# Stephen Smale

#418

Most Influential Person Now

American mathematician

## Stephen Smale's AcademicInfluence.com Rankings

Stephen Smalemathematics Degrees

Mathematics

#41

World Rank

#127

Historical Rank

#12

USA Rank

Differential Topology

#2

World Rank

#2

Historical Rank

#2

USA Rank

Algebraic Topology

#3

World Rank

#6

Historical Rank

#2

USA Rank

Measure Theory

#117

World Rank

#187

Historical Rank

#55

USA Rank

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Mathematics

## Why Is Stephen Smale Influential?

(Suggest an Edit or Addition)According to Wikipedia, Stephen Smale is an American mathematician, known for his research in topology, dynamical systems and mathematical economics. He was awarded the Fields Medal in 1966 and spent more than three decades on the mathematics faculty of the University of California, Berkeley , where he currently is Professor Emeritus, with research interests in algorithms, numerical analysis and global analysis.

## Stephen Smale's Published Works

### Published Works

- Differentiable dynamical systems (1967) (3050)
- Differential Equations, Dynamical Systems, and Linear Algebra (1974) (2951)
- On the mathematical foundations of learning (2001) (1556)
- Emergent Behavior in Flocks (2007) (1472)
- Differential Equations, Dynamical Systems, and an Introduction to Chaos (2003) (1306)
- Mathematical problems for the next century (1998) (1280)
- On a theory of computation and complexity over the real numbers: $NP$- completeness, recursive functions and universal machines (1989) (1144)
- An Infinite Dimensional Version of Sard's Theorem (1965) (656)
- Finding the Homology of Submanifolds with High Confidence from Random Samples (2008) (655)
- Learning Theory Estimates via Integral Operators and Their Approximations (2007) (573)
- Diffeomorphisms with Many Periodic Points (1965) (563)
- The Mathematics of Learning: Dealing with Data (2005) (553)
- Generalized Poincare's Conjecture in Dimensions Greater Than Four (1961) (512)
- On the mathematics of emergence (2007) (466)
- Topology and mechanics. I (1970) (466)
- On the Structure of Manifolds (1962) (458)
- Newton’s Method Estimates from Data at One Point (1986) (426)
- Convergent process of price adjust-ment and global newton methods (1976) (425)
- On Gradient Dynamical Systems (1961) (409)
- DIFFEOMORPHISMS OF THE 2-SPHERE (1959) (400)
- The fundamental theorem of algebra and complexity theory (1981) (373)
- On the differential equations of species in competition (1976) (308)
- On the efficiency of algorithms of analysis (1985) (306)
- Complexity of Bézout’s theorem. I. Geometric aspects (1993) (305)
- The Classification of Immersions of Spheres in Euclidean Spaces (1959) (285)
- A generalized Morse theory (1964) (283)
- Best Choices for Regularization Parameters in Learning Theory: On the Bias—Variance Problem (2002) (277)
- Complexity of Bezout's Theorem: III. Condition Number and Packing (1993) (277)
- Morse inequalities for a dynamical system (1960) (269)
- ESTIMATING THE APPROXIMATION ERROR IN LEARNING THEORY (2003) (268)
- Complexity of Bezout’s Theorem II Volumes and Probabilities (1993) (268)
- Shannon sampling and function reconstruction from point values (2004) (255)
- On the average number of steps of the simplex method of linear programming (1983) (250)
- Topology and mechanics. II (1970) (248)
- On the Structure of 5-Manifolds (1962) (226)
- Shannon sampling II: Connections to learning theory (2005) (216)
- Complexity of Bezout's theorem IV: probability of success; extensions (1996) (185)
- On a theory of computation and complexity over the real numbers; np-completeness (1989) (185)
- A Topological View of Unsupervised Learning from Noisy Data (2011) (171)
- The mathematics of time (1980) (169)
- Structurally Stable Systems are not Dense (1966) (167)
- Complexity theory and numerical analysis (1997) (166)
- Complexity of Bezout's Theorem V: Polynomial Time (1994) (164)
- On the topology of algorithms, I (1987) (159)
- Online Learning Algorithms (2006) (148)
