Recent Papers
- Hodge Theory on Metric Spaces
- Mathematics of Neural Response
- Hodge Decomposition and Learning Theory
- Topology and Data #2
- Derived Distance
- Geometry on Probbability Spaces
- Risk Bounds for Random Regression Graphs
- Emergent Behavior in Flocks
- Learning Via Integral Operators
- Shannon Sampling 2
- Shannon Sampling
- Online Learning Algorithms
- Obstructions to New Ideas
- Homology of Submanifolds from Data
- On the Mathematical Foundations of Learning
- Best Choices for Regularization Parameters in Learning Theory: On The Bias-Variance problem
- The Mathematics of Learning: Dealing with Data
- Modeling Language Evolution
- Estimating the Approximation Error in Learning Theory
- The Mathematics of Emergence
- Online learning with Markov sampling
List of Publications 1994-2004
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, pp321-339. Springer-Verlag 1993.
Some Autobiographical Notes In M. Hirsch, J. Marsden, M. Shub,(Eds.), From Topology to Computation: Proceedings of the Smalefest, pp3-21. Springer-Verlag 1993.
Complexity of Bezout's Theorem V: Polynomial Time (with Mike Shub), /Theoretical Computer Science/, *133* (1994), pp. 141-164.
- Postscript: http://www6.cityu.edu.hk/ma/people/smale/pap95.ps
- PDF: http://www6.cityu.edu.hk/ma/people/smale/pap95.pdf
Separation of Complexity Classes in Koiran's Weak Model (with Felipé Cucker and Mike Shub) /Theoretical Computer Science/, *133* (1994), pp. 3-14.
- Postscript: http://www6.cityu.edu.hk/ma/people/smale/pap96.ps
- PDF: http://www6.cityu.edu.hk/ma/people/smale/pap96.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).
- Postscript: http://www6.cityu.edu.hk/ma/people/smale/pap97.ps
- PDF: http://www6.cityu.edu.hk/ma/people/smale/pap97.pdf
Complexity of Bezout's Theorem IV: Probabability of success; Extensions (with Mike Shub), /SIAM Jour. of Numerical Analysis/, *33* (1996) pp. 128-148.
- Postscript: http://www6.cityu.edu.hk/ma/people/smale/pap98.ps
- PDF: http://www6.cityu.edu.hk/ma/people/smale/pap98.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.
- Postscript: http://www6.cityu.edu.hk/ma/people/smale/pap100.ps
- PDF: http://www6.cityu.edu.hk/ma/people/smale/pap100.pdf
Complexity theory and numerical analysis, /Acta Numerica/ (1997) pp.523-551.
- Postscript: http://www6.cityu.edu.hk/ma/people/smale/pap101.ps
- PDF: http://www6.cityu.edu.hk/ma/people/smale/pap101.pdf
Complexity and real computation (with L. Blum, F. Cucker, and M. Shub), Springer-Verlag 1998.
Finding a Horseshoe on the Beaches of Rio, /Mathematical Intelligencer/ Vol 20, (1998) No. 1, 39-44.
Mathematical Problems for the Next Century, /Mathematical Intelligencer/ Vol 20, (1998) No. 2, 7-15.
- Postscript: http://www6.cityu.edu.hk/ma/people/smale/pap104.ps
- PDF: http://www6.cityu.edu.hk/ma/people/smale/pap104.pdf
Some Lower Bounds for the Complexity of Continuation Methods (with J-P Dedieu) /Jour of Complexity/ Vol 14 (1998) 454-465.
- Postscript: http://www6.cityu.edu.hk/ma/people/smale/pap105.ps
- PDF: http://www6.cityu.edu.hk/ma/people/smale/pap105.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.
- Postscript: http://www6.cityu.edu.hk/ma/people/smale/pap106.ps
- PDF: http://www6.cityu.edu.hk/ma/people/smale/pap106.pdf
Finding a Horseshoe on the Beaches of Rio, /Mathematical Intelligencer/, 1998, vol 1.
- Postscript: http://www6.cityu.edu.hk/ma/people/smale/pap107.ps
- PDF: http://www6.cityu.edu.hk/ma/people/smale/pap107.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.
- Postscript: http://www6.cityu.edu.hk/ma/people/smale/pap108.ps
- PDF: http://www6.cityu.edu.hk/ma/people/smale/pap108.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.