First, a YouTube channel where I post theoretical/mathematical physics talks at the graduate level and above: GraduatePhysics
A graduate level Quantum Mechanics course, by Prof. Cumrun Vafa.
A course on Black Holes, by Prof. Andrew Strominger.
An introductory course on String Theory (without requiring knowledge of QFT), by Prof. Paul Townsend.
A lightning revision of a first course in QFT in preparation for loops and renormalisation, by Prof. Washington Taylor.
Notes on the path-integral, including the basics and some tricks.
Everything below is relevant to theoretical physics courses of Cambridge University’s Part III of the Mathematical Tripos. Some are intentionally left incomplete, and there can certainly be mistakes.
Past January DAMTP early PhD admission exams:
Past exam papers can be found here. Below are some solutions to past exams.
Symmetries, Particles, Fields: 2013.
String Theory: 2013.
Note that I often changed conventions to match those of the 2013-14 lecturers. In writing the solutions, I have made use of the lecture notes of the corresponding years.
Finally, here are some collections of definitions in differential geometry from my Symmetries (Nicolas Manton), General Relativity (Harvey Reall) and Applications of Differential Geometry (Maciej Dunajski) courses. These are not intended for mathematicians. :P