I am the professor who taught the epsilon geometry course in 2011. I taught from the beautiful Green Lion edition of the famous T.L. Heath translation of Euclid's Elements. This is the book I used when I taught upper level undergraduates and graduate students at UGA in Athens, GA. This is considerably above the level of the geometry course I myself had in high school and also what is normally available now, say from Harold Jacobs' book Geometry. These kids were stronger than a typical college class and I just loved teaching them, although it did take a good part of the first week for us to get to know and respect each other. I apologize for my primitive class management skills. But I think we mostly solved this within three days.
We went from scratch, Euclid's own postulates, and a critical examination of them in the light of modern theory, mostly due to Hilbert, and guided by the beautiful book by Robin Hartshorne, written for his college class at UC Berkeley. I ended up showing how the limit theory of volumes in Euclid and Archimedes allowed the computation of the volume of a sphere, and then leads to calculus, and even wrote up but did not present, an argument for how Archimedes could have computed the volume of a 4 dimensional sphere.
I conjecture there is no way the kids could have ever seen this material before, unless they went to college. Indeed I made up some of it myself just for them. But they are so bright, it went well, better even than when I taught it in college. Since this stuff was so intense and high powered, we had an afternoon class in which we did only hands on constructions of solids from cardboard, such as icosahedra and dodecahedra, and they really enjoyed that. For this I myself learned for the first time how to really construct a pentagon. I had never learned it in high school and not really grasped it in grad school even when i taught it until last summer. If you want to explain something to 9 year olds, or anyone, you should really understand it! In preparation for the course, we emailed about some advanced algebra, including Euler's explanation of how to solve both quadratic and cubic equations, something I had not understood that well before even though i had taught it to grad students.
This year I envisioned going forward with a more general geometry course on curved geometry, showing that Euclid's geometry fits in between positively curved and negatively curved geometry, as the case of flat geometry, i.e. zero curvature. Thus we would cover spherical geometry first ( a surface of positive curvature) and then geometry of negative curvature ("saddle surfaces") and show how the curvature is reflected in the angle sum of a triangle. I.e. on a sphere, the angle sum of a triangle is more than 180 degrees, and gets larger as the area of the triangle gets larger. For negatively curved surfaces the sum is smaller than 180 degrees, and there is an upper bound to the area of a triangle. if time allowed we would explore tessellations of the various planes by polyhedra, including negatively curved ones which allow more possibilities.
Ultimately I found I did not have have time to prepare adequately for this course since I was learning the material myself, and when I admitted this and withdrew, George hired an expert who is a college professor with research experience at the highest level in this material (he has published on it in the best research journal in America, Annals of Math.), and I believe it will be very exciting for the kids. The Euclidean geometry course will also be repeated for the newcomers. And there will be other courses as well.
Pardon me if my participation was unexpected, but I hoped to be able to answer some of the questions asked here. My experience was that the camp organizers are of the highest quality and focus primarily on mutual respect and mathematical expertise of the first order. I would send my kids. The expertise of the faculty is quite high. Basically they are research mathematicians qualified to teach graduate students in the area.
The demands on the kids for sitting and listening at length seemed high, but I was amazed at their ability to do it well. When I saw kids who were busying themselves doing something else, sometimes I called on them to keep them involved (I knew all their names) but the second week I eventually just gave them their space, and spoke mostly to those who were listening. At the end I really liked those young people and they made me feel they appreciated and liked me. i will never forget that experience. We live all our lives as teachers hoping to meet students whom we can help at this level.
If you want to see what we did mathematically, you may consult the books mentioned, and also my notes from the class which are on my webpage and were made available to the students along with the course. (See http://www.math.uga.edu/~roy/, near the bottom.)
I am willing to bow out here so the discussion can proceed candidly if desired.
Thank you for writing this! I just wanted to repeat how much my son LOVED these two weeks of math. We cannot attend this year, but will attend next year since he will still be young enough.
