It is really justified to say that this book fills a gap between the universe and geometry. Written by an outstanding (celebrated) researcher in geometry and his junior colleague ‒ an astrophysicist and astronomer ‒ it gives a carefully written description of the mainstream theories. Moreover, it includes a good account of historical and classical concepts derived from dependable sources of information. The effort that lies at the heart of this book is the proof of the Calabi conjecture by the first author. It is a great pleasure to review a work of such excellence, which does so much to promote the formation of a new geometric analysis branch of mathematics. The book is divided into fourteen chapters and includes also Preface, Prelude, Epilogue, Postlude and two poems. Each of the parts can be read independently but their full beauty unfolds only when reading the whole book. The chapters are excellent and give a thoughtful view of why certain notions and methods are useful for commonly encountered as well as various particular questions and problems. Completely self-contained, the book starts with a thorough discussion of geometry with respect to physics and the universe. In the Preface and the Prelude, the authors formulate the aims of the book, establish connections among mathematics, physics and their paths to the truth, express their thanks to their near relations and colleagues, and show the coming of the shapes of things. In chapter one, ‘A Universe in the margins’, the authors discuss Hubble, Planck and other possible volumes of our universe and give a historical sketch of its dimensional theories. Chapter two, ‘Geometry in the Natural order’, deals with main concepts of geometry and with applications of these concepts for probing the universe. Chapter three, ‘A new kind of hammer’, discusses geometric analysis tools used in mathematical investigations by C. Morrey, S. Donaldson, R. Hamilton, P. Li, G. Perelman and by the first author. These include topics on partial differential equations in geometry and curvature, minimal surfaces, advances in four-dimensional topology, geometric flows, Ricci flows, and the Poincaré conjecture. Chapter four, ‘Too Good to be True’, discusses the Calabi conjecture and demonstrates the third major success of geometric analysis related to the conjecture. Chapter five, ‘Proving Calabi’, contains the ideas and the description of the proof of the Calabi conjecture by the first author. Topics on geometric analysis with applications and with more general results in the field were presented by the first author [in: Essays in geometry in memory of S. S. Chern. Somerville, MA: International Press. Surveys in Differential Geometry 10, 275‒379 (2006; Zbl 1138.53004)]. Chapter six, ‘The DNA of String Theory’, applies the results of chapters 3‒5 to physics and string theory. The cases of zero, positive Ricci and negative curvature as well as Kähler-Einstein manifolds are discussed. Five separate string theories (Type I, Type IIA, Type IIB, Heterotic $SO(32)$, Heterotic $E8 \times E8$) and $M-$theory are briefly presented. The chapter contains an exceptionally interesting historical outline of the development of string theory up to our days. Chapter seven, ‘Through the Looking Glass’, deals with conformal invariance in a quantum setting, beta function and mirror symmetry. The SYZ (Strominger-Yau-Zaslow) conjecture and homological mirror symmetry are discussed. Chapter eighth, ‘Kinks in space time’, discusses singularities in our universe. These include black holes and their entropy, supersymmetry and Calabi-Yau manifolds, stable subsurfaces within the Calabi-Yau manifolds, developments of $AdS/CFT$ correspondence by {\it J. Maldasena} [Adv. Theor. Math. Physics 2,No. 2, 231‒252 (1998; Zbl 0914.53047)], and the black hole information paradox. Aspects of $AdS/CFT$ correspondence were considered by {\it D. Serban} [J. Phys. A: Math. Theor. 44, 83 p. (2011; Zbl 1228.81242)] and by {\it B. Vicedo} [J. Phys. A, Math. Theor. 44, No. 12, Article ID 124002, 183 p. (2011; Zbl 1270.81191)]. Chapter nine, ‘Back to the Real World’, takes up three topics. First, the Standard Model and its lacks are discussed. Then, relations between the Yang-Mills theory and Calabi-Yau manifolds are considered. These involve possibilities of getting the right particles and trying to compute their masses. The third topic concerns how to produce metrics for Calabi-Yau manifolds. Chapter ten, ‘Beyond of Calabi-Yau’, treats approaches to ‘a theory that works on all scales ‒ a theory that gives us both particle physics and cosmology’. In the framework, the shape moduli of a manifold with fluxes is considered. Further the authors present the conifold transition, non-Kähler manifolds, their properties and applications. Strominger equations which apply to non-Kähler manifolds are discussed. Chapter eleven, ‘The Universe Unravels’, discusses quantum tunneling, vacuum decays, thermal fluctuations, bubbles, compactification and decompactification of the extra dimensions. The de Sitter space and its entropy is considered. “With ten spacetime dimensions to play with, and six new directions in which to roam, life would have possibilities we can’t even fathom”. Chapter twelve, ‘The Search for Extra Dimensions’, is devoted to the discussion of questions: Do string theory ideas actually describe our universe? ‘Do we have a player if verifying any of this ‒ of gleaning any hints of extra dimensions, strings, branes and the like?’. In chapter thirteen, ‘Truth, Beauty, and Mathematics’, the authors ‘do believe the best chance for arriving at a successful theory lies in pooling the resources of mathematicians and physicists, combining the strengths of the two disciplines and their different ways of approaching the world. We can work on complementary tracks, sometimes crossing over to the other side for the benefit of both.’ The last chapter ‘The End of Geometry?’ is concerned with the comparison of classical and quantum geometry. Throughout the book there are many examples and figures which illustrate the considered concepts and theories. The reviewer believes that the book is an excellent, clear, well-written presentation of the key ideas of geometry, string theory, and the geometry of the universes hidden dimensions.

Reviewer:

Nikolaj M. Glazunov (Kyïv)