Name: A Course in Modern Mathematical Physics
ISBN: 9780521829601

A Course in Modern Mathematical Physics

This book provides an introduction to the major mathematical structures used in physics today. It covers the concepts and techniques needed for topics such as group theory, Lie algebras, topology, Hilbert space and differential geometry. Important theories of physics such as classical and quantum mechanics, thermodynamics, and special and general relativity are also developed in detail, and presented in the appropriate mathematical language. The book is suitable for advanced undergraduate and beginning graduate students in mathematical and theoretical physics, as well as applied mathematics. It includes numerous exercises and worked examples, to test the reader's understanding of the various concepts, as well as extending the themes covered in the main text. The only prerequisites are elementary calculus and linear algebra. No prior knowledge of group theory, abstract vector spaces or topology is required.

What Hacker News says

d_trOct 2026

"A Course in Modern Mathematical Physics: Groups, Hilbert Space and Differential Geometry" by Peter Szekeres has several chapters on linear algebra early on, starting with sets and groups and including tensors (which you absolutely need for QM).

I loved it.

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d_trJan 2023

A Course in Modern Mathematical Physics: Groups, Hilbert Space and Differential Geometry by Peter Szekeres

My intro to abstract math... Wide range of topics, very clearly written and very well structured. Sets, groups, vector spaces, tensors, topology, differential geometry, lie groups and more.

An Introduction to Category Theory by Harold Simmons

Very enjoyable read. You cannot go wrong with this as your first book on the subject.

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plaguepilledAug 2022

If you're willing to branch out into stuff that falls into the "not going to get you hired, but enriching personal experiences" bucket, then my number one recommendation will always be 'A Course in Modern Mathematical Physics' by Peter Szekeres. The book compresses so much into ~1000 pages without feeling like you're missing anything as you go.

You can get a lot from the David Tong lectures if you want something similar but free, but to me, Szekeres wins out on delivery.

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bargerJan 2019

While "all the high school math" is somewhat clear, the "all the college math" should be clarified a bit.

Are we talking about math for social studies majors, engineers, physicists, economists or mathematicians?

For example, there are books like [0] for engineers which serve as a boot camp of sorts. There are no theorems, proofs or deep math in it. There are many different kinds of engineers, so books like this don't include everything an engineer needs to know. For instance, there's no automata or group theory in [0]. It has a part at the beginning called Foundation Topics which could serve as "all the high school math one needs". In fact, this whole book could probably serve as "all the math an advanced high school student needs".

There are books like [1] for physicists. They are meant to introduce physics majors to a wide array of math topics in a relatively pain-free way. This book is much more rigorous and contains a LOT more material than [0]. Basically, a theoretical minimum for a physics major.

Economics majors also get somewhat rigorous math load where measure theory features a lot more prominently than it does in other majors (except mathematics proper).

List of subjects for math majors vary from place to place, but you'll have much easier time down the road if you master the (rigorous) rudiments of linear algebra (vector spaces), group theory, number theory and real analysis. Of course, knowing more math (say topology, complex analysis, category theory, combinatorics etc) is always good.

Before you get started with math for mathematicians, you'll want to learn their jive. A good intro is [2]. It's free and a really nice book. Another really nice book with non-existent pre-reqs is [3].

A pitfall that awaits a lot of people new to math is a concept of "multivariable calculus". This concept is a mess and means everything to everyone. Oftentimes it means surface level discussion of concepts in scalar fields (functions from R^n to R) and a little bit of talk about differential geometry of curves and surfaces (functions from reals and planes to R^n). The treatment is often not rigorous and n is limited to 2 and 3. After this laughable bullshit, one is thought ready to jump straight into the rigorous analysis of manifolds, granted they know a bit of real analysis. This is like jumping from 3rd grade straight to 9th grade. Along the way the most important thing missing is the rigorous treatment of vectors fields (say, at the level of Rudin). Some nice books here include [4]. Since diff geo (in particular, that of curves and surfaces) is its own thing entirely and there are a lot of really nice books for that like [5]. Also, note there's a nice new intro to "manifolds and stuff" [6] which is like what's calculus is to analysis.

Before I forget, most intro to stats books are written for science majors and are entirely inadequate for math majors, but there are elementary intro to stats books for math majors like [7].

Originally, I wanted to write a more fleshed out and huge comment, but I am running out of time.

Good Luck.

[0] Engineering Mathematics by Stroud/Booth

https://www.amazon.com/Engineering-Mathematics-K-Stroud/dp/0...

https://www.amazon.com/Advanced-Engineering-Mathematics-Kenn...

[1] A Course in Modern Mathematical Physics: Groups, Hilbert Space and Differential Geometry by Peter Szekers

https://www.amazon.com/Course-Modern-Mathematical-Physics-Di...

[2] Book of Proof by Richard Hammack

https://www.people.vcu.edu/~rhammack/BookOfProof/

[3] Linear Algebra: Step by Step by Kuldeep Singh

https://www.amazon.com/Linear-Algebra-Step-Kuldeep-Singh/dp/...

[4] An Introduction to Analysis by Wade