Michael Shub

Michael Ira Shub (born August 17, 1943) is an American mathematician who has done research into dynamical systems and the complexity of real number algorithms.

Career 1967: Ph.D. and early career In 1967, Shub obtained his Ph.D. degree at the University of California, Berkeley with a thesis entitled Endomorphisms of Compact Differentiable Manifolds. In his Ph.D. thesis, he introduced the notion of expanding maps, which gave the first examples of structurally stable strange attractors. His advisor was Stephen Smale.

From 1967 to 1985, he worked at Brandeis University, the University of California, Santa Cruz and the Queens College at the City University of New York. In 1974, he proposed the Entropy Conjecture, an open problem in dynamical systems, which was proved by Yosef Yomdin for mappings in 1987.

1985–2004: IBM research From 1985 to 2004, he joined IBM's Thomas J. Watson Research Center. In 1987, Shub published his book Global Stability of Dynamical Systems, which is often used as a reference in introductory and advanced books on the subject of dynamical systems. In 1993, Shub and Stephen Smale initiated a rigorous analysis of homotopy-based algorithms for solving systems of nonlinear algebraic equations, which has inspired much of the work in that area during the last two decades.

From 1995 to 1997, Shub was the founding chair of the Society for the Foundations of Computational Mathematics. In 2001, Shub became a founding editor of their journal, Foundations of Computational Mathematics.

1986: Blum Blum Shub

Shub, along with coauthors Lenore and Manuel Blum, described a simple, unpredictable, secure random number generator (see Blum Blum Shub). This random generator is useful from theoretical and practical perspectives.

1989: Blum–Shub–Smale machine

In 1989, he proposed with Lenore Blum and Stephen Smale the notion of Blum–Shub–Smale machine, an alternative to the classical Turing model of computation. Their model is used to analyse the computability of functions.

2004–2010: Post-IBM From 2004 to 2010, he worked at the University of Toronto. 2000: Fellow of the American Association for the Advancement of Science. 2015: Fellow of the American Mathematical Society "for contributions to smooth dynamics and to complexity theory." 2016: Fulbright Specialist.

Selected publications