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.^[1]

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 ${\displaystyle C^{\infty }}$ mappings in 1987.^[2]

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.^[3][4][5] 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.^[6]

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.^[7]

2004–2010: Post-IBM

From 2004 to 2010, he worked at the University of Toronto.^[7] After 2010, he became a researcher at the University of Buenos Aires and at the Graduate Center of the City University of New York.^[7] Since 2016, he has been Martin and Michele Cohen Professor and Chair of the Mathematics Department at City College of New York.^[7]