## Three out of Three

Davidson Institute for Talent Development announced their 2012 Winners. Out of 22 students, three were recognized for their math research. All three of them are ours: that is, they participated in our PRIMES and RSI programs:

- David Ding’s project, “Infinitesimal Cherednik Algebras of gl
_{n},” came out of his participation in the PRIMES program. - Sitan Chen’s project, “On the Rank Number of Grid Graphs,” came out of his participation in the RSI program.
- Xiaoyu He’ project, “On the Classification of Universal Rotor-Routers,” came out of his participation in the PRIMES program.

I already wrote about Xiaoyu’s project. Today I want to write about Sitan’s project and what I do as the math coordinator for RSI.

RSI students meet with their mentors every day and I meet with students once a week. On the surface I just listen as they describe their projects. In reality, I do many different things. I cheer the students up when they are overwhelmed by the difficulty of their projects. I help them decide whether they need to switch projects. I correct their mistakes. Most projects involve computer help, so I teach them Mathematica. I teach them the intricacies of Latex and Beamer. I explain general mathematical ideas and how their projects are connected to other fields of mathematics. I never do their calculations for them, but sometimes I suggest general ideas. In short, I do whatever needs to be done to help them.

I had a lot of fun working with Sitan. His project was about the rank number of grid graphs. A vertex *k-ranking* is a labeling of the vertices of a graph with integers from 1 to *k* so that any path connecting two vertices with the same label passes through a vertex with a greater label. The *rank number* of a graph is the minimum possible *k* for which a *k*-ranking exists for that graph. When Sitan got the project, the ranking numbers were known for grid graphs of sizes 1 by *n*, 2 by *n*, and 3 by *n*. So Sitan started working on the ranking number of the 4 by *n* graph.

His project was moving unusually fast and my job was to push him to see the big picture. I taught him that the next step, once he finishes 4 by *n* graphs is not to do 5 by *n* graphs, as one might think. After the first step, the second step should be bigger. He should use his insight and understanding of 4 by *n* graphs to try to see what he can do for any grid graphs.

This is exactly what he did. After he finished the calculation of the rank number of the 4 by *n* graphs, he found a way to improve the known bounds for the ranking number of any grid graph. His paper is available at the arXiv.

I just looked at my notes for my work with Sitan. The last sentence: “Publishable results, a potential winner.”