- The Prisoner's Dilemma and Dynamical Systems Associated to Non-Cooperative Games (1980) (147)
- A classification of immersions of the two-sphere (1959) (146)
- ONLINE LEARNING WITH MARKOV SAMPLING (2009) (142)
- A Vietoris mapping theorem for homotopy (1957) (138)
- On the mathematical foundations of electrical circuit theory (1972) (133)
- On Algorithms for Solving f(x)=0 (1979) (129)
- Global Analysis and Economics I: Pareto Optimum and a Generalization of Morse Theory† (1975) (125)
- Global analysis and economics III: Pareto Optima and price equilibria (1974) (121)
- Dynamics in General Equilibrium Theory (1976) (117)
- Morse theory and a non-linear generalization of the Dirichlet problem (1964) (108)
- A Mathematical Model of Two Cells Via Turing’s Equation (1976) (107)
- Regular curves on Riemannian manifolds (1958) (106)
- Global analysis and economics IIA: Extension of a theorem of Debreu (1974) (106)
- The Mathematics of Time: Essays on Dynamical Systems, Economic Processes, and Related Topics (1980) (102)
- Functional organization of the human 4D Nucleome (2015) (99)
- Stable Manifolds for Differential Equations and Diffeomorphisms (2011) (96)
- Global analysis and economics VI: Geometric analysis of Pareto Optima and price equilibria under classical hypotheses (1976) (95)
- Algorithms for Solving Equations (2010) (93)
- Computational Complexity: On the Geometry of Polynomials and a Theory of Cost: II (1986) (90)
- Finding a horseshoe on the beaches of Rio (1998) (89)
- Geometry on Probability Spaces (2009) (83)
- Computational complexity: on the geometry of polynomials and a theory of cost. I (1985) (83)
- Complexity estimates depending on condition and round-off error (1998) (82)
- Global analysis and economics V: Pareto theory with constraints (1974) (82)
- Modeling Language Evolution (2004) (81)
- Dynamics retrospective: great problems, attempts that failed (1991) (81)
- Lectures of Smale on differential topology (1963) (80)
- THE DYNAMICAL SYSTEMS APPROACH TO DIFFERENTIAL EQUATIONS (2007) (75)
- The Problem of the Average Speed of the Simplex Method (1982) (75)
- On a theory of computation over the real numbers; NP completeness, recursive functions and universal machines (1988) (73)
- Some Remarks on the Foundations of Numerical Analysis (1990) (68)
- Global analysis and economics (1975) (68)
- Mathematics of the Neural Response (2010) (65)
- The generalized Poincaré conjecture in higher dimensions (1960) (64)
- The qualitative analysis of a difference equation of population growth (1976) (63)
- Global analysis and economics IV: Finiteness and stability of equilibria with general consumption sets and production (1974) (59)
- COMPLEXITY AND REAL COMPUTATION: A MANIFESTO (1996) (56)
- A Polynomial Time Algorithm for Diophantine Equations in One Variable (1999) (54)
- On the Immersion of Manifolds in Euclidean Space (1958) (47)
- Problems on the nature of relative equilibria in celestial mechanics (1971) (42)
- Algebraic Settings for the Problem “P ≠ NP?” (1998) (42)
- Separation of Complexity Classes in Koiran's Weak Model (1994) (35)
- Dynamical systems and the topological conjugacy problem for diffeomorphisms (1963) (34)
- Discrete Dynamical Systems (2013) (34)
- The mathematics of numerical analysis (1996) (32)
- On the existence of generally convergent algorithms (1986) (31)
- A survey of some recent developments in differential topology (1963) (31)
- Stability and Isotopy in Discrete Dynamical Systems (1973) (29)
- Differentiable and Combinatorial Structures on Manifolds (1961) (27)
- A Structurally Stable Differentiable Homeomorphism with an Infinite Number of Periodic Points (2007) (27)
- Hodge Theory on Metric Spaces (2009) (26)
- The Gödel Incompleteness Theorem and Decidability over a Ring (1993) (25)
- Smooth solutions of the heat and wave equations (1980) (25)
- Self-Intersections of Immersed Manifolds (1959) (23)
- Review: E. C. Zeeman, Catastrophe theory: Selected papers, 1972–1977 (1978) (23)
- The mathematics of numerical analysis : 1995 AMS-SIAM Summer Seminar in Applied Mathematics, July 17-August 11, 1995, Park City, Utah (1996) (23)
