Professor Stephen Smale

PhD - University of Michigan, USA

Senior Fellow of Hong Kong Institute for Advanced Study, City University of Hong Kong

National Medal of Science Laureate

Member of the US National Academy of Sciences

Fields Medalist (1966)

Wolf Prize Laureate (2007)

Professor Stephen Smale

Contact Information

Web: Personal Homepage

Research Interests

  • Complexity and Computation
Earlier in his career, Professor Smale was involved in controversy over remarks he made regarding his work habits while proving the higher-dimensional Poincaré conjecture. He said that his best work had been done "on the beaches of Rio". This led to the withholding of his grant money from the NSF. He has been politically active in various movements in the past, such as the Free Speech movement. At one time he was subpoenaed by the House Un-American Activities Committee.

In 1960 Professor Smale was appointed an associate professor of mathematics at the University of California, Berkeley, moving to a professorship at Columbia University the following year. In 1964 he returned to a professorship at UC Berkeley where he has spent the main part of his career. He retired from UC Berkeley in 1995 and took up a post as professor at the City University of Hong Kong. He also amassed over the years one of the finest private mineral collections in existence. Many of Professor Smale's mineral specimens can be seen in the book—The Smale Collection: Beauty in Natural Crystals.


1930Born July 15 at Flint, Michigan
1952BS, University of Michigan
1953MS, University of Michigan
1957PhD, University of Michigan
Instructor, University of Chicago
1958–60Member, Institute for Advanced Study, Princeton
1960–62Alfred P. Sloan Research Fellow
1960–61Associate Professor of Mathematics, University of California, Berkeley
1961–64Professor of Mathematics, Columbia University
1962Visiting Professor, College de France, Paris (Spring)
1964–94Professor of Mathematics, University of California, Berkeley
1966Member, Institute for Advanced Study, Princeton (Fall)
1967–68Research Professor, Miller Institute for Basic Research in Science, Berkeley
1969–70Visiting Member, Institut des Hautes Etudes Scientifiques (Fall)
1972–73Visiting Member, Institut des Hautes Etudes Scientifiques (Fall)
1972–73Visiting Professor, University of Paris, Orsay (Fall)
1974Visiting Professor, Yale University (Fall)
1976Visiting Professor, Instituto de Matematica Pura e Aplicada, Rio de Janeiro
1976Visiting Professor, Institut des Hautes Études Scientifiques (May, June)
1976–Professor without stipend, Department of Economics, University of California, Berkeley
1979–80Research Professor, Miller Institute for Basic Research in Science, Berkeley
1987Visiting Scientist, IBM Corporation, Yorktown Heights (Fall)
1987Visiting Professor, Columbia University (Fall)
1988Visiting Professor, Instituto de Matematica Pura e Aplicada, Rio de Janeiro
1994Visiting Professor, Instituto de Matematica Pura e Aplicada, Rio de Janeiro
1994–Professor of Mathematics (and Economics) Emeritus, University of California, Berkeley
1995–2001Distinguished University Professor, City University of Hong Kong
2002–2009Professor, Toyota Technological Institute at Chicago
2009–2016University Distinguished Professor, City University of Hong Kong
2018–PresentHonorary Professor, City University of Hong Kong

Honours (Selected)

  • 1974 Honorary Degree, University of Warwick
  • 1987 Honorary Degree, Queen's University, Kingston, Ontario
  • 1994 Class of the Grand Cross of the Brasilian National Order of Scientific Merit
  • 1996 Honorary Doctor of Science, University of Michigan
  • 1996 National Medal of Science, Washington, D. C.
  • 1997 Doctor Honoris Causa, Université Pierre et Marie Curie, Paris
  • 1997 Honorary Doctor of Science, City University of Hong Kong
  • 1997 Honorary Member of the Moscow Mathematical Society
  • 1998 Honorary Member of the London Mathematical Society
  • 1999 Honorary Doctorate of Rostov State University
  • 2004 Honorary Doctorate of University of Genoa
  • 2005 Jurgen Moser Prize of SIAM, Dynamics Group
  • 2007 Wolf Prize

