Which natural numbers occur as the area of a right triangle with three rational sides? This is a very old question and is still unsolved, although partial answers are known (for example, five is the smallest such natural number). In this talk we will discuss this problem and recent progress that has come about through its connections with other important open questions in number theory.
Karl Rubin is the Thorp Professor of Mathematics at the University of California, Irvine. His research deals with elliptic curves and other aspects of number theory. Rubin attended Washington DC public schools, was a Putnam Fellow as an undergraduate at Princeton, and received his Ph.D. from Harvard. He was a professor at Ohio State, Columbia, and Stanford before moving to UC Irvine in 2004. Rubin received the Cole Prize in Number Theory from the American Mathematical Society, a National Science Foundation Presidential Young Investigator award, a Humboldt Research Award, and Guggenheim and Sloan fellowships.
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Mathematicians believe, correctly, that they are uniquely qualified to answer complicated questions in science and engineering. But it very often happens that such problems are unsolvable or intractable in their original form. Is it acceptable to say politely "I'm sorry; this problem is impossible" and then return to answering questions that can be answered? Or should we do more? How can we do more? This talk, intended for a general audience, will describe, with examples from the speaker's experiences in optimization, how mathematicians can become local heroes after they say they're sorry.
Doodling has many mathematical aspects: patterns, shapes, numbers, and more. Not surprisingly, there is often some sophisticated and fun mathematics buried inside common doodles. Vakil will begin by doodling, and there is no telling where it could take him.
What is the role of proof in mathematics? Most of the time, the search for proof is less about establishing truth than it is about exploring unknown territory. In finding a route from what is known to the result one believes is out there, the mathematician often encounters unexpected insights into seemingly unrelated problems. I will illustrate this point with an example of recent research into a generalization of the permutation matrix known as the "alternating sign matrix." This is a story that began with Charles Dodgson (aka Lewis Carroll), matured at the Institute for Defense Analysis, drew in researchers from combinatorics, analysis, and algebra, and ultimately was solved with insights from statistical mechanics. This talk is intended for a general audience and should be accessible to anyone interested in a window into the true nature of research in mathematics.
