features compare brain images. With respect to the field, Dr. Milnor commented: “Mathematical biology is a relatively new subject, though statistics was used to study genetics for many years. Certainly it is a very exciting field, especially data-mining and understanding how the many facts of the human genome give clues to many diseases.”

Mathematical Creativity When David Hilbert, a renowned German mathematician, was informed that one of his students was to leave the field of mathematics with a goal of studying poetry, Hilbert replied with positivity that the student was not creative enough to be a true mathematician. It is curious to understand the inconspicuous creativity involved in the study of advanced mathematics. When does math stop being math and start being an art form? When does the simple logic of equations stop being enough in generating proofs? According to Dr. Milnor, there is no clear distinction: “Math, when it is well done, tends to be an art form. There is the distinction of math done out of curiosity or math done to solve some practical problem. In either case it can be done in an elegant or beautiful way. Sometimes there are questions and ideas that are important yet no one has found a nice way of doing them. There may be proofs that are just very ugly and that is always an unfortunate situation.”

Education and Legacy Stony Brook University graduate mathematics has been recognized among the top national programs in the field. National as well as worldwide interactions with the established faculty have helped produce phenomenal alumni and progressive students. As of May 2011, 6% of Stony Brook students graduate with a Bachelor’s degree in mathematics. Alan Tucker, a professor in the mathematics department at Stony Brook has argued to revitalize the math major known to his father’s (Albert Tucker) and Milnor’s generation dating to Princeton. He notes that the rigorous mental training of math majors produces a broad range of success in fields ranging from economics to the natural sciences [9]. The quality of the Stony Brook mathematical training can be reflected in Dr. Milnor’s position at the university. Dr. Milnor commented: “I was at the Institute for Advanced Study [at Princeton University] for many years, which was an idyllic place with perfect working conditions, but I found myself getting more and more isolated. I was working by myself and missed having students and giving classes. When I had an opportunity to start a small institute here at Stony Brook, it seemed like a very nice idea to me.” It is important to underscore the achievements of a professor by the success of his students, whom Dr. Milnor was able to produce at Stony Brook. “I have been very lucky in having some excellent graduate students here at Stony Brook”, he says. Among them are Jan Kiwi, Associate Professor at Pontificia Universidad Catolica de Chile; Saeed Zakeri, Associate Professor at Queens College and Graduate Center of CUNY; and Rodrigo Perez, Assistant Professor at Cornell University. Michael Spivak, who received his Ph.D. under the

supervision of Dr. Milnor, is well known for publishing multiple textbooks.

Research Techniques How is mathematical research carried out? A series of steps involves investigating specific properties, determining their legitimacy in several scenarios, generating a conjecture, and finally attempting to prove these properties. With no more materials necessary than a blackboard and chalk, established theories have come into existence However, mathematicians working at the peak of John Milnor’s career, such as Stephen Smale, Dennis Sullivan, William Thurston, Rene Thom, and Michael Freedman were all faced with a new crucial tool: computer incorporation. While some were enthusiastic and saw computers as a way to facilitate research, the general consensus was rather hesitant. As Leslie G. Valiant, a fellow applied mathematician and theoretical computer scientist said at the International Congress of Mathematicians in 1986 regarding computers, “I should clarify my own position, I don’t use computers either” [4]. Dr. Milnor, on the other hand, conveyed his enthusiasm regarding the practical use of computers for simple tasks. “I am completely addicted. In the old days, just for the process of writing a paper I would have something typed up and I would scribble corrections all over it. The press secretary would have to change it again and a few times the secretary would get very grouchy. It is a different world now; we can make all the changes we want easily.” Dr. Milnor, however, does warn about the dangers of completely entrusting computers with respect to results. Computers do only as much as they are instructed to do, therefore, Dr. Milnor uses them with discretion. “There is a field called experimental mathematics in which you try to gather evidence about a problem which you cannot really solve,” Dr. Milnor said. “Computers are especially important to me in the area of dynamical systems when trying to figure out how some mathematical system changes with time. It is really complicated to work all the details out on paper. Thus, you try to work out what happens experimentally on a computer. There is always a danger of not knowing whether the computer model really grasps the original situation, so you have to be very cautious. Nowadays, solutions generated by computers are accepted as proof, but some people do not trust them. Classical proofs, if they are very complicated, can easily have mistakes as well. Therefore, no matter what kind of proof it is you have to be very cautious and examine the conjectures very carefully.”

Recognition In 2011, Dr. Milnor received the highly prestigious Leroy P. Steele Prize for Lifetime Achievement award. “It is very gratifying. As I said at the ceremony, it is wonderful to get a prize for doing what you like to do.”

The Stony Brook Young Investigators Review, Fall 2011

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