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

• Abstract Algebra by David S. Dummit and Richard M. Foote
• Algebra by Thomas W. Hungerford
• Algebra by Serge Lang
• Topics in Algebra by I. N. Herstein
• Advanced Modern Algebra by Joseph L. Rotman

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

• Linear Algebra Done Right by Sheldon Axler
• Visual Group Theory by Nathan Carter (hat tip to blog reader David for recommending this here!)

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.

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Books

Some classics:

• Principles of Mathematical Analysis and Real and Complex Analysis by Walter Rudin
• Real Analysis: Modern Applications and Their Techniques by Gerald B. Folland
• Real Analysis by H. L. Royden and P. M. Fitzpatrick
• Real and Functional Analysis by Serge Lang

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:  

• Real Mathematical Analysis by Charles Chapman Pugh
• Measure and Category: A Survey of the Analogies between Topological and Measure Spaces by John C. Oxtoby
• Counterexamples in Analysis by Bernard R. Gelbaum and John M. H. Olmsted

Links

• Shameless plug: Need to construct a Lebesgue measurable set which is not a Borel set? I've written out a step-by-step proof here!
• If you want to brush up on your undergraduate real analysis, take a look at Francis Su's video series.
• Terry Tao has a great collection of notes (and some problems/solutions) on metric spaces, sequences, convergence, Lebesgue integration and more. See part one and part two. Of course, don't forget to check out his blog, too!
• And here are some posts on the famed triad: Fatou's Lemma, the Monotone Convergence Theorem, and the Dominated Convergence Theorem. (Yes, this is another plug.)

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Books

Some classics

• Algebraic Topology by Allen Hatcher
• Topology and Elements of Algebraic Topology by James Munkres
• Algebraic Topology by William S. Massey
• General Topology by John L. Kelley
• Basic Topology by M.A. Armstrong
• An Introduction to Algebraic Topology by Joseph L. Rotman

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

• Homology Theory: An Introduction to Algebraic Topology by James W. Vick
• Counterexamples in Topology by Lynn A. Steen and J. Arthur Seebach Jr.
• Topology of Surfaces by L. Christine Kinsey
• Singular Homology Theory by William S. Massey (Note: Massey's A Basic Course in Algebraic Topology is the union of his two books, Algebraic Topology and Singular Homology Theory)
• Differential Forms in Algebraic Topology by Raoul Bott and Loring W. Tu

Links

• Shameless plug: I once compiled a list of six homotopy equivalent spaces. There are LOTS of pictures!
• Here are some videos on point-set topology (bases, metrics, sequences, etc.) in the style of Khan Academy.
• New to this whole algebraic topology thing? This video series by N J Wildberger is a good place to start. (On that note, I found this page about quotient spaces illuminating when I first learned about them.)
• Over at Roots of Unity, Evelyn Lamb has a wonderful collection of easy-to-read articles about different topological spaces: the topologist's sine curve, the long line, the house with two rooms, Cantor's leaky tent and more!
• Confused by simplicial homology? Jeremy Kun has written an excellent primer over at his blog. (And if you're wondering "What's homology, anyway?" you may find this little Instagram post helpful.)
• Here's a nice animation and commentary on why the real projective plane can be viewed as a Möbius strip and disc glued together.
• These videos on basic homotopy theory (deformation retracts, covering spaces, van Kampen's theorem, etc.) are excellent. And they pair quite nicely with the material in Hatcher's book.

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Books

Some classics

• Real and Complex Analysis by Walter Rudin
• Complex Analysis by Lars Ahlfors
• Functions of One Complex Variable I by John B. Conway
• Complex Analysis by Serge Lang

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

• A First Course in Undergraduate Complex Analysis by Richard Spindler (hat tip to Evelyn Lamb)
• Complex Functions: An Algebraic and Geometric Viewpoint by Gareth A. Jones and David Singerman
• Theory of Complex Functions by Reinhold Remmert

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

• Introduction to Smooth Manifolds by John M. Lee
• A Comprehensive Introduction to Differential Geometry by Michael Spivak (there are 5 volumes)

My favorite

• coming soon!

See also:

• Differential Forms in Algebraic Topology by Raoul Bott and Loring W. Tu
• Applied Differential Geometry by William L. Burke
• Vector Bundles and K-Theory by Allen Hatcher

Links

• Notes on Differential Forms by Lorenzo Sudan