Henryk Iwaniec

for their introduction and development of fundamental tools in number theory, allowing them as well as others to resolve some longstanding classical problems.

The Shaw Prize in Mathematical Sciences 2015 is awarded to **Gerd Faltings**, Managing Director at Max Planck Institute for Mathematics in Bonn, Germany, and **Henryk Iwaniec**, New Jersey Professor of Mathematics at Rutgers University, USA, for their introduction and development of fundamental tools in number theory, allowing them as well as others to resolve some longstanding classical problems.

Number theory concerns whole numbers, prime numbers, and polynomial equations involving them. The central problems are often easy to state but extraordinarily difficult to resolve. Success, when it is achieved, relies on tools from many fields of mathematics. This is no coincidence since some of these fields were introduced in attempts to resolve classical problems in number theory. **Faltings** and **Iwaniec** have developed many of the most powerful modern tools in algebra, analysis, algebraic and arithmetic geometry, automorphic forms, and the theory of zeta functions. They and others have used these tools to resolve longstanding problems in number theory.

The theory of numbers is one of the oldest branches of mathematics, going back more than two thousand years in China, Greece, and India. It is concerned with the study of whole numbers, prime numbers, and polynomial equations involving them. The third of these goes by the name Diophantine equations after the Alexandrian/Greek mathematician Diophantus. Gauss, who laid many of the foundations of modern number theory, called it the “Queen of Mathematics”. At the time, and for many years after, it was considered as very much on the theoretical and pure side of mathematics. However, in our modern digital/discrete world the deeper truths and techniques that have been developed to study whole numbers play an increasingly significant role in applications.

Many of the central problems in the theory of numbers are elementary and easy to state; but the experience of generations of mathematicians shows that they can be extraordinarily difficult to resolve. Success, when it is achieved, often relies on sophisticated tools from many fields of mathematics. This is no coincidence, since aspects of these fields were introduced and developed in efforts to resolve classical problems in number theory.

**Gerd Faltings **was born in 1954 in Germany and is currently the Managing Director at the Max Planck Institute for Mathematics in Bonn, Germany. He obtained his PhD in Mathematics from the University of Münster in 1978. He then spent a year doing postdoctoral work as a Research Fellow at Harvard University from 1978 to 1979. He was an Assistant Professor at the University of Münster from 1979 to 1982. From 1982 to 1984, he was Professor at the University of Wuppertal. After that he was appointed Professor at Princeton University from 1985 to 1994. Since 1995 he has been a Director of the Max Planck Institute for Mathematics.

**Henryk Iwaniec **was born in 1947 in Poland and is currently New Jersey Professor of Mathematics at Rutgers University, USA. He graduated from the University of Warsaw in Mathematics in 1971, and obtained his PhD there in 1972. He then held positions at the Institute of Mathematics of the Polish Academy of Sciences until 1983 when he left Poland. Before being appointed Professor of Mathematics at Rutgers University in 1989, he held visiting positions at the Institute for Advanced Study, Princeton, the University of Michigan, and the University of Colorado at Boulder.