**Vladimir Arnold**, together with Andrei Kolmogorov and Jurgen Möser, made fundamental contributions to the study of stability in dynamical systems, exemplified by the motion of the planets round the sun. This work laid the foundation for all subsequent developments right up to the present time.

Arnold also produced extremely fruitful ideas, relating classical mechanics to questions of topology. This includes the famous "Arnold Conjecture" which has only recently seen important progress.

In classical hydrodynamics the basic equations of an ideal fluid were derived by Euler in 1757 and major steps towards understanding them were taken by Helmholtz in 1858, and Kelvin in 1869. The next significant breakthrough was made by Arnold a century later and this has provided the basis for more recent work.

**Ludwig Faddeev** has made many important contributions to quantum physics. Together with Victor Popov he showed the right way to quantize the famous Yang-Mills equations which underlie all contemporary work on sub-atomic physics. This led in particular to the work of 't Hooft and Veltman which was recognized by the Nobel Prize for Physics of 1999.

Faddeev also developed the quantum version of the beautiful theory of integrable systems in two dimensions which has important applications in solid state physics as well as in recent models of string theory.

In another application of the scattering theory of differential operators, Faddeev (jointly with Boris Pavlov) discovered a surprising link with number theory and the famous Riemann Hypothesis.

Mathematical Sciences Selection Committee

The Shaw Prize

10 June 2008, Hong Kong