Alice Silverberg presents "Cryptography: How to Keep a Secret" in the MAA Carriage House on December 8, 2010. Her presentation was a part of the MAA's Distinguished Lecture Series, sponsored by the National Security Agency. Read more about her lecture and listen to a full podcast.
Where did you grow up?
I grew up in Queens, in New York City.
When did you first show an interest in mathematics?
When I was in elementary school, I learned that I was unusually fast and accurate on math tests.
When my brother went off to college, his letters home included math problems for me to solve. I think he didn't want to be bothered sending me real letters, and sending me math problems was easier. I wanted the attention of an older brother, so I sent him back my solutions.
It was probably also important that my mother was good in mathematics. My mother was the one to go to when I had a math problem, while my father helped me with my English assignments. My mother would have been a mathematician, if she had had the right opportunities.
In high school I was eventually captain and top scorer of my high school's Math Team. I was mentored and trained by the students one and two years ahead of me. Some of them had gone to Arnold Ross's summer program at Ohio State, and they encouraged me to apply.
What persuaded you to major in mathematics?
Well, given how much my parents were spending for me to go to Harvard, I didn't think I could justify majoring in Folklore and Mythology. While there's some truth to that answer, a slightly more serious answer is that I was good at it and, thanks to the Ross Program, I had a community of friends whom I respected who were majoring in math, and it seemed like the thing to do.
What appeals to you about mathematics? About the research that you do?
One thing that appeals to me about mathematics is that it's a search for the truth.
In addition, I like to solve problems, and mathematics gives me a chance to solve interesting and difficult problems.
I definitely gravitate towards the discrete side of mathematics, not the continuous. The number theory in the Ross Program appealed to me. That's one reason that I like cryptography and the number theory that it uses.
Have your research interests changed over the years?
Much of my work has had something to do with the number theory of abelian varieties, which are higher dimensional generalizations of elliptic curves. I started out very theoretical and later moved in the direction of applications. My research focus has expanded to include very applied aspects of number theory, especially cryptography.
How important to you is the teaching component of being a mathematics professor?
Teaching can be one of the most rewarding aspects of being a professor. I enjoy teaching, especially when I'm given enough freedom and institutional support to do things right. I think that universities should do more to encourage and reward good teaching.
Can you give an example of a particularly rewarding teaching experience?
Last winter I taught the undergraduate number theory course, and enjoyed it immensely. I think what was most satisfying about it was that I got the students to participate a lot in class, and become enthusiastic about what they were learning. Mathematics has to be learned actively, not passively. I had a lot of freedom about what material to cover and how to cover it, so I could do what worked for me and for this group of students.
Another experience that was particularly rewarding was getting two Japanese elementary school students to understand and be interested in Fermat's Last Theorem, even though I didn't know Japanese and they didn't know English. The extent to which mathematics is a universal language is truly amazing.
It's also satisfying to reach the students that are bored with the regular class, by getting them excited about some special question. A problem that I've had a lot of success with in discrete math classes is a red hat/blue hat problem that I first read about in an MAA publication.
Who have been major influences on your career?
Arnold Ross is probably the person who had the greatest impact on my decision to go into mathematics. His summer program was a very important part of my life for several years. I thought it was great to have a community of friends who wanted to help each other learn to, as Arnold Ross would say, "think deeply of simple things". That community was also a major influence on me.
At Harvard, I took many courses from Lynn Loomis, and his encouragement was important.
My Ph.D. thesis advisor, Goro Shimura, had a very positive impact on my career. He influenced me greatly mathematically, and was a very good thesis advisor. I learned a lot from him.
One of my role models for giving good, clear lectures and writing mathematics well was Jean-Pierre Serre.
My co-author Yuri Zarhin was a major mathematical influence, and taught me how to collaborate on mathematics. I learned from him that doing research with other mathematicians can be fun.
I have heard from other women who started out at Harvard in mathematics that they found it unwelcoming or uncomfortable and ended up having to leave, even in quite recent times. One person noted that many of the men, even if not unwelcoming, tended to be condescending. Your experience seemed somewhat different. What might have made it so?
Yes, many female students (including me) found the Harvard Math Department to be an unfriendly place for women. I was slow to realize how pervasive sexism is in the mathematical community, which is why I stayed in the field when others left who didn't feel welcome. While there was sexism at Harvard, I noticed it more at Princeton when I was a grad student, and at Ohio State when I was on the faculty there. I think there's a cumulative effect. If I had known what it would be like, I might have gone into computer science, where I think that women have greater opportunities, even when the climate isn't completely welcoming.
What value do you place on the communication of mathematics, particularly to audiences beyond immediate research colleagues?
I try to convince my students that communication is one of the most important aspects of doing mathematics. If you can't communicate what you've done, what good is it? It might give you pleasure, but it's not useful to anyone else.
There are many good reasons to communicate mathematics to the public. One is to let students know that there is interesting mathematics to do, and that there are interesting professions for them to consider that involve mathematics. Another is that it seems to fill a need – people seem to be interested in depictions of mathematicians and mathematics in film and on TV, and want to find out more.
Improving mathematical literacy is in everyone's best interest. Few people are proud to be illiterate, but surprisingly many people are not at all embarrassed about being mathematically illiterate. We need to change that.
Where do you see yourself, say, 10 years from now?
Mathematics and the world can change very quickly. I think it's dangerous to predict what will be useful or interesting in 10 years. As a problem solver, I tend to go where the problems are. Having said that, I think that I will continue to be interested in both the deep theory and how to apply it to the real world.