Here is an anti-recommendation list by the site owner. Of course, lists of this kind are subjective in nature. For a compensation, the site owner tries to be specific on
- why that one is not recommended, and
- possible alternatives. Caveat, theses alternatives are by no means exhaustive.
Abstract Algebra
“Algebra“, S. Lang — Dry and tasteless, typical of the textbooks written in 1960’s. Will serve as source of information if you already have a fair knowledge on the subject, but the beginners may feel at a loss as to what-is-going-on-for-what.
Possible alternatives:
First step:
- “Introduction to Abstract Algebra”, P. Garett https://www-users.cse.umn.edu/~garrett/m/intro_algebra/index.shtml
- “Abstract Algebra“, I. N, Herstein
- “Classic Algebra“, P. M. Cohn
- “Algebra: Notes from the Underground“, P. Allufi
- “A First Course in Abstract Algebra“, J. Fraleigh
- “Abstract Algebra: The Basic Graduate Year“, R. Ash
The site owner recommends any of the above. If you know practically nothing about the subject, you can start with 1 or 2 or 3 or 4 and then can proceed to 1 or 2 or 3 or 4 (accordingly) in Second step below.
Second step:
- “Abstract Algebra“, P. Garrett
- “Topics in Algebra“, I. N. Herstein
- “Basic Algebra“, P. M. Cohn
- “Algebra: Chapter 0“, P. Allufi
- “Abstract Algebra“, D. S. Dummit, R. M. Foote
- “First Course in Abstract Algebra“, J. J. Rotman
- “Basic Algebra I“, N. Jacobson
- “Basic Algebra“, A. W. Knapp
- “Abstract Algebra“, P. A. Griller
Herstein’s notation is a bit outdated (he writes right-to-left for maps including homomorphisms), but if you can endure that, it will turn out to be pedagogically wonderful. Rotman’s books are (in)famous for errors including typo, so if you use his book(s), you should look up his homepage for errata.
Third step:
- “Further Algebra and Applications“, P. M. Cohn
- “Advanced Modern Algebra“, J. J. Rotman
- “Basic Algebra II“, N. Jacobson
- “Advanced Algebra“, A. W. Knapp
Algebraic Topology
“Algebraic Topology“, E. H. Spanier — Like “Algebra” by S. Lang, this one is dry and tasteless, too. Another typical of the textbooks written in 1960’s.
Possible alternatives:
- “Algebraic Topology“, A. Hatcher
- “Topology and Geometry“, G. Bredon
- “Algebraic Topology“, T. tom Dieck
- “A Concise Course in Algebraic Topology“, J. P. May
4 is more abstract than 1, 2, and 3 in treatment.
Addendum: If you find that any of the above to be too abstract, or that you need more general topology at hand, you may want to start with “Elementary Topology”, O. Ya. Viro, O. A. Ivanov, N. Yu. Netsvetaev, and V. M. in Kharlamov.
Homotopy Theory
“Homotopy Theory“, S-T Hu — Actually, the site owner has no intention to discourage reading this pioneer book. To make the long story short, students of mathematics don’t have to stick to this, as there are many textbooks on this subject. But (this is a huge “but”) at the time of the publishing of this book, there was virtually nobody who dared to write a textbook on homotopy theory. The situation is succinctly summarised in the review by W. S. Massey who says “He (= the author S-T Hu) has succeeded in doing what no other living topologist has even really seriously attempted.”
Possible alternatives:
As the site owner wrote above, now there are many textbooks on homotopy theory. Among those, the site owner recommends the following two books by Russian topologists.
- “Beginner’s Course in Topology: Geometric Chapters“, D. V. Fuks and V. A. Rokhlin
- “Basic Homotopy Theory” and “Homotopy Theory of Cellular Spaces” M. M. Postnikov
1 includes not only homotopy theory but also differential topology necessary to investigate homotopy theory along the line of Pontryagin-Thom construction.
2 the two books comprise a comprehensive treatment. (An English translation is provided by the site owner –> https://mathmusic52.com/2024/01/26/basic-homotopy-theory-by-m-m-postnikov-translated-by-the-site-owner/,
Algebraic Number Theory
“Algebraic Number Theory“, S. Lang — Barely more than a collection of memoires. Good for reference, but extremely user-hostile. You should already have a fair knowledge of the subject to appreciate the content.
