Great articles and books
In spring 2008, during one of our algebraic geometry lunches, we discussed how to write mathematics well. I find that learning by example is more helpful than being told what to do. In this spirit, we tried to name as many examples of “great writing” as possible. Asking for “the best article you’ve read” isn’t reasonable or helpful. Instead, we asked ourselves the question “what is a great article”, and implicitly, “what makes it great”?
If you have suggestions of great writing, please let me know, and I’ll add them here periodically. (I’ve lost much of the original list coming out of that lunch unfortunately.) I won’t attach names to the suggestions, although I’ll keep track of the number of times each article or book was proposed. I refuse to give criteria for greatness; that’s your job. But please don’t propose writing that has a major flaw unless it is outweighed by some other truly outstanding qualities. In particular, “great writing” is not the same as “proof of a great theorem”. I won’t veto any suggestions (except you are not allowed to recommend anything by yourself or me, because we are both such great writers, and it just wouldn’t be fair), so if you see something terrible listed here, don’t blame me; blame the anonymous contributor. No more than three per person please! Feel free to pick things already on this list (so they will appear with multiplicity).
Not acceptable reasons:
This paper is really very good.
This book is the only book covering this material in a reasonable way.
This is the best article on this subject.
This paper changed my life.
This book inspired me to become a topologist. (Ideally in this case it should be a book in topology, not in real analysis…)
Anyone in my field who hasn’t read this paper has led an impoverished existence.
I wish someone had told me about this paper when I was younger.
This list is quite arbitrary, and a function of the people contributing to it. It will necessarily be weighted toward algebraic geometry for this reason. You should read it not as the opinion of the mathematical community, but as a sampling of the diverse tastes of a number of mathematicians. Many great papers and books are not included (including yours). What can be considered: undergraduate texts and up.
Atiyah-Bott, “The moment map and equivariant cohomology”
Keith Ball, “An elementary introduction to modern convex geometry”, in Flavors of geometry, 1–58, Math. Sci. Res. Inst. Publ., 31, Cambridge Univ. Press, Cambridge, 1997.
Einstein’s original papers on relativity.
Grothendieck, Tohoku paper
Hamilton’s first paper on Ricci flow
Huber-Sturmfels, “A pollyhedral method for solving sparse polynomial systems”
Kapranov-Sturmfels-Zelevinsky, “Quotients of toric varieties and Chow polytopes and general resultants”
Nick Katz, “Nilpotent connections and the monodromy theorem: Applications of a result of Turrittin”
Nick Katz, “Algebraic solutions of differential equations (p-curvature and the Hodge filtration)”
Kleiman, “Transversality of a general translate”
Lazarsfeld, Brill-Noether using K3 surfaces without degeneration techniques
Milnor, “Construction of universal bundles I and II”
Mori, tangent bundle paper
Mumford, “Towards an enumerative geometry of the moduli space of curves”
Mumford, “The Picard group of moduli problems”.
B. Riemann, “Theorie der Abelschen Funktionen”, J. Reine Angew. Math., 54 (1857). English translation 2004 from Kendrick Press: “Theory of abelian functions”.
Serre, “FAC” (Faisceaux Algebriques Coherents)
Serre, “The Duke Paper” (Sur les representations modulaires de degre 2 de Gal(Qbar/Q))
Tate, John T., The arithmetic of elliptic curves. Invent. Math. 23 (1974), 179–206.
Tate, John T., “Number theoretic background”, in the Corvallis volume.
Witten, “Supersymmetry and Morse theory”.
Shou-wu Zhang, “Equidistribution of small points on abelian varieties”.
Adams, Infinite Loop Spaces
Bosch-Lutkebohmert-Raynaud, Neron Models
H. Cohn, “Introduction to the construction of class fields”
Roy Dubisch, Introduction to Abstract Algebra
Fulton, “Intersection theory”
Gel’fand-Kapranov-Zelevinsky, “Discriminants, Resultants, and Multidimensional Deterimants”
Walther Greiner, “Lehrbuch der theoretischen Physik”
Kempf, Algebraic Varieties
G. Kempf, Abelian Integrals, monografias del instituto de matematicas #13, UNAM.
T. Y. Lam’s book: The algebraic theory of quadratic forms, The Benjamin/Cummings Publishing Company, 1973 (second printing 1980).
Maclane, Categories for the working mathematician
Melcher, “Relativitätstheorie in elementarer Darstellung mit Aufgaben und Lösungen”
Milnor, Topology from a Differentiable Viewpoint
Milnor, Characteristic Classes
Milnor, Morse Theory
Mumford, Abelian Varieties
Mumford, Curves on an Algebraic Surface
Mumford, Red Book of Varieties and Schemes
Samuel, Algebraic theory of numbers
Serre, A Course in Arithmetic (x 2)
Serre, Linear Representations of Finite Groups
Serre, Local Fields
Ziegler, Lectures on Polyltopes