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(Summary) There will be a reading course using Differential Geometry: Bundles, Connections, Metrics and Curvature by C.Taubes. What is the prerequisite for this book? Is its contents something that every mathematician should know? Do you recommend that I take this course?


(Details) I am currently an undergraduate student, and there will be a reading course on differential geometry that I can take. I am wavering in my decision in taking that course, and I would like some advice.

The text used for the reading course is the book by C.Taubes mentioned above. I am not sure if I should take this reading course because I feel like the contents in this book is too advanced for me. I glanced over its table of contents, but I have almost no idea what this book actually does. But the professor chose this book, so it must be something very important and what everyone should know.

I asked the professor if I have the prerequisite for reading this book, but he gave me an unsatisfactory answer, which was that if I understand the topics covered in the course on smooth manifolds, which is described below, then I am ready for this book. (To be fair, he has not met me before, so I believe he couldn't say anything more than that.)

Maybe, as the professor told me, I might be ready. I would love it if I can take this course. However, I know that I am quite dumb (in the sense that a lot of the books recommended in my past courses did not quite work well with me, and I often ended up working thorough one of the friendliest introductions to the subjects taught in the courses), so if there is a slightest concern, I would like to avoid taking this course. (I cannot opt out once I decide to take the course, so I am very cautious...)

I do not want to miss this opportunity for the reading course is offered only for students with good grades, and I worked hard to earn the "good grades". But I am unsure if I am ready to take this course. Do you recommend taking this course?


My background: I have completed a semester-long course on smooth manifolds, which covered essentially all the topics up to Chapter 16 of Introduction to Smooth Manifolds by J.Lee (a wonderful book!). I also worked through all the exercises and problems of this book up to Chapter 16, and I am, at least, not uncomfortable working with smooth manifolds. I've also read the its prequel Introduction to Topological Manifolds except for chapter 13, which covers homology.

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  • $\begingroup$ I cannot assess if you have the necessary prerequisites to take this course and appreciate all fine details of the theory, but I can tell you of some similar personal experience, of when I took a Ph.D. level course on C*-algebras in my last undergrad year. Just as you did, I spoke to the professor beforehand and studied some prerequisite material on Functional Analysis and Operator Algebras, but of course there were some points which were not completely natural to me at the time. The professor took this in consideration in his evaluation so it didn't make me get bad grades (cont.) $\endgroup$ – Luiz Cordeiro Sep 17 at 3:06
  • $\begingroup$ If there is any prerequisite material you have not seen, it is more important to try and get an intuitive idea of why some results are true than to know all the details of all the proofs of the prerequisite material. After some time you will probably be able to come up with the proofs yourself anyway. I would recommend taking this course. $\endgroup$ – Luiz Cordeiro Sep 17 at 3:13
  • $\begingroup$ @Luiz Cordeiro Thank you for sharing your experience and giving me an advice. Can you elaborate on why you recommend taking this course? $\endgroup$ – Ken Sep 17 at 3:38
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    $\begingroup$ If you manage to follow the course well (as I said, relying on your intuition whenever your formal studies may be lacking), it will allow you to skip steps and have a deeper understanding of the subject early on, which is always good. A reading course also usually has a more lenieng grading scheme, so it is a place where you can take some "risk". It should require more work than your other courses, though, so keep this in mind. $\endgroup$ – Luiz Cordeiro Sep 17 at 3:59
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    $\begingroup$ Also, I just had a look at Taubes' table of contents. You really shouldn't have any problem following it, since you've read Lee's Introduction to Smooth Manifolds (as you said, a wonderful book). Taubes' book up to chapter 16 (which is what I can understand) does not seem to cover anything too far out of reach for you. Perhaps you're lacking a bit of intuition of what a bundle is? $\endgroup$ – Luiz Cordeiro Sep 17 at 4:05

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