Possible alternatives:
First step:
- “Théorie Algébrique des Nombres“, P. Samuel
- “Introductory Algebraic Number Theory“, S. Alaca, and K. S. Williams
Both 1 and 2 are easy to read with many examples.
Second step:
- “Algebraic Number Theory“, J. S. Milne https://www.jmilne.org/math/CourseNotes/ant.html
- “Number Fields“, D. A. Marcus
- “Algebraic Number Theory“, J. Neukirch
1 and 2 are comparable both in coverage and readability.
3 includes class field theory.
Complex Analysis
“Complex Analysis“, L. Ahlfors — An overrated classic, period! Actually, there are better textbooks including:
- “Complex Analysis“, J. Bak and D. J. Newman
- “Théorie élémentaire des fonctions analytiques d’une ou plusieurs variables complexes“, H. Cartan
- “Theory of Complex Functions“, R. Remmert
- “Complex Analysis“, E. M. Stein and R. Shakarch
Galois Theory
“Galois Theory”, E. Artin — Another overrated classic. The site owner advises to stay away from this book unless you are interested in (a) historical treatment of the subject.
Possible alternatives:
Galois theory is treated in most textbooks of abstract algebra; see 4, 5, and 6 of First step, and all the textbooks in Second step, of Abstract Algebra – Possible alternatives: above.
Of course, there are textbooks dedicated to field theory including Galois theory.
- “Foundations of Galois Theory“, M. M. Postnikov
- “Fields and Rings“, I. Kaplansky
- “Galois Theory“, S. H. Weintraub
- “Field and Galois Theory“, P. Morandi
1 is written by a world-renowned topologist. It is self-contained in the sense that necessary field theory and group theory are developed in the text.
2 comprises three parts and the part one is dedicated to Galois theory.
3 and 4 are examples of modern treatment of field theory.
Addendum: “Algebraic Extensions of Fields“, P. J. McCarthy Often overlooked, this is a unique book covering field theory and valuation theory necessary to pursue algebraic number theory. Probably this should better be titled “Field Theory for an Algebraic Number Theorist “. It is recommended to associate this when reading one of the books in First step, Possible alternatives Algebraic Number Theory.
Differential Topology
“Elementary Differential Topology“, J. R. Munkres — This book does not give the reader a perspective in differential topology. It has two chapters:
- the first chapter is a collection of auxiliary results in muti-variate calculus and differential topology as required to read “Differential Topology“, J. W. Milnor. A retyped version of Milnor’s book is listed in this site,
- the second chapter discusses the triangulation of smooth manifolds.
Thus this book does not offer you the basic ideas of differential topology like cobordism or surgery. Still worse, it does not discuss fundamental components such as Sard’s theorem, Whitney’s imbedding theorem, Morse theory, etc.
Possible alternatives:
- “Introduction to Differential Topology“, T. Bröcker, and K. Jänich
- “Beginner’s Course in Topology: Geometric Chapters “, D. B. Fuks and V. A. Rokhlin
- “Notes on Cobordism Theory“, Robert E. Stong
- “Differential Topology“, C. T. C. Wall
1 is an accessible textbook covering embedding, isotopy and transversality theorems, Sard’s theorem, partitions of unity, dynamical systems, connected sums, tubular neighbourhoods, collars and the glueing.
2 has a detailed discussion of embeddings and immersions, smoothings and approximations and smooth bundle together with a brief introduction to Morse theory, cobordism and surgery.
3 extensively discusses, as the title suggests, cobordism with necessary material on differential topology in the appendix.
4 is, to the site owner’s knowledge, the most comprehensive textbook on differential topology.
Algebraic Geometry
“Algebraic Geometry“ R. Hartshorne — Once regarded as “bible”. Actually, it is dry and tasteless, and not for weakly motivated (i. e., having no intention to major in algebraic geometry) students.
