Google Scholar metrics 2013

Earlier this month Google released the 2013 edition of Google Scholar metrics. Top 20 journals (and non-journals) in Mathematical Analysis category:

  1. Nonlinear Analysis: Theory, Methods & Applications
  2. Journal of Mathematical Analysis and Applications
  3. arXiv Analysis of PDEs (math.AP)
  4. Journal of Functional Analysis
  5. Fixed Point Theory and Applications
  6. arXiv Functional Analysis (math.FA)
  7. SIAM Journal on Mathematical Analysis
  8. Journal of Differential Equations
  9. Abstract and Applied Analysis
  10. Journal of Inequalities and Applications
  11. Annales de l’Institut Henri Poincare (C) Non Linear Analysis
  12. arXiv Classical Analysis and ODEs (math.CA)
  13. Calculus of Variations and Partial Differential Equations
  14. Discrete and Continuous Dynamical Systems
  15. arXiv Operator Algebras (math.OA)
  16. Indiana University Mathematics Journal
  17. Journal de Mathématiques Pures et Appliquées
  18. Communications in Partial Differential Equations
  19. arXiv Complex Variables (math.CV)
  20. ESAIM: Control, Optimisation and Calculus of Variations

That’s one weird (and Elsevier-infested) list.

The ratings are based on the total number of citations, not citations per article. The more bloated a journal is, the better it looks. Maintaining an industry of Ctrl-C Ctrl-V generalizations helps too. JMAA scores high on both counts.

44 influential journal articles

Same idea as 44 Great Books, but for journal articles. Influence is measured by the number of citations on MathSciNet, and as such is subject to chronological and geographical bias. Italic indicates I took a look at the paper, whether or not I got anything out of it.

The earliest paper on the list was published in 1938 (see Geometry), the latest in 2000 (see K-theory). Half were published in 1977 or later.

Area Title Author Year
General A level-set approach for inverse problems involving obstacles Santosa 1995
History and biography Mathematical problems for the next century Smale 1998
Mathematical logic and foundations Linear logic Girard 1987
Combinatorics On the evolution of random graphs1 Erdős, Rényi 1960
Order, lattices, ordered algebraic structures A lattice-theoretical fixpoint theorem and its applications Tarski 1955
General algebraic systems Algebras whose congruence lattices are distributive Jónsson 1967
Number theory Modular elliptic curves and Fermat’s last theorem Wiles 1995
Field theory and polynomials Factoring polynomials with rational coefficients Lenstra, Lenstra, Lovász 1982
Commutative algebra Tight closure, invariant theory, and the Briançon-Skoda theorem Hochster, Huneke 1990
Algebraic geometry Resolution of singularities of an algebraic variety over a field of characteristic zero Hironaka 1964
Linear and multilinear algebra; matrix theory The invariant theory of n×n matrices Procesi 1976
Associative rings and algebras On the deformation of rings and algebras Gerstenhaber 1964
Nonassociative rings and algebras Lie superalgebras Kac 1977
Category theory; homological algebra Des catégories abéliennes Gabriel 1962
K-theory The coarse Baum-Connes conjecture for spaces which admit a uniform embedding into Hilbert space Yu 2000
Group theory and generalizations Representations of Coxeter groups and Hecke algebras Kazhdan, Lusztig 1979
Topological groups, Lie groups L’algèbre de Fourier d’un groupe localement compact Eymard 1964
Real functions Analysis on measure chains—a unified approach to continuous and discrete calculus Hilger 1990
Measure and integration Fractals and self-similarity2 Hutchinson 1981
Functions of a complex variable Quasiconformal maps in metric spaces with controlled geometry Heinonen, Koskela 1998
Potential theory A symmetry problem in potential theory Serrin 1971
Several complex variables and analytic spaces Real hypersurfaces in complex manifolds Chern, Moser 1974
Special functions Differential-difference operators associated to reflection groups Dunkl 1989
Ordinary differential equations Ordinary differential equations, transport theory and Sobolev spaces DiPerna, Lions 1989
Partial differential equations User’s guide to viscosity solutions of second order partial differential equations Crandall, Ishii, Lions 1992
Dynamical systems and ergodic theory A shallow water equation on the circle Constantin, McKean 1999
Difference and functional equations Approximate homomorphisms Hyers, Rassias 1992
Sequences, series, summability A contribution to the theory of divergent sequences Lorentz 1948
Approximations and expansions Basic calculus on time scales and some of its applications Agarwal, Bohner 1999
Harmonic analysis on Euclidean spaces H^p spaces of several variables Fefferman, Stein 1972
Abstract harmonic analysis Banach spaces related to integrable group representations and their atomic decompositions Feichtinger, Gröchenig 1989
Integral transforms, operational calculus On singular integrals Calderón, Zygmund 1956
Integral equations Traveling waves in a convolution model for phase transitions Bates, Fife, Ren, Wang 1997
Functional analysis Dual variational methods in critical point theory and applications Ambrosetti, Rabinowitz 1973
Operator theory Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces Amann 1976
Calculus of variations and optimal control The concentration-compactness principle in the calculus of variations2 Lions 1984
Geometry A theorem in finite projective geometry and some applications to number theory Singer 1938
Convex and discrete geometry On general minimax theorems Sion 1958
Differential geometry On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation Yau 1978
General topology A generalization of Tychonoff’s fixed point theorem Fan 1960
Algebraic topology On the structure of Hopf algebras Milnor, Moore 1965
Manifolds and cell complexes Differentiable dynamical systems Smale 1967
Global analysis, analysis on manifolds Spectral asymmetry and Riemannian geometry Atiyah, Patodu, Singer 1975
Probability theory and stochastic processes A mathematical theory of communication Shannon 1948


