Resources for Intro-Level Graduate Courses

In recent months, several of you have asked me to recommend resources for various subjects in mathematics. Well, folks, here it is! I've finally rounded up a collection of books, PDFs, videos, and websites that I found helpful while studying for my intro-level graduate courses. Now before we jump in, let me preface the list by saying

it's FAR from comprehensive!

Most of the items below are accessible at the advanced undergraduate/early graduate level, so if you're looking for more advanced resources, you might be disappointed. Also, if you read my ramble about qualifying exams you'll know that my quals were in algebra, real analysis, and topology. So naturally my list is a little biased. But, hey, I figured a biased list is better than no list at all! In particular, I have very few items under differential geometry. That's simply because I haven't taken the course. But I am taking it this year, so I'll update this post as the months go on. (Bear with me, friends!)

And I've undoubtedly omitted several great resources either because I forgot to include them or I'm simply unaware of their existence. For this reason, I strongly encourage you to share your recommendations in the comments below! What books/links/etc. do you think are gems? Let me know! I'd love to hear.

Lastly, you'll see that I've included 'my favorite' books below. These aren't necessarily the most popular or 'the best.' They are simply the ones I enjoyed reading because of their accessibility and intuitive explanations.

And now, without further ado, I present to you The List!

 

Books

Some classics

My favorite

  • Basic Abstract Algebra by P. B. Bhattacharya, S. K. Jain and S. R. Nagpaul (I especially found the discussion on modules over a PID, Smith normal form, and rational canonical form helpful. See chapters 20-21.)

See also

Links

  • Shameless plug: I once wrote a non-technical introduction to Galois theory. It's one of the blog's more popular posts!
  • Keith Conrad has a goldmine of expositions (on pretty much everything) on his website. The reading is fairly easy, and there are plenty of worked-out examples.
  • Here is a helpful collection of videos by Matthew Salomone on Galois theory (field extensions, minimal polynomials, Galois groups, field automorphisms, etc.)
  • CUNY's own MathDoctorBob also has a nice collection of videos on everything from the definition of a group to Sylow's Theorems to polynomial rings to Galois theory!
  • Need some motivation for tensor products? Jeremy Kun can help you conquer your tensorphobia here.
  • What's the big deal with conjugation in group theory? I really like the accepted answer on this StackExchange post.
 

Books

Some classics:

My favorite: 

  • Real Analysis by N. L. Carothers (This is really a fantastic book (see my review here)! I found the discussion on absolute continuity especially illuminating. See chapter 20.)

See also:  

Links

 

Books

Some classics

My favorite

  • The Shape of Space by Jeffrey R. Weeks. (Filled with pictures and easy-to-read explanations, this book is a great resource for those first learning algebraic topology. I've written a review about it here!)

See also

Links

 

Books

Some classics

My favorite 

  • Visual Complex Analysis by Tristan Needham (The intended audience for this book is undergraduate students in math, physics, and engineering, but it's so wonderfully written that I'm compelled to recommend it. I've also written a review about it here.)

See also

Links

  • Shameless plug: Computations got you in the blues? Here's a motivational post for why we care about the automorphisms of the unit disc (upper half plane), complex plane, and Riemann sphere.
  • Here's a video lecture series on advanced complex analysis: part 1 (Rouche's theorem, the open mapping theorem, Schwarz's Lemma, the Riemann mapping theorem, etc.) and part 2 (the Casorati-Weierstrass theorem, Picard's theorems, the Arzelà-Ascoli theorem, Montel's theorem etc.)
  • And don't forget about this lovely video which beautifully illustrates the action of Möbius transformations! 
 

Books

Some Classics

My favorite

  • coming soon!

See also:

Links