Andrei Okounkov used contemporary imagery to bring his plenary to a large assembly of international mathematicians today. His presentation, ‘On the crossroads of enumerative geometry and geometric representation theory’, focussed on Lie Theory and its applications. Okounkov won a Fields medal in 2006 for his contributions to ‘bridging probability, representation theory, and algebraic geometry’.
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Lie Theory touches a number of mathematical areas including classical, differential, and algebraic geometry, topology, ordinary and partial differential equations, complex analysis(one and several variables), group and ring theory, number theory, and physics, from classical to quantum and relativistic. Okounkov’s work touches on several areas of math but it focuses on the mathematical ramifications of modern high-energy physics. “Lie Theory is among the main building blocks of mathematics and mathematical physics,” he said in his presentation.
Joanna Kania-Bartoszynska, Program Director for Topology and Geometric Analysis at the National Science Foundation, was in audience and was astounded at the work presented. “I study a grain of sand, but his work is the whole beach,” she said afterward.
Unlike many math geniuses, Okounkov found his way to the subject in a roundabout way— through economics and army service. When he was completing his Ph.D. in Moscow, he was much older than his colleagues and already had a family. “I didn’t go through special schools and Olympiads,” he told the American Math Society in 2007. “I came via studying economics and army service. I had a family before papers. As a result, my mind is probably not as quick as it could have been with an early drilling in math. But perhaps I also had some advantages over my younger classmates. I had a broader view of the universe and a better idea about the place of mathematics in it. This helped me form my own opinion about what is important, beautiful, promising, etc.”
“I personally don’t know how one can understand something without both thinking about it quietly over and over and discussing it with friends,” Okounkov said. “When I feel puzzled, I like long walks or bike rides. I like to be alone with my computer playing with formulas or experimenting with code. But when I finally have an idea, I can’t wait to share it with others. I am so fortunate to be able to share my work and my excitement about it with many brilliant people who are at the same time wonderful friends.”
Okounkov is the Samuel Eilenberg Professor of Mathematics at Columbia University and academic supervisor of the HSE International Laboratory of Representation Theory and Mathematical Physics. He received his doctorate at Moscow State University in 1995 under Alexandre Kirillov and Grigori Olshanski and was professor at Princeton University from 2002 to 2010. He also worked as assistant and associate professor at the University of California, Berkeley, and instructor at the University of Chicago.
In 2004, Okounkov received the European Mathematical Society Prize. In 2016 he became a fellow of the American Academy of Arts and Sciences.
Check out his presentation from ICM below.