![\"05e74735-3f90-4d43-aab8-b99734257b01\"](\"http:\/\/blogs.ams.org\/mathgradblog\/files\/2016\/01\/05e74735-3f90-4d43-aab8-b99734257b01.jpeg\")
<\/a>Goldmakher tells Ai about the importance of creativity and ownership in mathematics.<\/p><\/div>\n
How did you get interested in mathematics? <\/span><\/i><\/p>\n When I was in 7th grade, in Boston, I enrolled in an experimental program called The Math Circle, where students discover math on their own. We, the students, came up with all sorts of ideas about number theory, argued with each other, proposed conjectures, shot each other down, and eventually came up with all sorts of proofs. No one told us these things<\/span>\u2013<\/span>we made them up. Turns out other people had discovered them earlier, but that wasn\u2019t the point. The point was that we owned them<\/span>\u2013<\/span>they were ours because we invented them! This experience showed me that math was creative and could be created by me. From then on, I was totally hooked. \u00a0<\/span><\/p>\n <\/p>\n Do you take this kind of curiosity into your teaching at Williams College?<\/span><\/i><\/p>\n Absolutely! I think it\u2019s important to learn things before you know why they\u2019re good for you. I also think it\u2019s really important to have motivation from something other than an authority figure. That is what the Math Circle is all about. The teacher showed us this weird thing, and it was up to us to decide what we wanted to prove or discover. This greatly informed my ideas of teaching.<\/span> My goal is not to transmit information; it’s to get people excited<\/span>. Information is everywhere. There are textbooks and Wikipedia; we don\u2019t need professors for information. The point is to excite people and get people to start exploring. I go in, I teach, I interact with students, I see a twinkle in an eye, and that\u2019s it, I\u2019ve made my day. In a university setting, you can\u2019t exactly do a math circle model since there is a set amount of material that must be covered, but I try to make my classes student driven. <\/span><\/p>\n I guess Math Circles really started \u00a0you along the number theory route. Could you talk a little about your research in number theory?<\/span><\/i><\/p>\n My research is in classical number theory. At a fundamental level, I am trying to understand structure and randomness within the primes. \u00a0I think of primes as atoms: you take any whole number and break it down into parts, which are the primes. The question is if there is any sort of rhyme or reason to them. \u00a0Is there a way to find out what the millionth prime is? Well, you could just go one by one and find the next prime and the next prime and eventually say, that\u2019s the millionth one! This is obviously not a great way to do it. Is there a shortcut or a formula for figuring out the <\/span>n<\/span><\/i>th<\/span> prime? \u00a0My theorems are about subsets of the primes and trying to understand how noisy and how structured they are. <\/span><\/p>\n How was your grad school experience?<\/span><\/i><\/p>\n I was extremely lucky to have an amazing PhD advisor. His name is Kannan Soundararajan<\/span>\u00a0(he goes by Sound). He not only met individually with each of the students every week to talk about their progress, but also met with all of his students once a week all at the same time to talk about half-baked pre-ideas. They were along the lines of \u201cWhat if I could do this?\u201d or \u201cHere\u2019s a set-up of what I would like to work on.\u201d We would shoot out all these pre-ideas. \u00a0It was fascinating to see how people thought about and approached ideas. It was especially amazing to see how Sound thought and how his intuition worked.<\/span><\/p>\n Was there a most difficult moment?<\/span><\/i><\/p>\n I\u2019d say all of it was difficult. There’s the course part of grad school, but at least the goals are well-defined. Then comes the research, which is frustrating. \u00a0You have no idea what the goals are, and you don\u2019t have a concrete, specific statement you are trying to prove. When I was out there, I really started getting demoralized by my lack of progress. I had been working for a while and I hadn\u2019t really discovered much, if at all. It became clear to me that not only was I nothing compared to the faculty, but also I did not measure up to the students. In some ways it was inspiring, but also very demoralizing. I was getting pretty close to quitting.<\/span><\/p>\n How did you overcome this impasse?<\/span><\/i><\/p>\n I remember I was at lunch sitting next to Ravi Vakil at Stanford, and he asked me how things were going. And I said, \u201cYou know, not too well, I\u2019m thinking of maybe quitting. I might be at the end.\u201d <\/span><\/p>\n He said, \u201cWhy is that?\u201d I said, \u201cYou know, it’s clear. I look at Sound, I look at myself. He\u2019s supposed to be a mathematician, I\u2019m not.\u201d Then he said, \u201cYou know, most mathematicians aren\u2019t as good as Sound.\u201d I said, \u201cYea, but I look at the other students, and they\u2019re on another level.\u201d And he said, \u201cYou know, most mathematicians aren\u2019t as good as the other students either.\u201d His words had a huge impact on me. Later in the year, I ended up making my first progress and proved a theorem.<\/span> Once I knew that it was possible for me to discover a result, there was a psychological shift.<\/span><\/p>\n Leo Goldmakher is Assistant\u00a0Professor of Mathematics at Williams College. Stephen Ai is a current sophomore at Williams studying mathematics and music; his favorite piece is JS Bach\u2019s Mass in B Minor.<\/span><\/i><\/p>\n