Mathematical structures in economics and finance

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Applications of mathematics have traditionally been sought in the natural sciences, and much of the historical progress of our discipline comes from answering questions raised by physics, engineering and, more recently, biology. I have always been convinced that the social sciences have the same potential as the natural ones, and that a true understanding of the problems raised by economics, for instance, will raise new mathematical problems, both at the theoretical and numerical level, and open new directions for mathematics (and new positions for mathematicians). On the other hand, old problems should not be neglected, because (a) they serve as a benchmark against which to test new mathematical methods which (witness, for instance, the tremendous advances of symplectic geometry in recent years, which is largely due to the progress of nonlinear analysis), and (b) there is a deep unity in mathematics, and results obtained in one field can turn out to be useful in another (stochastic analysis, for instance, including its most recent development, such as Malliavin calculus, has deeply influenced modern finance).