Speech by Isadore Singer

Your majesties, members of the Norwegian Academy and The Abel Board, colleagues, friends and family:

I am honored to receive the Abel Prize and to share it with my friend Sir Michael Atiyah. I want to thank the Academy and The Abel Prize Committee for finding our work so worthy.

I must admit that outside the university, it is difficult to be a pure mathematician. No one in my family understands what I do. My neighbors wonder why I spend all my time in my study scribbling with pencil on a yellow pad of paper instead of going outside to mow the lawn. At parties, when someone learns I am a mathematician, they frown and say, "Ok, I could never understand calculus!" and they turn away. And so often I stumble finding words to answer the common question, "but what is all this abstract math good for?"

It's true we pure mathematicians are connected to a different world. But it is a very real world nevertheless. Our roots go back to antiquity. The ancient Greeks understood that much around them could be expressed in mathematical terms. Euclid used logic to codify geometry. Gallileo recognized that mathematics was THE basic language of science. Einstein needed geometry to describe his theory of relativity.

I sometimes think the public is in awe of mathematics because it cannot comprehend how mathematicians can explore phenomena that are not detectable by our senses or by high-tech equipment we invent. And the public does not understand why it can take so long before mathematical developments result in practical applications.

What I am trying to say is that there is a unity in mathematics that we are connected to. We are entranced by its beauty and its power. It is frustrating not to be able to communicate that to the public.

Forty-two years ago, Sir Michael and I transformed the mathematical landscape by discovering the Index Formula and proving the Index Theorem. As Professor Størmer said, solutions to differential equations determine how things move. The Atiyah-Singer Index Theorem broke new ground by calculating the number of solutions to such equations in geometric terms. To do so, we had to employ techniques from different fields such as analysis, geometry, and topology; and we could get new results not possible by older methods. Many earlier theorems in geometry and topology hinted at and proved to be illustrations of our general theorem, which thus unified different areas of mathematics. This was indeed a revolution.

But at the time, we had no idea of the Index Theorem's coming applications to physics. It turned out later that the theorem gave new insights into gauge theory and string theory. The breakthroughs in physics needed new mathematics; index theory frequently supplied what was needed. Mathematicians and physicists began talking to each other once again. Now we take for granted this new vigorous discipline of mathematical physics.

Niels Henrik Abel was the first modern mathematician, as Sir Michael said. His views were strikingly different from those of his peers in that he -??- computation in favor of an abstract approach to what we now call Abelian integrals. Abel was 26 years old when he died, the same age as Sir Michael and I were when we received our Ph.D.'s. Had Abel lived a few more years, we can only imagine what major discoveries he would have made. We are particularly honored to receive this prize named after Abel because he was a remarkable mathematician so far ahead of his time.

In choosing our Index Theorem, the Norwegian Academy lends its support to the idea of mathematical research as a deep intellectual achievement. We are grateful to the Norwegian government and to the Academy for its inspired way of honoring one of its great sons. By establishing the Abel Prize, you are calling the attention of the whole world to the fundamental importance of mathematics.

Thank you,
I. M. Singer

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Your majesties, members of the Norwegian Academy and The Abel Board, colleagues, friends and family: