able to follow a master's in math or in physics with a bachelor's in medicine. This does not mean that you would succeed. Apr 8 '15 at 9:22
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You should contact potential advisors who work in the areas in which you are interested. Introduce yourself, explain any work you've done, and ask if they have specific feedback for moving on in your career. You should contact several people. If you live near a university, arranging a time to meet with some current faculty members is also a good way to go about this.

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Start by taking a class at a university as a non-matriculated student. You can audit if you need to save money. Choose the class carefully. Don't forget to look at ratemyprofessor.com. This class is a way for you to get your feet wet, and find out if you enjoy institutional learning as much as you've enjoyed your autodidactic learning.

The instructor of that class is going to be your prime target for looking for a recommendation letter.

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