2022 Mathematics undergraduate prize recipients

Don’t be afraid to ask questions! I remember when I first went to office hours, I was super intimidated by the ULAs and TAs. Now, as a ULA myself, I can’t wait to hear any and all student questions! While you can do math alone, it’s so much more fun to do it with others — and more often than not, seemingly simple questions can lead down really fascinating rabbit holes.

Slow down. It’s natural to feel pressure to get ahead, and as a result, skip prerequisites in order to take advanced classes. But those classes are always going to be there, and they’ll be much more fun and more satisfying when you have a strong foundation in the basics. It’s always important to stop and smell the roses—not just because nailing the little details will make you a better mathematician, but because math is made to be experienced slowly and collaboratively, rather than as a race to the finish line.

I think it’s important to remember to take things at your own pace. Being surrounded by so many amazing students, it’s easy to feel pressured to have to catch up or keep up with our peers. Everyone, however, comes from different backgrounds and has different circumstances, so it’s important not to lose sight of what helps us as individuals do our best, both in and outside of math.

Don’t worry too much about not understanding concepts on your first tries at learning them! In my experience, I’ve had to revisit the same topics over and over again before things finally started falling a bit more into place. And while it is valuable to spend time working independently, I think my most rewarding mathematical experiences have come from reaching out to my professors and discussing with my peers. Attend office hours, form study groups, and most importantly, don’t be afraid to ask questions (to emphasize oft-stated advice: no matter how basic my questions felt, I think there was always at least one other person wondering the same thing). Finally, don’t underestimate the importance of physical health. As tempting as it may be to stay up until 5 am working on a tantalizing problem (I might be a bit of a hypocrite here…), it’s not worth it - sleep!

The research project is on the (p, q)-problem for hypersurfaces. The (p, q)-problem for convex sets is a famous topic in discrete geometry, which studies the minimal number of points (the Hadwiger-Debrunner number) that pierce an arbitrary family of n convex sets in R^d satisfying the (p, q)-property, i.e. every p sets of the family contain q sets with non-empty intersection. Surprisingly, Alon and Kleitman showed that such number is finite and depends on p, q, d only, independent of n, the size of the family of convex sets. Following this, the asymptotic bounds for the Hadwiger-Debrunner number keeps getting improved. We hope to find a similar result when the (p, q)-property is applied to hypersurfaces, which restricts to the roots of polynomials of d variables and degree at most D, and see if the minimal piercing number depends only on p, q, d, D in this case. The hypersurface case seems promising, for Hilbert Nullstellensatz translates the geometric problem of intersection of hypersurfaces into an algebraic one studying the ideal generated by their corresponding polynomials, and this could be handled by algebraic methods similar to those developed for convex sets. If the conjecture is true, we will also adapt the tools for studying convex sets to find asymptotic bounds.

You may feel that each statement in lectures looks clear to you, but you are unsure about the big picture and where these are proceeding. Active exploration with patience may help you gather your knowledge, find connections among areas within math or between math and other subjects, and inspire you to go further.