To Count the Natural Numbers

2016 AWM Essay Contest: High School Level Winner

By: Emily Jia Illinois Math and Science Academy (Aurora, Illinois)

Tanya Khovanova won two medals as part of the USSR team at International Math Olympiad; earned her PhD in Mathematics from Moscow State University; and worked at renowned academic institutions such as MIT, Princeton, Bar Ilan (Israel), and the Council on Cybernetics of the USSR. She has authored/co- authored 65 papers and serves as head mentor in the two most competitive high school math research programs: RSI math and MIT PRIMES.

But to me, she has been much more than a mathematician of stunning intellect and productivity. In the time that I’ve known her, she’s been Tanya: the person who listened to my rambling about graph theory research; the person who offered me cookies while I attempted to articulate my dreams and fears; the person who has advised me to always do what feels right. Interviewing her felt right.

Her earliest memory of mathematics involves a summer night in the USSR. Tanya was on the farm with her parents and sister, lying in bed, when she began to think about numbers. Specifically, she imagined big numbers: how there is a large number, and she could add one, and there’d be a next, larger number. At some point, she realized that the number of numbers was infinite. Retelling this tale, Tanya laughs bemusedly. “I was five, and I was euphoric.”

Although Tanya’s parents were educated—her father was a physics teacher, and her mother a chemistry teacher—she developed an interest in mathematics by doing math by herself during her elementary school classes. The math teachers would give her two versions of each test, and Tanya remembers that this was a source of pride. “I would finish both tests in five minutes, and the teachers didn’t know what to do with me!” In middle school, she transferred to a math-specific school where she was still the best in her class, and her teachers let her work through the textbooks at her own blistering pace.

Seeking greater challenges, Tanya began participating in the Soviet Math Olympiads, a series of proof- based problems that lasts several hours each round. She performed very well on these competitions, and was selected to train and compete as part of the USSR International Mathematics Olympiad (IMO) team, winning a silver medal in 1975 and then a gold in 1976. In high school, Tanya was also invited to the Gelfand seminar, one of the most famous seminars in the USSR which lasted from September 1946 until May 1989. There, Gelfand—one of the greatest Russian mathematicians—approached Tanya about becoming her adviser.

When speaking about her study under Gelfand, Tanya prefaces with his contributions to academia, support for collaborators, and positive influence on many mathematicians. However, she also describes this experience as a particularly unpleasant mentorship. During lectures, Gelfand asked Tanya to explain the speaker’s material, and when she made a mistake, he would call her a fool in front of the audience. Gelfand had also told her that she had 1.5 years to understand the seminar lectures; otherwise, she would never be a mathematician. As a result, Tanya was always afraid to ask Gelfand for help, and was left thinking that she could never be a mathematician.

Despite these hardships, Tanya went on to complete a Ph.D. in mathematics at Moscow State University and came to the United States to conduct research as an NSF Postdoctoral Fellow at MIT from 1993-1995. During this time, she struggled to balance her responsibilities as the matriarch of her family with the demands of her doctorate. Reflecting on her graduate and postdoc experience, Tanya speaks candidly.

“The gender bias was there. It wasn’t people treating me differently for mathematics, or not respecting me. It was the culture: in Russia, the woman takes care of cooking, the child, and shopping. I’ve heard of research comparing how much women and men worked in that time, and women had four hours more each day, because they have to do the home chores after their jobs. It was unfair, and it happened to me.”

Used to doing everything for the children, Tanya eventually stopped research and took two breaks in her career to take care of her sons. Although she had the option of pursuing faculty positions in universities, she no longer wanted to move her family; her oldest son already had to grow up in three different countries. Tanya quit academia.

At this point, Tanya pauses to ask me, “Emily, what do you want to do?” While a professorship is highly competitive and geographically demanding, she reassures me that mathematics is a great skill to have, and a very transferable profession in several areas– science, data analysis, technical industries, government, education. A mathematician can enjoy many lines of work and find a job in small communities and big cities alike.

Yet at that point in her life, Tanya found herself miserable: single, sick, and alone, she found herself thinking, “What will I do, I will soon be starving!” She found a job in telecommunications, and then moved to Boston to create algorithms for the military sector. However, this contributed to a war effort that she opposed, and she found her team’s tasks to be uninteresting. In her free time, Tanya started collecting interesting properties of numbers. She quit her job and went back to academia as soon as her children were ready for college.

Having struggled with writers block, Tanya started a blog that changed her life. She began to take English lessons, and stopped being afraid of writing papers. When she wrote about mathematical topics on her blog, she could write 3-4 posts and have enough material for a paper. Finally, she realized, “I wasn’t successful before as a mathematician because I was always doing what people told me to do.” Gelfand gave her the problem for her first publication, and afterwards she followed her then-husbands’ interests. She had picked a job in industry that she didn’t enjoy but, finally, this blog was a chance to turn this around. For the first time, she learned to follow her heart. And her heart led her to recreational mathematics: a mix of combinatorics, geometry, probability theory, and number theory that resembles puzzles instead of abstract math.

Meanwhile, she was contacted by Ingrid Daubechies—the first female professor of mathematics at Princeton—to work at a Women in Mathematics program. Working for four years at a program that she describes as nothing short of amazing, Tanya met women who became a support system for the rest of her life:

“When these girls started discussing their issues, I realized that they were exactly the same issues I had when I was young. If I had a woman friend who would guide me through these experiences, my life as a mathematician would have been so much easier. All my problems with my adviser, I couldn’t discuss with everyone, and it was a mistake. I didn’t have any support. So this program for women in mathematics—it changed my life. I started to think about my own problems and understanding how gender makes it more difficult to study mathematics. It was very eye-opening.”

Currently, Tanya lectures at MIT, coaches for math olympiad at a charter school, posts regularly about life and math on her blog at blog.tanyakhovanova.com, and helps high school students find their love for mathematics through the RSI and PRIMES research programs. Her advice for others? Look around, look inside yourself, and find what you like. Math is beautiful, and you shouldn’t be scared—go for what you want, and do what is best for yourself.

About the Student:

Emily fell in love with mathematics when discovering its tricks and charm through competitions in middle school. However, her interests shifted to research in high school, and she has conducted 2 years of research in quantum topological computation and graphs at the University of Chicago and the Illinois Math and Science Academy. In the summer of 2015, Emily was one of 51 junior selected nation-wide to conduct all-expenses-paid research at MIT through the Research Science Institute (RSI). Under the guidance of the MIT math department, she researched extremal graph theory and met Dr. Tanya Khovanova, head mentor of RSI math projects. Emily has been admitted as part of Harvard College’s Class of 2020, and plans to continue math research as an undergraduate.