- Emergence of function from coordinated cells in a tissue (2017) (22)
- Genome Architecture Mediates Transcriptional Control of Human Myogenic Reprogramming (2018) (22)
- Reproducing kernel hilbert spaces in learning theory (2006) (21)
- NP and mathematics – a computational complexity perspective (2006) (19)
- Sufficient conditions for an optimum (1975) (19)
- ON INVOLUTIONS OF THE 3-SPHERE.* (1959) (18)
- On the complexity of path-following newton algorithms for solving systems of polynomial equations with integer coefficients (1993) (17)
- ON THE PROBLEM OF REVIVING THE ERGODIC HYPOTHESIS OF BLOTZMANN AND BIRKHOFF (1980) (17)
- A NOTE ON OPEN MAPS (1957) (15)
- The Collected Papers of Stephen Smale (2000) (15)
- The story of the higher dimensional poincaré conjecture (what actually happened on the beaches of rio) (1990) (14)
- Mathematics of the Genome (2015) (13)
- Dynamics of the Genome (2015) (13)
- Continuation methods. (1979) (12)
- Generalization and Properties of the Neural Response (2010) (12)
- Derived Distance : towards a mathematical theory of visual cortex (2007) (12)
- The Lorenz System (2013) (12)
- First-Order Equations (2013) (11)
- Stability and genericity in dynamical systems (1971) (11)
- From Topology to Computation: Proceedings of the Smalefest (2011) (11)
- Lecture III Dynamical systems and turbulence (1977) (11)
- Genome Architecture Leads a Bifurcation in Cell Identity (2017) (10)
- On How I Got Started in Dynamical Systems (1980) (10)
- Chapter 8 Global analysis and economics (1981) (10)
- On the steps of Moscow University (1984) (9)
- The Pitchfork Bifurcation (2016) (9)
- ABSTRACT AND CLASSICAL HODGE–DE RHAM THEORY (2012) (8)
- A Dynamic Theory of Learning (2008) (8)
- Some Dynamical Questions in Mathematical Economics (1980) (6)
- CBCL Paper April 4 , 2008 On a model of visual cortex : learning invariance and selectivity from image sequences (2008) (6)
- What is Global Analysis (1969) (6)
- On the Morse index theorem (1967) (6)
- An Approach to the Analysis of Dynamic Processes in Economic Systems (1977) (6)
- Theory of computation (1992) (6)
- On a model of visual cortex: learning invariance and selectivity (2008) (5)
- The Mathematics of Learning: Dealing With Data (2005) (5)
- Some Lower Bounds for the Complexity of Continuation Methods (1998) (5)
- Global Analysis in Economic Theory (1989) (5)
- RKHS Representation of Measures Applied to Homogeneity, Independence, and Fourier Optics (2008) (5)
- Global stability questions in dynamical systems (1969) (5)
- On How I Got Started In Dynamical Systems, 1959–1962 (1993) (5)
- Risk Bounds for Random Regression Graphs (2007) (5)
- ON COMPARATIVE STATICS AND BIFURCATION IN ECONOMIC EQUILIBRIUM THEORY (1979) (5)
- Modelling Language Evolution ∗† (2003) (4)
- Smale horseshoe (2007) (3)
- Neurons That Confuse Mirror-Symmetric Object Views (2010) (3)
- Gerard Debreu wins the Nobel Prize (1984) (3)
- On the average cost of solving polynomial equations (1983) (3)
- Review: Marston Morse, Global variational analysis: Weierstrass integrals on a Riemannian manifold (1977) (3)
- On the Problem of Reviving the Ergodic Hypothesis of Boltzmann and Birkhoff: Dedicated to the memory of Rufus Bowen (1980) (2)
- The Condition Number in ℙ(H(d)) (1998) (2)
- Personal Perspectives on Mathematics and Mechanics (1980) (2)
- Beyond Hyperbolicity (2019) (2)
- Essays in mathematics and its applications : in honor of Stephen Smale's 80th birthday (2012) (2)
- The Condition Number for Nonlinear Problems (1998) (2)
- Existence and Uniqueness Revisited (2013) (2)
- Modelling Language Evolution ⁄y (2003) (2)
- The Mathematics of Learning: Dealing With Data (2005) (1)
- Learning Theory: An Approximation Theory Viewpoint: Foreword (2007) (1)
- Newton’s Method (1998) (1)
- Vladimir I. Arnold (2014) (1)
- Notes on the Shannon Entropy of the Neural Response (2009) (1)
- Phase Portraits for Planar Systems (2013) (1)
- Surveys in Modern Mathematics: On problems of computational complexity (2005) (1)
- Closed Orbits and Limit Sets (2013) (1)
- Data and learning (2006) (1)