  • (1) In combinatorics, the Erdős-Rényi paper is slightly behind Emergence of scaling in random networks by Barabási and Albert (1999). I disqualified the latter for being published in Science.
  • (2) The 1981 paper by Hutchinson on fractals was classified under Calculus of variations, and would be the most cited in that category. However, this is an artifact of MSC which did not have a classification for Fractals until 1991. Now they are under Measure and Integration (28A80).
  • (2) If I did not move Hutchinson’s paper to Measure and Integration, the winner there would be A relation between pointwise convergence of functions and convergence of functionals by Brézis and Lieb (1983). Notably, this is a 5-page paper in the Proceedings of the AMS.

Finally, some trivia based on the combined lists of 44 books and 44 papers.

  • Pierre-Louis Lions is the only person with three entries: most cited papers in ODE, PDE, and Calculus of Variations
  • Juha Heinonen, Walter Rudin, and Stephen Smale have entries in two different MSC areas:
    • Heinonen: most cited book in Potential Theory and most cited paper in One Complex Variable (another artifact of MSC: analysis on metric spaces is under MSC 30)
    • Rudin: most cited books in General and in Several Complex Variables
    • Smale: most cited papers in History/Biography and in Manifolds/Cell Complexes.
  • Victor Kac and Eliah Stein occupy both top spots (book and paper) in the same area:
    • Kac: Nonassociative Groups and Algebras
    • Stein: Harmonic Analysis on Euclidean Spaces

44 Great Books

Using MathSciNet, I found the most cited book in each of 44 “pure” areas of Mathematical Subject Classification, from General (00) to Probability (60). The results are below. Bold font indicates I own the book. Italic: used to own, but not anymore.

Area Title Authors(s)
General Real and complex analysis Rudin
History and biography A course of modern analysis. Whittaker, Watson
Mathematical logic and foundations Classical descriptive set theory Kechris
Combinatorics Symmetric functions and Hall polynomials Macdonald
Order, lattices, ordered algebraic structures Lattice theory Birkhoff
General algebraic systems A course in universal algebra Burris, Sankappanavar
Number theory The arithmetic of elliptic curves Silverman
Field theory and polynomials Local fields Serre
Commutative algebra Commutative algebra Eisenbud
Algebraic geometry Algebraic geometry Hartshorne
Linear and multilinear algebra Matrix analysis Horn, Johnson
Associative rings and algebras Hopf algebras and their actions on rings Montgomery
Nonassociative rings and algebras Infinite-dimensional Lie algebras Kac
Category theory; homological algebra Categories for the working mathematician MacLane
K-theory Cyclic homology Loday
Group theory and generalizations Endliche Gruppen Huppert
Topological groups, Lie groups Groupes et algèbres de Lie Bourbaki
Real functions Convex analysis Rockafellar
Measure and integration Geometric measure theory Federer
Functions of a complex variable Bounded analytic functions Garnett
Potential theory Nonlinear potential theory of degenerate elliptic equations Heinonen, Kilpeläinen, Martio
Several complex variables and analytic spaces Function theory in the unit ball of Cn Rudin
Special functions Special functions Andrews, Askey, Roy
Ordinary differential equations Introduction to functional-differential equations Hale, Verduyn Lunel
Partial differential equations Elliptic partial differential equations of second order Gilbarg, Trudinger
Dynamical systems and ergodic theory Random dynamical systems Arnold
Difference and functional equations Stability of functional equations in several variables Hyers, Isac, Rassias
Sequences, series, summability Divergent Series Hardy
Approximations and expansions Constructive approximation DeVore, Lorentz
Harmonic analysis on Euclidean spaces Harmonic analysis Stein
Abstract harmonic analysis Abstract harmonic analysis Hewitt, Ross
Integral transforms, operational calculus The classical moment problem and some related questions in analysis Akhiezer
Integral equations Evolutionary integral equations and applications Prüss
Functional analysis Sobolev spaces Adams
Operator theory Semigroups of linear operators and applications to partial differential equations Pazy
Calculus of variations and optimal control Optimization and nonsmooth analysis Clarke
Geometry Projective geometries over finite fields Hirschfeld
Convex and discrete geometry Convex bodies: the Brunn-Minkowski theory Schneider
Differential geometry Differential geometry, Lie groups, and symmetric spaces Helgason
General topology General topology Engelking
Algebraic topology Algebraic topology Spanier
Manifolds and cell complexes Introduction to compact transformation groups Bredon
Global analysis, analysis on manifolds Introduction to the modern theory of dynamical systems Katok, Hasselblatt
Probability theory and stochastic processes Convergence of probability measures Billingsley