Japan is a tiny country that occupies only 0.25% of the world's total land area. However, this small country is the world's third largest in economy: the Japanese GDP is roughly equivalent to the sum of any two major countries in Europe as of 2012.

This book is a first attempt to ask leaders of top Japanese companies, such as Toyota, about their thoughts on mathematics. The topics range from mathematical problems in specific areas (e.g., exploration of natural resources, communication networks, finance) to mathematical strategy that helps a leader who has to weigh many different issues and make decisions in a timely manner, and even to mathematical literacy that ensures quality control. The reader may notice that every article reflects the authors' way of life and thinking, which can be evident in even one sentence.

This book is an enlarged English edition of the Japanese book What Mathematics Can Do for You:  Essays and Tips from Japanese Industry Leaders. In this edition we have invited the contributions of three mathematicians who have been working to expand and strengthen the interaction between mathematics and industry.

The role of mathematics is usually invisible when it is applied effectively and smoothly in science and technology, and mathematical strategy is often hidden when it is used properly and successfully. The business leaders in successful top Japanese companies are well aware of this invisible feature of mathematics in applications aside from the intrinsic depth of mathematics. What Mathematics Can Do forYou ultimately provides the reader an opportunity to notice what is hidden but key to business strategy.

Yoshikazu Giga is a Professor of Mathematics at the University of Tokyo and one of the leading experts in mathematical analysis for nonlinear parabolic partial differential equations. His research interests include analysis on the Navier-Stokes equations, blow-up phenomena for semilinear heat equations, surface evolution equations, and viscosity solutions. His previous books include "Surface Evolution Equations" (Birkhaeuser 2006) and "Nonlinear Partial Differential Equations"(with M. Giga and J. Saal, Birkhaeuser 2010). He has been awarded the MSJ Autumn Prize and the Medal with Purple Ribbon.Toshiyuki Kobayashi, a mathematician, was born in 1962. When he was 25, he published the article on the necessary and sufficient condition for Calabi-Markus Phenomenon, which led to a new theory of discontinuous groups beyond the framework of Riemannian geometry. He was invited by Institute for Advanced Study, Princeton, for his world first accomplishment. His achievements in later career consists of many new theories of mathematics, such as the theory of discrete decomposable branching laws on infinite-dimensional spaces and visible actions on the unified theory of multiplicity-free representation, to name a few. His academic honors and awards include Spring Prize (the Mathematical Society of Japan), Osaka Science Prize, JSPS Prize (the Japan Society for the Promotion of Science), Sackler Distinguished Lecturer (Israel) and Humboldt Research Award (Germany). After working as a professor of Research Institute for Mathematical Sciences (RIMS) in Kyoto University and as a visiting professor of Harvard University, he was appointed to the University of Tokyo's Graduate School of Mathematical Sciences. He is also working as Principal Investigator of Kavli Institute for the Physics and Mathematics of the Universe (Kavli IPMU), the University of Tokyo.