Here I compile a list of resources I’ve found incredibly useful during my studies separated by discipline as best as I could. I do not plan on posting anything here I haven’t spent a lot of time on, even if that resource might be useful. A note of warning: I tend to have difficulty understanding many resources that other people might find useful. Furthermore, I have a slight bias towards shorter books.
- F. William Lawvere and Stephen H. Schanuel “Conceptual Mathematics: A First Introduction to Categories.”
Higher category theory
- Any of John C. Baez and co-authors’ articles in the “Higher Dimensional Algebra” series. These are good for a conceptual idea behind higher categories. These articles were written before the theory of infinity-categories was established.
- Venkataraman Balakrishnan’s video lectures on classical mechanics. Professor Balakrishnan has a spin on classical mechanics in terms of dynamical systems, one of his specialties. His lectures are the best I’ve ever seen on an advanced undergraduate classical mechanics course. A link for the first video is at https://www.youtube.com/watch?v=Q6Gw08pwhws
- V. I. Arnold, “Mathematical Methods of Classical Mechanics.” Arnold is an excellent expositor. The book contains precision, clarity, and excellent pictures.
- Ramamurti Shankar, “Principles of Quantum Mechanics.” This was the only book on quantum mechanics that I could understand as an undergraduate. It is more for a physics audience.
- Venkataraman Balakrishnan’s video lectures on quantum mechanics. A link for the first video is at https://www.youtube.com/watch?v=TcmGYe39XG0
- Brian C. Hall, “Quantum Theory for Mathematicians.” This book is a gem. It is suitable for a mathematician as well as a physicist who wishes to understand the mathematical foundations of quantum mechanics. It not only gives good physical intuition, but also gives precise statements and mathematical reasons for physical concepts. If I ever taught a first semester graduate course on quantum mechanics, it might be from this book.
- Frederick Reif, “Fundamentals of Statistical and Thermal Physics.” This is the only introductory book I could learn statistical physics from. It is meant for undergraduates. The exercises are great and the physical insight is apparent.
- E. T. Jaynes, “Information Theory and Statistical Mechanics,” The Physical Review 106 (1957), no. 4, 620–630. This is an excellent paper and every graduate student should have this as required reading.
- Peter Woit’s notes, which are available at https://www.math.columbia.edu/~woit/LieGroups-2013/ under “Old Lecture Notes.” This is where I learned representation theory. Reading these notes and filling in the gaps or wording things in ways that I can understand helped a lot.
- Jeffrey R. Weeks, “The Shape of Space.” I wish I knew about this book before I took a topology course. A high school student who has taken some geometry should be able to read this, but it is also suitable for anyone thinking of taking (or currently taking) a course in topology. The exercises are worthwhile and fun.
Differential topology and geometry
- John W. Milnor, “Topology from a Differentiable Viewpoint.” This is the first book you should read if you’re going to study smooth manifolds. I also recommend supplementing it with Milnor’s videos from 1965 https://www.math.sunysb.edu/Videos/IMS/Differential_Topology/
- John C. Baez and Javier P. Muniain, “Gauge Fields, Knots and Gravity.” This is probably my favorite textbook. It is the best concise introduction to differential geometry I’ve ever come across. Furthermore, it introduces the concepts from a physical perspective. I also learned about Lie groups, bundles, classical Chern-Simons theory, and many more things from this book. I highly recommend this book for anyone interested in studying quantum gravity as a prerequisite.
- Shigeyuki Morita, “Geometry of Differential Forms.” This book is quite approachable and covers quite a bit of bundle theory and characteristic classes.
- Raoul Bott and Loring W. Tu, “Differential Forms in Algebraic Topology.” This is one of those rare books that are true gems. Every student who wishes to pursue algebraic topology should get this book and go through it.
Quantum field theory
- Ludwig D. Fadeev and Andrei A. Slavnov “Gauge Fields: An Introduction to Quantum Theory.” This is a short book on quantum field theory, focusing on gauge theory and using the Feynman path integral. It is for a physics audience.
General audience books
- Lee Smolin, “The Trouble With Physics: The Rise of String Theory, the Fall of Science.” This is a well-written text (albeit with a rather heavy bias against string theorists) outlining the shortcomings of string theory during the time the author was a researcher uncovering the field of loop quantum gravity and the time before. I highly recommend this book for anyone interested in quantum gravity or anyone who wants to know what some of the biggest questions in physics are.
- Leonard Susskind, “The Black Hole War: My Battle With Stephen Hawking to Make the World Safe for Quantum Mechanics.” This is a good introduction to the information paradox, a key point for understanding quantum gravity. It contains some cutting-edge physics (Hawing radiation, the black hole entropy area law, and holography).