## Professor Israel M. Sigal

### Department of Mathematics

### Appointed a University Professor in 1997

Professor Sigal brings the problems of physics and chemistry to mathematics, especially the deep problems of the nature of matter. Professor Sigal is a mathematical physicist, and as such works in the very important part of mathematics concerned with rigorous description of basic physical phenomena.

While the problems themselves can be formulated in deceptively simple ways, it turns out to be extraordinarily difficult to establish that they do in fact give a faithful description of experimentally known phenomena, an effort which has driven the development of large parts of mathematics. Mathematical physicists are mathematicians, but their intuition and ideas come from an understanding of the physics. Within mathematical physics, Professor Sigal is the outstanding figure in Canada and, on an international scale, one of the leaders of the field. His work goes to the very heart of quantum theory, studying stability of atoms and molecules, scattering in systems of many quantum particles and their interaction with quantized radiation.

Professor Sigal's work has primarily centered on the Schroedinger equation, which in turn is at the heart of mathematical models of atoms, molecules and solids and, to some extent, nuclei and stars. This equation, discovered by the Austrian physicist Erwin Schroedinger in 1926, is one of the four known fundamental equations of physics (the other three are Newton's, Maxwell's and Einstein's equations). It has led to a whole new field of mathematics.

For fifty years one major unsolved problem remained - to describe completely the long time behaviour of an arbitrary quantum system of many particles. Experimental evidence suggested an obvious conjecture which was formulated precisely in the 1940s. A theorem, established through a series of papers by Sigal and his former postdoctoral student Soffer, provided the first rigorous proof of this conjecture. It is one of the most celebrated results in the field and is critical to establishing firmly the mathematical foundations of Quantum Mechanics.

In recent years, Professor Sigal has made groundbreaking contributions to the mathematical theory of the interaction between light and matter, known as Quantum Electrodynamics. An example of a question this theory addresses is why and how an atom (or a molecule) in a vacuum emits and absorbs radiation. An attempt to answer such a question led to the birth of quantum theory at the beginning of the twentieth century.

Einstein was the first to propose that light consists of quanta, called photons (he was awarded the Nobel prize for this and related work). Quantum equations for light were derived by Physics Nobel laureate Dirac in the late 1920's (and further developed by Physics Nobel laureates Feynmann, Schwinger and Tomonaga around 1950). These equations represent one of the great scientific achievements of the 20th century. This work created a need for a precise, consistent mathematical theory of matter interacting with radiation. For over sixty years this task seemed to be beyond reach. In 1998-2000 Professor Sigal, jointly with Professors Volker Bach and Jürg Fröhlich, concluded the first convincing attempt to provide such a theory. Their theory gave a mathematical description of the processes of emission and absorption of radiation by systems of non-relativistic matter such as atoms and molecules.

Professor Sigal's work has been rewarded by many honours including several invited lectures to the International Congress on Mathematical Physics and the International Congress of Mathematics and I.W. Killam fellowship. He is also a Fellow of the Royal Society of Canada and received the John L. Synge Award as outstanding Canadian mathematician in 1993 and the CRM-Fields prize in 2000. He was named the Norman Stuart Robertson Chair in Applied Mathematics in 1997. Professor Sigal came to the University of Toronto in 1985 after holding positions as a Senior Scientist at the Weizmann Institute of Science, an assistant professor at Princeton University and a postdoctoral fellow at the Swiss Institute of Technology in Zurich. In his work here in the Department of Mathematics, Professor Sigal has assembled an active group of talented junior faculty and post-doctoral fellows who have attracted many graduate students into their field of research. Their work is bringing renown to the University of Toronto.