Preface: What is Mathematics? -- Welcome -- Structure and Randomness in the Prime Numbers -- How to Solve a Diophantine Equation -- From Sex to Quadratic Forms -- Small Divisors: Number Theory in Dynamical Systems -- How do IMO Problems Compare with Research Problems? -- How do Research Problems Compare with IMO Problems? -- Graph Theory Over 45 Years -- Communication Complexity -- Ten Digit Problems -- The Ever-Elusive Blowup in the Mathematical Description of Fluids -- About the Hardy Inequality -- The Lion and the Christian, and Other Pursuit and Evasion Games -- Three Mathematics Competitions -- Complex Dynamics, the Mandelbrot Set, and Newton´s Method.

This Invitation to Mathematics consists of 14 contributions, many from the world's leading mathematicians, that introduce the readers to exciting aspects of current mathematical research. The contributions are as varied as the personalities of active mathematicians, but together they show mathematics as a rich and lively field of research.   The contributions are written for interested students at the age of transition between high school and university who know high school mathematics and perhaps competition mathematics and who want to find out what current research mathematics is about. We hope that it will also be of interest to teachers or more advanced mathematicians who would like to learn about exciting aspects of mathematics outside of their own work or specialization.   Together with a team of young ``test readers'', editors and authors have taken great care, through a substantial ``active editing'' process, to make the contributions understandable by the intended readership.