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I'm a PhD student trying to learn the math required to understand Gauge theory. From some reading over the past few weeks, I think I need to learn the following:

  • Lie algebra / Lie groups
  • Algebraic and geometric Topology
  • Differential geometry

I only need understand the material enough to be literate (my research hopes only to use some results from these fields and not to contribute to them). Does anyone have any good recommendations for textbooks that assume little mathematical knowledge beyond some undergrad. courses in linear algebra, vector calc., probability, and PDEs?

In particular, I'm hoping to find a textbook that assumes little prior knowledge while being somewhat brief its exposition (in my reading so far I'm finding myself getting lost in the volume of definitions).

Thanks for reading, I appreciate any suggestions!

edit: hardmath asked where I feel I've been getting lost in definitions. I've started reading "A Course in Modern Mathematical Physics" by Peter Szekeres (suggested by a friend). This seems like a great book and I've learnt a lot in the first few chapters but I'm struggling with the amount of terminology. For example, by chapter 3 my notes contained about 70 definitions.

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    $\begingroup$ walk thru amazon.com/Gauge-Theory-Variational-Principles-Physics/dp/… $\endgroup$ – janmarqz Dec 3 '20 at 18:47
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    $\begingroup$ How much do you want to know about these areas? This question is very broad unless you be more specific about this. For example, "An Introduction to Manifolds" by Loring Tu is an introductory book that covers the basics of smooth manifolds, including introducing the basics about Lie groups (and their corresponding Lie algebras) and de Rham cohomology (algebraic topology). How far beyond what is in this book do you need to go? (with some google searching you can probably find some previews of what's in this book; try amazon, for example) $\endgroup$ – Will R Dec 3 '20 at 20:01
  • $\begingroup$ @WillR thanks for the response. Looking at the preview for the book you suggested, it looks like they cover all the topics I need! My hope is to find a textbook that covers these topics which is brief in its exposition while assuming little prior knowledge outside of multivariable calc. & linear algebra. $\endgroup$ – KCQs Dec 3 '20 at 20:14
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Here are my picks for best intro books:

Lie Algebras: https://www.amazon.com/Introduction-Algebras-Springer-Undergraduate-Mathematics/dp/1846280400

Differential Geomtry: https://www.amazon.com/First-Steps-Differential-Geometry-Undergraduate/dp/146147731X

Geometric Topology:

https://www.amazon.com/Introduction-Theory-Translations-Mathematical-Monographs/dp/0821810227

Algebraic Topology:

Fundamental group / covering spaces: https://www.amazon.com/Introduction-Topological-Manifolds-Graduate-Mathematics/dp/1441979395

Homology/Cohomlogy: https://www.amazon.com/Elements-Algebraic-Topology-James-Munkres/dp/0201627280/ref=sr_1_2?dchild=1&keywords=munkres+algebraic+topology&qid=1607026571&s=books&sr=1-2

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    $\begingroup$ Thanks for the really detailed suggestions. Truth be told I think I'm suffering from the fact that I don't know what I don't know. Can these books be read in parallel? Or do you suggest starting from the top and working my way down? PS: I appreciate that your suggestions are reasonably priced. (learning is getting pricey :) ) $\endgroup$ – KCQs Dec 3 '20 at 21:05
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    $\begingroup$ I'd start with the top book, it's overall probably the easiest. Type introduction to lie algebras erdmann pdf into google and the third link down is a free PDF. As far as you don't know what you don't know, the truth is you know nothing, and I've read all these books and still know nothing, Math goes deep brah. Study it because you enjoy learning it not for any other reason haha. Also, if you haven't studied point-set topology yet then you need to drop everything your doing and read through Munkres "point set topology", or at least the first four and last chapters $\endgroup$ – Mutated Penguin Dec 3 '20 at 22:35
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As I suggested in comments, "An Introduction to Manifolds" by Loring Tu is a good place to start. Points in favour:

  1. From the preface the author wrote it with the rationale of providing "a readable but rigorous introduction that gets the reader quickly up to speed, to the point where for example he or she can compute the de Rham cohomology of simple spaces." He goes on to explain that there is a "self-imposed absence of point-set topology in the prerequisites... [His] solution is to make the first four sections of the book independent of point-set topology and to place the necessary point-set topology in an appendix." The intention is that the reader study the first four sections and Appendix A simultaneously.
  2. The book is the first volume of a series, so if you need more than what is in this book then you might find it in another book by the same author. It is the prequel to the famous "Differential Forms in Algebraic Topology" by Bott and Tu, and there is also "Differential Geometry" by Tu (ostensibly the third book but written with only the contents of the "Introduction" as prerequisites).
  3. It's a Springer Universitext book, which at the time of writing makes it pretty cheap as far as textbooks go.

Point against: a perhaps-more-standard set of references for similar material is the series of books by John M Lee. Lee's books make better references in part because they are written to be more encyclopaedic, which might mean that Tu's books are slightly more approachable to someone learning on their own.

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  • $\begingroup$ Thanks so much for the extremely thoughtful response. I've taken a quick look at John M Lee's books and I agree that Tu's book looks more approachable. Really appreciate it. $\endgroup$ – KCQs Dec 4 '20 at 14:07
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    $\begingroup$ @KCQs: FYI, my suggestion can certainly be read in one semester, but you will have to work quite hard if you don't know point-set topology. I would advise you to make sure you can do the end-of-section exercises, which are quite well-chosen to emphasise the most important material, and to not overthink differential forms; they are basically just ways of stringing together determinants in an algebraically clever way, so that the notion of "integral of a form" makes sense via the change of variable formula. It all makes sense once you've done some simple examples; before that it's alphabet soup. $\endgroup$ – Will R Dec 4 '20 at 15:20

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