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.