Perhaps, it is not useless to remark that his book was written for students who wish to tackle EGAs by A. Grothedieck and his school. The following remark by M. Reid (pp 122-123 in “Undergraduate Algebraic Geometry“) may be of interest:
On the other hand, the Grothendieck personality cult had serious side effects: many people who had devoted a large part of their lives to mastering Weil foundations suffered rejection and humiliation, and to my knowledge only one or two have adapted to the new language; a whole generation of students (mainly French) got themselves brainwashed into the foolish belief that a problem that can’t be dressed up in high powered abstract formalism is unworthy of study, and were thus excluded from the mathematician’s natural development of starting with a small problem he or she can handle and exploring outwards from there. (I actually know of a thesis on the arithmetic of cubic surfaces that was initially not considered because ‘the natural context for the construction is over a general locally Noetherian ringed topos’. This is not a joke.) Many students of the time could apparently not think of any higher ambition than Étudier les EGAs. The study of category theory for its own sake (surely one of the most sterile of all intellectual pursuits) also dates from this time; Grothendieck himself can’t necessarily be blamed for this, since his own use of categories was very successful in solving problems.
Possible alternatives:
First step:
- “Lectures on Curves, Surfaces and Projective Varieties” M. C. Beltrametti, E. Carletti, D. Gallarati, G. M. Bragadin
- “Algebraic Curves, an Introduction to Algebraic Geometry” W. Fulton
- “Undergraduate Algebraic Geometry” M. Reid
Second step:
- “Basic Algebraic Geometry” vol. 1 and 2 I. R. Shafarevich
- “A Royal Road to Algebraic Geometry” A. Holme
Third step:
- “Algebraic Geometry I, Schemes with Examples and Exercises” U. Görtz, T. Wedhorn
- “Algebraic Geometry and Arithmetic Curves” L Qing
Category Theory
“Categories for the Working Mathematician” S. Mac Lane
An extremely dry and tasteless book. It practically gives no insight or motivation to study the subject for beginners. Probably, that may be inevitable, as category theory is like grammar and gives no magnificent results per se. In fact, to really appreciate this book, you should have a fair amount of background in abstract algebra or algebraic topology, say: in other words, this book is suitable for someone who wishes to reorganise his/her understanding of mathematics.
Possible alternatives:
First step:
“The Joy of Abstraction”, E. Cheng
A really elementary account of the subject. Some people may not like her “woke opinions” on politics scattered here and there, but apart from that, this makes a good read.
Second step:
- “An Introduction to Category Theory” H. Simmons
- “Basic Category Theory” T. Leinster
- “An Introduction to the Language of Category Theory” S. Roman
These should be accessible for students who have read Cheng’s book. The site owner prefers 3 over 1 and 2.
Addendum: “Algebra: Chapter 0” P. Aluffi is a unique textbook of abstract algebra which tries to (re-)explain things in the category-theory context.
Sheaf theory
“Sheaf theory”, B. R. Tennison — the site owner has nothing to add to the conclusion of the review by R. T. Hoobler in “Bulletin of the American Mathematical Society”,
Volume 83, Number 4, July 1977.
Here is an excerpt:
It would have been better, however, if the book had never been written. After all, the basic definitions and properties of sheaves are not very difficult to grasp. Sheaf theory should be a chapter in a book on several complex variables or algebraic geometry or differential geometry or … With the applications immediately at hand, it is much easier to maintain a proper perspective.
Possible alternatives:
As Hoobler remarks, sheaf theory should be studied in a book on several complex variables or algebraic geometry.
- “Function Theory of Several Complex Variables” —- S. Krantz
- “Basic Algebraic Geometry” — I. R. Shafarevich
However, the site owner feels scheptical in the use of sheaf theory in differential geometry. His scheptisim is shared in the answer by Emerton in MathOverflow as excerpted below:
If X is a manifold, and E is a smooth vector bundle over X (e. g., its tangent bundle), then E is again a manifold. Thus working with bundles means that one doesn’t have to leave the category of objects (manifolds) under study; one just considers manifold with certain extra structure (the bundle structure). This is a big advantage in the theory; it avoids introducing another class of objects (i. e., sheaves), and allows tools from the theory of manifolds to be applied directly to bundles too.
Here is a longer discussion, along somewhat different lines:
The historical impetus for using sheaves in algebraic geometry comes from the theory of several complex variables, and in that theory sheaves were introduced, along with cohomological techniques, because many important and non-trivial theorems can be stated as saying that certain sheaves are generated by their global sections, or have vanishing higher cohomology. (I am thinking of Cartan’s Theorem A and B, which have as consequences many earlier theorems in complex analysis.)
