# 2008 President's medal

Professor Sir Michael Atiyah

*University of Edinburgh*

*In recognition of his outstanding contributions to a broad range of topics in mathematics, many of which have provided highly significant foundations to the development of theoretical physics; and of his eminent leadership within the scientific community.*

This year’s **President’s medal** is awarded to **Professor Sir Michael Atiyah** for his outstanding contributions to mathematics and theoretical physics.

Atiyah's ideas on the role of topology in quantum field theory have been influential for a generation of theoretical physicists and mathematicians, and he was the among the first to realise the mathematical importance of the problems that arise in quantum field theory.

Atiyah, in collaboration with I M Singer, established the ‘index theorem’ which calculates the ‘index’ - the number of solutions of a differential equation minus the number of restrictions which it imposes on the values of the quantities being computed - in terms of the geometry of the surrounding space.

The index theorem can be interpreted in terms of quantum theory and has proved an invaluable tool for theoretical physicists. It was the beginning of the understanding that global properties have important physical consequences, and topology is now seen as playing an important role in quantum field theory.

Gauge theories involve deep and interesting nonlinear differential equations and in particular, the Yang-Mills equations have turned out to be particularly fruitful for mathematicians. Atiyah played a leading role in the study of the non-linear solutions that govern non-perturbative physics, finding general Yang-Mills instanton solutions and studying the dynamics of magnetic monopoles. His ideas have played an important role in the work of others using ideas from physics in mathematics, most notably the work of Witten on knot theory and the work of Donaldson, who was his graduate student at the time, on the structure of four-dimensional spaces. His ideas and influence continue in areas of current research such as topological field theory and string theory.