# PRIMES: Michael Zhang's Story

PRIMES has been without question one of the most intellectually stimulating experiences of my high school career. When I first heard about the program, I had known better than most what math research is like, in that I had a sense from past interaction and participation in math summer programs of what mathematicians did, but I had no idea what it felt like to actually be doing it nor did I think I had the capability. When I saw people putting out ridiculously technical research in news of science competitions and the like I just saw the incomprehensible project titles and asked "how in the world could they do that?" There is certain mysticism to the mathematics profession propagated by the media and popular culture that I wanted to be able to defog. And I knew I liked math, I knew I wanted to learn more, I knew I wanted to see what original work I could put out and I knew I wanted to be able to use the dubiously ingenious pick-up line of "I'm a published mathematician" that indubitably would have scores of women falling at my feet, so I applied, got in, and dived in head first.

Originally I was part of the 6-person Computer Algebra lab, which was a crash course on mathematical computer programming in SAGE, GAP and MAGMA as well as linear algebra, abstract algebra and representation theory. It was easily one of the most enjoyable classes I've taken in the interestingness of the material, the speed in which it was executed and the quality of instruction by our instructor, David Jordan. Having only taken math through precalculus and not having any prior experience with computer programming, the class gave me the feeling that I was actively learning more than any I had yet taken. The class ended after a few months of instruction and we were set loose on our own research projects. On the research side, the mentoring we received was superb. I owe much gratitude to professor Pavel Etingof for providing us with a challenging, interesting, motivated, but sufficiently tractable problem and my mentor David Jordan, who struck a balance between excellent explanations, invaluable guidance and a manner of setting goals that let us accomplish things independently. To put it more succinctly, they were awesome.

Using the computer algebra system MAGMA we were able to generate computer data and then conjectures on our problem, pieces of which we were able to prove using more theoretical methods. Perhaps just as important for me during the course of the project was learning the background material and theory behind our problem, which resulted in the exposure to lots of new interesting mathematics.

A few things surprised me over the course of doing mathematics research. The first is how collaborative mathematics as a practice is. The image that I had conjured up of math research prior to PRIMES was the image of Wiles in his attic, working alone for 7 years. This was shown to be an anomaly to me over the course of the project, as I worked with an excellent partner, Yongyi, and had long discussions with my mentor and partner in which all sides offered contributions.

The second is that I learned how necessary it is to not care about feeling stupid over the course of the project. I quickly learned the necessity of asking questions when I was confused and that being completely lost is a relatively common occurrence. To future students – it does not matter how smart people think one is; all that matters is a ruthless pursuit of the truth. One is liable to feel stupid along the way but with every new thing learned one feels increasing clarity.

Third was how much of an equalizer hard work is. When one thinks of math research one thinks of the Taos and the Erdoses and the idea that everything falls easily for a select few with some unique perspective when the reality is that just as and perhaps more important than natural talent is willingness to put in the time and the energy and the persistence to keep thinking about something until it makes sense. Math research seems to be much more like the arduous trek of climbing a mountain than the sports-like point-scoring of a fast-paced competition; one may never reach the summit but with every few thousands of feet gained and theorem proved is a feeling of great satisfaction.

Skill-wise, research was invaluable in giving me experience in navigating seemingly intractable math papers and past literature, writing mathematics and communicating scientific ideas in presentations. Where PRIMES is unique is an opportunity to work with active researchers, receive excellent guidance, tractable problems and the opportunity to be on the frontiers of mathematics where nothing is set in stone like in a textbook, and where everybody is confused.

So seriously, apply to PRIMES now. You already read this far, didn't you?

**Michael Zhang**,
together with Yongyi Chen,
worked on the project
Poisson
homology in characteristic p under
the mentorship of David Jordan.

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