List of Publications (1994 - 2009)

  • The Godel Incompleteness Theorem and Decidability Over a Ring. (with Lenore Blum) In M. Hirsch, J. Marsden, and M. Shub (Eds.)
  • From Topology to Computation: Proceedings of the Smalefest, pp. 321–339. Springer-Verlag 1993.
  • Some Autobiographical Notes In M. Hirsch, J. Marsden, M. Shub, (Eds.), From Topology to Computation: Proceedings of the Smalefest, pp. 3–21. Springer-Verlag 1993.
  • Complexity of Bezout's Theorem V: Polynomial Time (with Mike Shub), /Theoretical Computer Science/, *133* (1994), pp. 141–164. [ ps | pdf ]
  • Separation of Complexity Classes in Koiran's Weak Model (with Felipe Cucker and Mike Shub), /Theoretical Computer Science/, *133* (1994), pp. 3–14. [ ps | pdf ]
  • On the Intractibility of Hilbert's Nullstellensatz and an Algebraic Version of "NP≠P?" (with Mike Shub), /Duke Math Jour/, *81* pp. 47–54 (1995). [ ps | pdf ]
  • Complexity of Bezout's Theorem IV: Probabability of success; Extensions (with Mike Shub), /SIAM Jour. of Numerical Analysis/, *33* (1996) pp. 128–148. [ ps | pdf ]
  • Complexity and Real Computation: A Manifesto (with L. Blum, F. Cucker and M. Shub) /International Journal of Bifurcation and Chaos/, *6* (1996) pp. 3–26.
  • Algebraic Settings for the Problem "P≠NP?" (with L. Blum, F. Cucker, M. Shub), Lectures in Applied Mathematics vol. 32, ed. J. Renegar, M. Shub and S. Smale, pp. 125–144, /Amer. Math. Soc./ 1996. [ ps | pdf ]
  • Complexity theory and numerical analysis, /Acta Numerica/, (1997) pp. 523–551. [ ps | pdf ]
  • Complexity and real computation (with L. Blum, F. Cucker, and M. Shub), Springer-Verlag 1998.
  • Finding a Horseshoe on Beaches of Rio,/Mathematrical Intelligencer/, Vol. 20, (1988) No. 1, 39-44.
  • Mathematical Problems for the Next Century, /Mathematical Intelligencer/, Vol. 20, (1998) No. 2, 7–15. [ ps | pdf ]
  • Some Lower Bounds for the Complexity of Continuation Methods (with J-P Dedieu), /Jour of Complexity/, Vol. 14 (1998) 454–465. [ ps | pdf ]
  • The Work of Curtis T. McMullen, in Proceedings of the International Congress of Mathematicians Berlin 1998 Vol. 1 Documenta Mathematica Bielefeld, Germany, 1998 pp. 127–132. [ ps | pdf ]
  • A Polynomial Time Algorithm for Diophantine Equations in One Variable (with F. Cucker and P. Koiran), /Jour. Symbolic Computation/, Vol. 27 (1999) 21–29. [ ps | pdf ]
  • Cucker, Felipe; Smale, Steve
  • Complexity estimates depending on condition and round-off error. J. ACM 46 (1999), no. 1, 113–184.
  • Batterson, Steve Stephen Smale
  • The Mathematician Who Broke the Dimension Barrier. American Mathematical Society, Providence, RI, 2000. xviii+306 pp.
  • Smale, Steve
  • Mathematical Problems for the Next Century. Mathematics: Frontiers and Perspectives, 271–294, Amer. Math. Soc., Providence, 2000.
  • Smale, Stephen
  • The Collected Papers of Stephen Smale. Vol. 1–3. Edited by F. Cucker and R. Wong. Singapore University Press, Singapore; World Scientific Publishing Co., Inc., River Edge, NJ, 2000. Vol. 1: xxxiv+488 pp.; Vol. 2: pp. i–xii and 489–1031; Vol. 3: pp. i–xii and 1033–1677.
  • Cucker, Felipe; Smale, Steve
  • On the Mathematical Foundations of Learning. Bull. Amer. Math. Soc. (N.S.) 39 (2002), no. 1, 1–49.
  • Cucker, Felipe; Smale, Steve
  • Best Choices for Regularization Parameters in Learning Theory: On the Bias-Variance Problem. Found. Comput. Math. 2 (2002), no. 4, 413–428.
  • Smale, Steve; Zhou, Ding-Xuan
  • Estimating the Approximation Error in Learning Theory. Anal. Appl. (Singap.) 1 (2003), no. 1, 17–41.
  • Poggio, Tomaso; Smale, Steve
  • The Mathematics of Learning: Dealing with Data. Notices Amer. Math. Soc. 50 (2003), no. 5, 537–544.
  • Smale, Steve; Zhou, Ding-Xuan
  • Shannon Sampling and Function Reconstruction from Point Values. Bull. Amer. Math. Soc. (N.S.) 41 (2004), no. 3, 279–305.
  • Cucker, Felipe; Smale, Steve; Zhou, Ding-Xuan
  • Modeling Language Evolution. Found. Comput. Math. 4 (2004), no. 3, 315–343.
  • Hirsch, Morris W.; Smale, Stephen; Devaney, Robert L.
  • Differential Equations, Dynamical Systems, and an Introduction to Chaos. Second edition. Pure and Applied Mathematics (Amsterdam), 60. Elsevier/Academic Press, Amsterdam, 2004.
  • With F. Cucker, "On the mathematics of emergence", Japanese Jour of Mathematics (The first Takagi Lectures) Vol. 2, No. 1, March 2007. 197–227.
  • With A. Caponnetto, "Risk Bounds for Random Regression Graphs", Jour Foundations of computational mathematics (2007), vol 7, #4, 495–528.
  • With F. Cucker, "Emergent behavior in Flocks", IEEE Transactions on automatic control, Vol 52, issue 5 (May, 2007) 852–862.
  • With P. Niyogi and S. Weinberger, "Finding the homology of submanifolds with high confidence from random samples", Discrete and Computational geometry (2008) 39: 419–441.
  • With Y. Yao "Online learning algorithms", Foundations of computational mathematics vol. 6 no. 2 145–170 (2006).
  • With D.-X. Zhou, "Learning theory estimates via integral operators and their approximations", Constructive Approximations (2007) 26: 153–172.
  • with D.-X. Zhou, "Shannon 2: Connections to learning theory", Applied and Computational harmonic analysis 19 (2005) 285–302.

