features ries allowed for manifolds with corners or very ragged surfaces. There was a general belief that one definition was convenient in certain cases, and another definition was convenient in others. Sometime in the 1950’s when I was trying to study examples of manifolds, I suddenly hit a snag: In one way, I proved that a certain manifold could exist, and in another, I proved that it could not. This was very baffling. Then I realized after a lot of worry that I had been passively assuming that it did not matter which definition one used. The manifold in question could exist if you allowed angles and strange shapes; it could not exist if everything was smooth and differentiable. This was a shock to me and many other people. They somehow never asked a question or imagined that such a thing was possible. In a way it started a new industry with the goal of trying to understand these things.”

Influences and Great Beginnings There were several important influences on Dr. Milnor’s important accomplishments. At the roots of his beginnings in topology were his teachers at Princeton, Ralph Fox and Norman Steenrod. René Thom, a French mathematician, also had a large influence. However, the area of topology had many different people working in it and many were influential. Therefore, it is hard to list only a few. Jean-Pierre Serre, another French mathematician, made very important contributions, and Friedrich Hirzerbruch from Germany did as well.” Dr. Milnor described his time at Princeton at the start of a new area of mathematics as a very exciting time. Certain questions were being asked by mathematicians that no one thought to ask before and new ideas developed on how to answer these questions. “Suddenly,” Dr. Milnor commented, “great human efforts were being made to attain a goal; it was certainly a very wonderful time.”

Applications of Topological Research A concern that troubles those studying the applied sciences is the possible applications of mathematical theories. A popular,

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primary understanding of the sequence of studies is that mathematics underlies physics, while physics underlies chemistry and chemistry underlies the biological sciences. Mathematics is the strong fundament, at the bottom of the pyramid. Dr. Michael Freedman of the University of California, Santa Barbara, a Fields Medalist famous for his proof of the Poincare Conjecture in dimension four, explains the general work of mathematicians: “Theirs is a way of thinking that thrives by disdaining the need for practical applications. Let the applications come later by accident - they always do” [4]. The main applications are rooted in physics. For instance, linear algebra, operator theory, and group theory contribute to the theory of special relativity; differential geometry to general relativity; algebraic topology and algebraic geometry to quantum fields and the physics of elementary particles, among many others [5].

Mathematical Biology Dr. Milnor’s recent lecture at the Institute for Advanced Study at Princeton entitled Geometry of Growth and Form, based on a book of the same title by D’Arcy Thompson, explored the nature of organisms through dimensional analysis [6]. The discussion demonstrated how organisms can be compared based on their physical characteristics. An organism can grow and retain its form while varying in size, a phenomenon that can be understood from a topological standpoint. One of the central questions posed was the possibility of a conformal transformation (a map that preserves angles) of one closely related species to another without losing any of the species’ considerable corresponding features [7]. When do such transformations exist? Within the lecture, Dr. Milnor discussed a paper by Xianfeng Gu of the Computer Science Department at Stony Brook and Shing-Tung Yau of Harvard University, on conformal mapping of the brain surface to a sphere [8]. This application can be considered an extremely useful tool in medical imaging. Mathematical methods can be used to more efficiently analyze and

The Stony Brook Young Investigators Review, Fall 2011

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