I met Ian Cavey, an undergraduate at Boise State, about a year ago. He took my Communication in the mathematical sciences course. It was a pleasure to have him as a student. (You can see the slides of one of his group projects here.)
Last Spring, Ian took my advanced linear algebra class. Also, he told me he was interested in trying some independent research under my guidance. He proved to be quite independent (he chose the problem he wanted to work on) and resourceful (for instance, finding workarounds when his direct approach would not work). The result of this was a paper (Volumes of Sphere-Bounded Regions in High Dimensions) that he presented at the recent MAA centennial meeting (see page 18 of the program). You can see his slides here.
At the conference, Ian not only presented his talk. He also competed and won first place in the US National Collegiate Mathematics Championship. Some additional details can be found here.
Ian’s talk is about his approach to estimate the -dimensional volume of the central region of a unit -cube bounded by -spheres centered at the vertices. Ian proves that this volume quickly approaches 1. His slides detail his nice combinatorial argument that circumvents the need for explicit computations of unwieldy iterated integrals.