As the 20th century begins to draw to a close, mathematical research is in an extremely healthy state, with many long-standing problems being solved and many new areas of research emerging. Perhaps the most encouraging aspect of all is that the relation between mathematics and the applied sciences is becoming markedly stronger, with ideas being traded in both directions. The importance of mathematics lies not in its applications to any particular area of science, but in its ability to 'transfer technology' between different areas of application. The technology being transferred is mathematical concepts and techniques, not hardware; but the effect is just as strong. Because mathematical concepts are general, ideas that first become apparent in one area can readily be transferred to another, apparently quite different, area. Examples discussed later will include new developments in knot theory, arising from mathematical physics and applied to molecular biology; a musical problem whose solution has illuminated the theory of waves; an optimization problem that has led to fundamental questions about computability; and a new kind of geometry that originated in classical mechanics and is now of central importance in quantum physics.

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