MAA FOCUS August/September 2008 It’s Easy to Get Interested in Number Theory An Interview With Karl Rubin By Ivars Peterson

Educated at Princeton and Harvard, at Harvard that I learned that there was Professor Karl Rubin held a number of something called Iwasawa theory, which positions before becoming the Thorp has been central in a lot of my work. Professor of Mathematics at the Univer- The direction of my research in graduate sity of California, Irvine. Stops at Ohio school and thereafter is due mostly to State, Columbia, and Stanford led Rubin my advisor, Andrew Wiles, and to John to Irvine, where his main research focus Coates. More recently my work has is on number theory and elliptic curves. been influenced by two of my coauthors, A former Putnam and Sloan Fellow, Alice Silverberg (who introduced me to Rubinʼs accomplishments also include cryptography) and Barry Mazur. the AMS Cole Prize in Number Theory, a Guggenheim Fellowship, and the NSF IP: How would you describe your main Presidential Young Investigator Award. research areas? Why are these areas par - The author of nearly 60 published papers, ticularly exciting to you? Rubin delivered an MAA Distinguished Lecture in May. KR: I work in algebraic number theory and arithmetic algebraic geometry. A Ivars Peterson: Did you become inter- major focus of number theory is solv- ested in mathematics at a young age? ing polynomial equations in integers or What attracted you to mathematics? porting a lot of summer programs for rational numbers, and algebraic geom- high school students. I think the Ross etry is a natural tool for approaching Karl Rubin: I grew up in a scientific program was unique in the way it went such problems. I am especially inter- family. My mother and sister are astrono- very deeply into one subject, which hap- ested in elliptic curves. Elliptic curves mers, and both of my brothers are geolo- pened to be number theory. For the eight have genus one, and are defined by gists. My father was trained in chemistry weeks of the program, the daily routine cubic polynomials. They fall in between and physics but was essentially an ap- Monday through Friday was a one-hour curves of genus zero (conic sections, plied mathematician, and he especially lecture at 9 a.m., followed by a problem defined by quadratic polynomials), where encouraged me in mathematics. set that took most of the next 23 hours to we know almost everything we want, and complete. The problem sets would lead curves of genus two or more (defined by Beginning when I was in junior high the students from numerical discoveries higher-degree polynomials), which are school my father would bring home to conjectures to proofs. more complicated. Elliptic curves have a mathematics books from the library for very rich structure, so progress is pos- me. For a long time I just ignored them, I spent two summers in the Ross program sible, but there are still many unanswered but eventually I started reading them. as a student while in high school, two questions about them. What is Mathematics , by Courant and more as a counselor while in college, and Robbins is one that I remember. Hardy I have continued to be involved with the In recent years I have also been working and Wrightʼs The Theory of Numbers program off and on up to the present. In on applications of number theory and is another. By the time I finished high addition to receiving an introduction to algebraic geometry to cryptography. school I was pretty sure I wanted to be a real mathematics, and a great example mathematician. of how to teach mathematics, it was very One attractive thing about number theory valuable to meet a group of like-minded is that there are many questions that are I always enjoyed puzzles, and my en- students. easy to state, but with solutions that are joyment of mathematics was a natural very deep. Fermatʼs Last Theorem is a outgrowth of that. I was also fortunate to have a number of good example. The fact that modern, good mathematics teachers in the Wash- highly abstract mathematics can be used IP: What people or experiences most ington, D.C., public schools. I had many to solve such an old problem indicates influenced the direction of your studies? good teachers later as well, but I was to me that mathematics is moving in the Your subsequent career? pretty well set on my path by then. right direction. Itʼs easy to get interested in number theory at a young age, be- KR: In addition to my parents, the Arnold When I was an undergraduate at Princ- cause the questions are relatively acces- Ross summer program at Ohio State had eton, the professor who had the greatest sible. In my case, I never moved away. a big influence on me. In the 1970s the influence on me was Kenkichi Iwasawa. National Science Foundation was sup- But it wasnʼt until I got to graduate school

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IP: Are applications, in cryptography, the Birch and Swinnerton-Dyer conjec- KR: There are lots of them. The most for example, now an important force in ture. This conjecture is one of the Clay famous are two of the Clay Millennium driving number theory research? Mathematics Institute one million dollar problems, the Riemann Hypothesis and Millennium problems, and has been a the Birch and Swinnerton-Dyer conjec- KR: Applications are one important driving force behind a lot of modern ture. Other questions that are particularly force, but certainly not the only one. number theory and arithmetic algebraic interesting to me include questions about About twenty years ago people discov- geometry. rational points on curves. ered that elliptic curves have applications to cryptography. I have done some work Computations can also play a role after For example, we still donʼt know in on problems inspired by cryptography, a theorem has been proved. To me, an general how to decide whether a curve with applications to cryptography. I find abstract result becomes much more excit- of genus one has a rational point. There it great fun when the things Iʼm interested ing if one can use it to produce interesting are also many interesting questions about in turn out to have useful and surprising concrete examples. Examples can give ranks of elliptic curves that we have no applications. I suspect many other “pure a better understanding of a theorem, idea how to answer. mathematicians” feel the same way. But and can lead to a better or more useful I still spend most of my time on problems result. In another direction, there are interesting with no currently known applications. computational questions where prog- Personally, I enjoy such calculations. Iʼm ress would have immediate real-world IP: What role, if any, can (computational) grateful that algorithmic number theo- impact. How hard is it to factor large experiments play in number theory? rists, clever programmers, and modern integers? What is the fastest way to computers have given all of us the op- solve the discrete logarithm problem in KR: Experiments are often very impor- portunity to perform computations that a finite field, or on an elliptic curve over tant, and they play some part in most of would have been unimaginable when I a finite field? my work. One important role is as an was a student. indication of what to expect, what to try to prove. For example, in the 1950s, IP: What do you see as key questions Birch and Swinnerton-Dyer carried out worthy of future mathematical explora- some numerical experiments with el- tion in number theory? liptic curves, which led to what is now

Classroom Capsules Online The committee working to get more articles in the Class- room Capsules data bank met in the Carriage House at the MAA Headquarters June 2 – 4, 2008. The intention mirrors that of the long running section in the College Mathematics Journal : to offer ideas that can be used in the teaching of mainline undergraduate mathematics courses. These include new proofs, new connections to applications or other areas of mathematics, examples, historic tidbits, and more, all kept short and presented in a form ready to use.

Those working on the project, shown from left to right in the picture are Danrun Huang, Paul Zorn, Byungchul Cha, Lang Moore (Executive Editor of MathDL), Olaf Stackelberg, Sue Doree, and Wayne Roberts (Editor of Classroom Capsules).

To visit the Classroom Capsules site, go to http://www. maa.org and scroll down until you see the MAA Re- sources box.

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