Recent Publications

  • With Ding-Xuan Zhou, "Online Learning with Markov Sampling",1 "Analysis and Applications"., Vol. 7, Jan, 2009
  • With Ding-Xuan Zhou, "Geometry on Probability Spaces" 1"Constructive Application"., (2009) 30. 311-323.
  • With Partha Niyogi and Shmuel Weinberger, "A Topological View of Unsupervised Learning from Noisy Data", SIAM Journal on Computing 2011 40 (3) 646-663.
  • With L. Rosasco, J. Bouvrie, A. Caponnetto and T. Poggio, Mathematics of the Neural Response, 2010, Foundations of Computational Mathematics 10, 67-91.
  • with L. Bartholdi, T. Schich, N. Smale Hodge theory on Metric Spaces, 2012, 12, 1-48.
  • Journal Foundations of Computational Mathematics
  • with Nathan Smale, Abstract and Classical Hodge-DeRham theory 2010, Analysis and applications, 18 pp
  • with W. Shen, H. S. Wong, Q.-W Xiao, X. Guo, Introduction to the Peptide Binding Problem of Computational Immunology; New Results1 Journal Foundations of Computational Mathematics 2013, 34 pp
  • with M. Andreatta, S. Liaplagne, S. C. Li, 1Prediction of Residue-residue contacts from Protein famulies using similarity Rerrels and least squares regularization1 Bioinformatics, 2013 submitted.
  • MHC Binding Prediction with Kernel RLS pan and its variations with W. Shen, Y. T. Wei, X. Guo, H.-S. Wong and S. C. Li, submitted.