The scar of race Book

I read this little book some years ago and I really enjoyed. It shows you all the mathematical bricks you need to know in order to build your understanding some of the most advanced theories in Physics. The explanations are clear and the style is one that physicists will certainly appreciate (also mathematicians, but perhaps they would prefer more theorem and proof style, absent here). It can be read by advanced undergraduates, as it assumes nothing more than advanced calculus and linear algebra, but as it is usually said, some "mathematical maturity" would also help.

The book begins discussing the math tools needed for advanced Physics, basically differential geometry and group theory, and it does it in a very fast way, so don't expect anything about the Gauss-Bonnet theorem or discussion of exotic groups (SU(5) or even more exotic ones). The example of the electromagnetism, rewritting the Maxwell equations in the language of differential forms is very well done. The explanation of the concept of fiber bundle is also brilliant, taking into consideration that it's a difficult concept.

There's a section of knot theory that it wasn't specially interesting for me but that it's very well explained, beginning with the basics. It seems that knots could be involved in the unification of General Relativity and Quantum Mechanics.

The final section of the book is about Einstein's relativity, perhaps the most beautiful theory of Physics, trying to link it to what it has been showed previously: a geometric theory that can be described with Cartan's formalism. Curious point: all the groups involved in the electromagnetic, weak and strong forces are compacts, and when a group is compact is equivalent to a sum of irreducible representations (not proved in the text, and probably an advanced proof, but we physicists have faith). The irreducible representations are the particles. The exception is General Relativity, that it is not compact. A change of a topological property that creates a really big problem and literally divides physics in two.

Overall, a very nice book if you're serious about learning advanced theories in Physics but you're also a beginner.