This volume is dedicated to Stephen Smale on the occasion of his 80th birthday. Besides his startling 1960 result of the proof of the Poincaré conjecture for all dimensions greater than or equal to five, Smale’s ground breaking contributions in various fields in Mathematics have marked the second part of the 20th century. Stephen Smale has done pioneering work in differential topology, global analysis, dynamical systems, nonlinear functional analysis, numerical analysis, theory of computation and machine learning as well as applications in the physical and biological sciences and economics.

The volume consists of 18 chapters written by 27 outstanding mathematicians, among which we can find names as S.K. Donalson, M. Freedman, M. Gromov, Y. Eliashberg, P. Mihailescu, Y. Sinai, J. Mather, A. Iserles, R. Pérez-Marco or Th.M. Rassias, just to name a few. The editors of the volume, Panos M. Pardalos (University of Florida, USA) and Themistocles M. Rassias (National Technical University of Athens, Greece), have made a great effort to collect articles from such a number of high level mathematicians, as the honored person certainly deserves.

The articles contained in the book are research articles of high level in all areas related to the work of Stephen Smale. The list of articles is the following:

- Transitivity and Topological Mixing for C^1 Diffeomorphisms. F. Abdenur, S. Crovisier

- Recent Results on the Size of Critical Sets. D. Andrica, C. Pintea

- The Fox-Li operator as a Test and a Spur for WienerHopf Theory. A. Böttcher, S. Grudsky, A. Iserles

- Kahler Metrics with Cone Singularities along a Divisor. S. Donaldson

- The Space of Framed functions is Contractible. Y. Eliashberg, N. Mishachev

- Quantum Gravity via Manifold Positivity. M. Freedman

- Parabolic Explosions in Families of Complex Polynomials. E. Gavosto, M. Staviska

- Super Stable Kahlerian Horseshoe?. M. Gromov

- A Smooth Multivariate Interpolation Algorithm. J. Guckenheimer

- Bifurcations of Solutions of the 2-Dimensional Navier-Stokes System. D. Li, Y. Sinai

- Arnold Diffusion by Variational Methods. J. Mather

- Turning Washington's Heuristics in Favor of Vandiver's Conjecture. P. Mihailescu

- Schwartzman Cycles and Ergodic Solenoids. V. Muñoz, R. Pérez-Marco

- An Additive Functional equation in Orthogonality Spaces. C. Park, Th. Rassias

- Exotic Heat PDEs.II. A. Prástaro

- Topology at a Scale in Metric Spaces. N. Smale

- Riemann, Hurwitz and Hurwitz-Lerch Zeta Functions and Associated Series and Integrals. H. Srivastava

- Gyrations: The Missing Link Between Classical Mechanics with its Underlying Euclidean Geometry and Relativistic Mechanics with its Underlying Hyperbolic Geometry. A. Ungar

This book is a fantastic recopilation of nicely written papers by outstanding mathematicians, and deserves to be in every mathematical library.