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When I study a new topic, I never feel satisfied until I have spent some time solving a long list of problems.

I am looking for either a problem book or a list of problems on graduate math topics. While there is an abundance of problem books on undergraduate math topics (such as various websites on quals or books like Berkeley Problems in Mathematics), there seems to be fewer books at the graduate level with a lot of problems. There are books like Evan's PDE book or do Carmo's Riemannian Geometry book which has a good number of problems, but again, I feel like they are in the minority.

The closest thing to what I am looking for is the Cambridge Tripos III.

To clarify, the following is what I am looking for:

  • A book with ≥ 10 problems for a particular topic.
  • By "graduate topic," I mean anything that requires standard undergraduate curriculum (single/multivariate calculus, basic/Fourier analysis, ODEs, linear/basic algebra, point set topology, basic manifold theory, curves and surfaces, say) as a prerequisite. I am particularly interested in problem books for "advanced topics" whose prerequisites are standard graduate topics (algebraic/differential topology/geometry, measure theory, real/complex analysis, commutative algebra, representation theory, say).
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    $\begingroup$ What's the advantage of specific "problem books" as opposed to the more traditional exercises within a textbook (which are also common even at the graduate level)? $\endgroup$
    – Sam Hopkins
    May 19, 2021 at 3:07
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    $\begingroup$ What about solving old phd qualifyling exams from various universities? There is a lot of content online, and not all universities evaluate the same topics. $\endgroup$
    – efs
    May 19, 2021 at 4:54
  • $\begingroup$ @EFinat-S Sure, but qual exams usually don't go far beyond the "standard graduate topics" I described above. I was hoping there are lists of problems for more advanced topics as well. $\endgroup$
    – user676464327
    May 19, 2021 at 9:40
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    $\begingroup$ @SamHopkins Well for one, it is nice to have everything in one place; I can also work with one consistent notation/convention. I also think longer "problem books" have more interesting problems, especially towards the end of the list. (Just compare any two undergraduate textbooks on, say abstract algebra.) $\endgroup$
    – user676464327
    May 19, 2021 at 9:42
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    $\begingroup$ The Cambridge Part III Guide to Courses 2020-21 has suggested literature for each course, though the courses rarely follow a particular book. Past Tripos questions are also available, though the courses change year to year. $\endgroup$
    – Henry
    May 21, 2021 at 22:19

9 Answers 9

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Pólya-Szegő seems unsurpassed as a graduate level problem book on classical function theory. Other classic examples are:

P. Halmos, A Hilbert space problem book,

A. Kirillov and A. Gvishiani, Theorems and problems in functional analysis, available in English and French, bseides the Russian original, and

I. Glazman and Yu. Lyubich, Linear analysis in finite-dimensional spaces, translated from the Russian.

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You could try M. Ram Murty and Jody Esmonde, Problems in Algebraic Number Theory and/or M. Ram Murty Problems in Analytic Number Theory.

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    $\begingroup$ There's also "Problems in the Theory of Modular Forms" by M. Ram Murty, Michael Dewar and Hester Graves which was written in the same spirit. $\endgroup$
    – Anurag Sahay
    May 26, 2021 at 3:32
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Just yesterday I was looking at Clark Barwick's 121 Exercises on Locally Compact Abelian Groups: An Invitation to Harmonic Analysis. The opening sentence is "This is a collection of challenging exercises designed to motivate interested students of general topology to contemplate Pontryagin duality and the structure of locally compact abelian groups."

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László Lovász, Combinatorial Problems and Exercises: Second Edition. Over 600 pages, divided into Problems, Hints, and Solutions.

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An extensive list is at MSE, including pointers to dedicated web sites, such as this one. A particularly comprehensive collection is in Problems and Solutions in Mathematics, with a list of advanced topics (including Galois theory, homotopy/homology, differential geometry of manifolds, measurability and measure, PDE's).

I might add

The Stanford Mathematics Problem Book by George Polya and Jeremy Kilpatrick.

These 20 sets of intriguing problems test originality and insight rather than routine competence. They involve theorizing and verifying mathematical facts; examining the results of general statements; discovering that highly plausible conjectures can be incorrect; solving sequences of subproblems to reveal theory construction; and recognizing "red herrings," in which obvious relationships among the data prove irrelevant to solutions.

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    $\begingroup$ What's the level of mathematics you need to solve these problems? Do they go beyond the standard undergraduate math? $\endgroup$
    – user676464327
    May 19, 2021 at 9:57
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For differential geometry, there is (now in its second edition):

Gadea, Pedro M.; Muñoz Masqué, Jaime; Mykytyuk, Ihor V., Analysis and algebra on differentiable manifolds: a workbook for students and teachers, Problem Books in Mathematics. London: Springer (ISBN 978-94-007-5951-0/hbk; 978-94-007-5952-7/ebook). xxv, 617 p. (2013). ZBL1259.53002.

From the preface of the 2009 (revised first) edition: “This book is intended to cover the exercises of standard courses on analysis and algebra on differentiable manifolds for advanced undergraduate and graduate years, with specific focus on Lie groups, fibre bundles and Riemannian geometry.”

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    $\begingroup$ Note that this “Problem Books in Mathematics” series has quite a few more volumes, most with reviews linked from the ZBL page above. $\endgroup$
    – Francois Ziegler
    May 19, 2021 at 13:10
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There is John Dixon's book Problems in Group Theory. But I do not know whether you will classify it as graduate level or not. (It has the advantage that it is quite cheap.)

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Instead of Cambridge Part III, you can look at Oxford's Part C and post-graduate courses.

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Some of the books that I can remember from my PhD reading list under Peter Petersen at UCLA were: Characteristic Classes, by Stasheff, Morse Theory, Milnor, Dimension Theory, Hurewicz and Wallman, The Topology of Fibre Bundles, Steenrod.

We both agreed that Spivak's $5$ volume Comprehensive Introduction to Differential Geometry was quite useful. There was Galot, Hulin and La Fontaine's Riemannian Geometry.

I'll update this if I remember anything else.

It seems to me that Milnor's Topology from the Differentiable Viewpoint might have been also, and Spivak's Calculus on Manifolds probably wasn't, but I enjoyed it.

I also had a copy of his Riemannian Geometry in manuscript form, which is now available in the GTM series.

A good place to get these was from "Book Scientific", where Spivak himself used to pick up the phone. They had an $800$ number.

Perhaps finally, I don't think it's necessarily that important whether there are a lot of problems, because you should be primarily grappling with ideas and concepts, as opposed to just working problems. He did also give me a list of unsolved problems, which I have long since lost (maybe that's more what you were looking for!)

He also rated the journals, and recommended reading those. Annals of Mathematics and Journal of Differential Geometry, there were journals from Duke and Indiana university, there was the Pacific Journal of Mathematics, among others. Inventiones Mathematicae is also one of the best.

I think Lang's Algebra was probably on the list too. If not, it probably should have been.

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    $\begingroup$ When you say "grappling with ideas and concepts," what specifically are you doing? Are you working through examples and calculations that you come up with while reading the book? $\endgroup$
    – user676464327
    May 20, 2021 at 2:20

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