Speech by Erling Størmer

Your Majesties

The Abel Prize for 2004 is being shared by Sir Michael Atiyah from the University of Edinburgh and Isadore Singer from the Massachusetts Institute of Technology. They are receiving the prize for having discovered and proved the index theorem, which links together topology, geometry and analysis, and for playing an extraordinary role in building new bridges between mathematics and theoretical physics.

Allow me to give you a simplified picture of the basic ideas of the index theorem, to give you feeling for what it states.

In topology, we translate the essential properties of geometric objects into algebraic formulas. To a topologist, a circle is the same as a triangle, which in turn is the same as a rectangle. All three can be drawn by moving the pencil from one point and around the figure without crossing the figure anywhere. In differential geometry we avoid figures with corners, like triangles and rectangles. Instead we concentrate on geometric objects that are nice and smooth, like circles and ellipses. For these we study finer properties, such as curvature, which we can determine through the derivation of functions.

In the part of mathematics called analysis, we look at quantities that change. If we take the example of a car driving on a road, the speed is a measurement of the change in the length of road the car has driven, whereas acceleration measures the change in the speed. In mathematical language, this sort of thing is formulated using differential equations. The theory of such equations is an important part of mathematical analysis.

These three disciplines, topology, differential geometry and analysis, come together in the Atiyah-Singer index theorem. This theorem states that on the basis of the shape of the geometric area where the equations are defined, we can determine a numerical quantity - the index - that provides information about the solutions. The index can tell us whether any solutions exist, and if so, how many. This information is crucial, since it is often extremely difficult to find explicit solutions to differential equations, and so it is good to know whether there are any and how many.

The index theorem was proved in the early 1960s and is one of the most important mathematical results of the twentieth century. It has had an enormous impact on the further development of topology, differential geometry and theoretical physics. The theorem also provides us with a glimpse of the beauty of mathematical theory in that it explicitly demonstrates a deep connection between mathematical disciplines that appear to be completely separate.

After it was found in the 1970s that the index theorem had applications in theoretical physics, Atiyah and Singer were tireless in their attempts to build bridges between mathematics and physics. They showed physicists how they could apply differential geometry and analysis, especially in the areas of quantum field theory and string theory, and they taught mathematicians how the insight of physicists can be utilised in the study of mathematical problems.

I would like to conclude by saying that Sir Michael Atiyah and Isadore Singer have demonstrated mathematics at its very best and are worthy winners of the Abel Prize.

Thank you.

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Your Majesties