- Remembering Raoul Bott (1923–2005) (2013) (0)
- Applications in Mechanics (2013) (0)
- Applications in Circuit Theory (2013) (0)
- Deterministic Lower Bounds (1998) (0)
- On Involutions of the 3-Sphere Author ( s ) : (2017) (0)
- Classification of Planar Systems (2013) (0)
- 2006 Fields Medals Awarded, Volume 53, Number 9 (2006) (0)
- The Smale Collection: Beauty in Natural Crystals (2006) (0)
- The mathematics of time : essays on dynamical systems, economics processes, and related topics / Steve Smale (1980) (0)
- Computation over a Ring (1998) (0)
- The mathematics of theories of learning and intellegence (2000) (0)
- Condition Numbers and the Loss of Precision of Linear Equations (1998) (0)
- Letter to the Editor: A planetary classification scheme (2006) (0)
- Catastrophe theory : Selected papers , 1972-1977 (2007) (0)
- Remembering Raoul Bott (2013) (0)
- Data-Guided Controllability : Learning from the Human Genome (2015) (0)
- Complexity aspects of numerical analysis (1986) (0)
- Definitions and First Properties of Computation (1998) (0)
- Comparative Statics and Algorithms for Finding Economic Equilibria (1995) (0)
- Global Nonlinear Techniques (2013) (0)
- On a Theory of Cost for Equation Solving (1983) (0)
- Otopy classes of equivariant maps (2010) (0)
- The Class NP and NP-Complete Problems (1998) (0)
- Finding a I lorseshoe on the Beaches of Rio What is Chaos ? (2009) (0)
- Hierarchical Kernel Machines: The Mathematics of Learning Inspired by Visual Cortex (2013) (0)
- Is the Mandelbrot Set Decidable (1995) (0)
- Professional Biography, Bibliography, and Graduate Students (1993) (0)
- Higher-Dimensional Linear Algebra (2013) (0)
- Real‐Valued Computation and Complexity (2006) (0)
- Planar Linear Systems (2013) (0)
- Emergence of Function (2016) (0)
- Basic elements of the qualitative theory of ordinary differential equations (2019) (0)
- Robert Edward Bowen (1980) (0)
- The impact of Thom's cobordism theory (2004) (0)
- Mathematics of the Genome (2016) (0)
- Complexity and RealComputation : A Manifesto Lenore (1995) (0)
- Applications in Biology (2013) (0)
- On the Evolution of Language (2002) (0)
- THE STRUCTURE OF COMPLETE MANIFOLDS OF NONNEGATIVE CURVATURE BY JEFF CHEEGER AND DETLEF GROMOLL (2007) (0)
- [After 100 years of stagnation, some progress]. (1983) (0)
- Equilibria in Nonlinear Systems (2013) (0)
- REGULAR CURVES ON RIEMANNIAN MANIFOLDS ( J ) (2010) (0)
- Complexity and the Condition Number (1998) (0)
- Foundations of computational mathematics : proceedings of the Smalefest 2000, Hon Kong, 13-17 2000 (2002) (0)
- Some Separations of Complexity Classes (1998) (0)
- 1 Differential equations and solutions 2 . 1 . 1 Existence and uniqueness of solutions (2019) (0)
- Some Autobiographical Notes (1993) (0)
- ALGEBRAIC TOPOLOGY AT THE STEKLOV MATHEMATICAL INSTITUTE OF THE ACADEMY OF SCIENCES OF THE USSR (2008) (0)
- Higher-Dimensional Linear Systems (2013) (0)
- Algebraic Settings for the Problem P 6 = Np?" 1. Statement of Main Theorems (1996) (0)
- Hodge Theory on Metric Spaces (2011) (0)
- PSEUDO-DIVISION MACHINE (I): A MODEL OF SYMBOLIC COMPUTATION (2007) (0)
- THE CLASSIFICATION OF IMMERSIONS OF SPHERES (2016) (0)
- COMMENTARY ON “DIFFERENTIABLE DYNAMICAL SYSTEMS” BY STEPHEN SMALE (2018) (0)
- AFRL-OSR-VA-TR-2013-0116 Hierarchical Kernel Machines : The Mathematics of Learning Inspired by Visual CortexTitle , i . e , Synthetic Aperture Ladar for Tactical Imaging (2013) (0)
- Nonuniform Complexity Classes (1998) (0)
- Decision Problems and Complexity over a Ring (1998) (0)
- S . S MALE Stable manifolds for differential equations and diffeomorphisms (2018) (0)
- A Dynamic Formulation of Ring Oscillator as Solitary-Wave Propagator (1998) (0)
- Mathematical and Computational Foundations of Learning Theory (Dagstuhl Seminar 11291) (2011) (0)
- Continuous Direct Adaptive Control with Saturation Input Constraint (1994) (0)
- DIFFERENTIABLE AND COMBINATORIAL STRUCTURES (2016) (0)
- Fundamental Theorem of Algebra: Complexity Aspects (1998) (0)

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