If you read Zariski’s fantastic report on sheaves in algebraic geometry, from the 50s, you will see a discussion by a master geometer of how sheaves, and especially their cohomology, can be used as a tool to express, and generalize, earlier theorems in algebraic geometry. Again, the questions being addressed (e.g. the completeness of the linear systems of hyperplane sections) are about the existence of global sections, and/or vanishing of higher cohomology. (And these two usually go hand in hand; often one establishes existence results about global sections of one sheaf by showing that the higher cohomology of some related sheaf vanishes, and using a long exact cohomology sequence.)
These kinds of questions typically don’t arise in differential geometry. All the sheaves that might be under consideration (i.e. sheaves of sections of smooth bundles) have global sections in abundance, due to the existence of partions of unity and related constructions.
There are difficult existence problems in differential geometry, to be sure: but these are very often problems in ODE or PDE, and cohomological methods are not what is required to solve them (or so it seems, based on current mathematical pratice). One place where a sheaf theoretic perspective can be useful is in the consideration of flat (i.e. curvature zero) Riemannian manifolds; the fact that the horizontal sections of a bundle with flat connection form a local system, which in turn determines the bundle with connection, is a useful one, which is well-expressed in sheaf theoretic language. But there are also plenty of ways to discuss this result without sheaf-theoretic language, and in any case, it is a fairly small part of differential geometry, since typically the curvature of a metric doesn’t vanish, so that sheaf-theoretic methods don’t seem to have much to say.
If you like, sheaf-theoretic methods are potentially useful for dealing with problems, especially linear ones, in which local existence is clear, but the objects are suffiently rigid that there can be global obstructions to patching local solutions.
In differential geometry, it is often the local questions that are hard: they become difficult non-linear PDEs. The difficulties are not of the “patching local solutions” kind. There are difficult global questions too, e. g., the one solved by the Nash embedding theorem, but again, these are typically global problems of a very different type to those that are typically solved by sheaf-theoretic methods.
Commutative Algebra
“Steps in Commutative Algebra”, R. Y. Sharp
Another dry and tasteless book which gives no insight or motivation to study the subject for beginners.
According to the authour, the said book should give “stepping stones” to study Atiyah-McDonald and Matsumura. If no alternatives are available, the site owner would rather go directly to Atiyah-McDonald than to first read this, using this textbook as a supplement to A-M for bridging the gaps.
Possible alternatives:
Listed in the order of thickness (= the number of pages):
“A Primer of Commutative Algebra“ J.S. Milne
Retrievable from CA — J.S. Milne (jmilne.org)
“Commutative Algebra” P. Clark
Retrievable from Pete L. Clark’s Expositions (uga.edu) (the second entry)
“A Term of Commutative Algebra“, Allen Altman, Steven Kleiman
Retrievable from A term of Commutative Algebra (mit.edu)
Addendum:
Mel Hochster has a wonderful set of lecture notes as well as research papers here Mel Hochster’s Homepage (umich.edu).
General Topology
“General Topology”, John L. Kelley
This textbook should have been titled “General Topology for an analyst”.
While analytic topics are comprehensively organised, it covers few topics desirable for students of geometry (and algebra). For example, path-connectedness is not treated. So for those students this textbook would make a frustrating read.
Possible alternatives : (for non-analysts)
“Topology and Geometry“, G. Bredon
While being a comprehensive textbook on algebraic topology, its chapter 1 is a concise introduction to general topology required in studying geometry (especially algebraic and differential topology.)
Both covers more topics (in terms of general topology, of course) than Bredon. Good for geometry and algebra.
“Topology: A Categorical Approach“, T-D. Bradley, T. Bryson, J. Terilla
As the title suggests, this textbook stresses the categorical point of view, in the same spirit as “Algebra: Chapter 0” by P. Aluffi.
“Elementary Topology”, O. Ya. Viro, O. A. Ivanov, N. Yu. Netsvetaev, and V. M. in Kharlamov
A comprehensive collection of exercises in general topology with a strong geometric (or, algebraic-topological, for that matter) flavour.
Should one want more detailed account(s),
- “Topology“, J. Dugundji
- “General Topology“, S. Willard
Both are classical and (contrary to Kelley) covering non-analytic topics as well.
For profinite (group) topology as in infinite Galois theory,
“Profinite Groups“, L. Ribes, P. Zalesskii
Chapter 1 is a concise introduction to the (inverse) limit of “finite” objects and related topological considerations.
MMM 