Jolly good show...jolly good!

I'm halfway into a Friday filled with student presentations: the 280 folks have offered up three inventive exhibitions, and all did pretty well (especially when one norms out for the nervousness and jitters that accompany what's likely the first semiformal mathematics presentation most of these people have ever given).

The first course on the menu was a deep dish of induction (one within another, like two thirds of a turducken): three of the students worked up a careful verification of the fact that the gamma function generalizes the factorial function, complete with gorgeous LaTeXed slides!

The next offering was a quick course on multinomial coefficients and their usefulness in solving a number of enumeration problems. A few typos aside, their presentation was clear and correct, and interactive! Worksheets are never a bad idea.

Finally, we finished things off with a multimedia derivation of the closed form for the sum of the first N cubes, assuming the formula for the corresponding sums of squares and linear terms. Though there was insufficient time to develop the full proof in class, the method these three folks used was identical to the inductive proof one would use to derive the closed form for the sum of the first N nth powers.

All of today's talks (aside from minor bubbles and burps) were clear, correct, and mostly well-composed. I'm happy. Overall I'm mightily impressed by the maturity of the problems selected by the students in this class, especially since several of the choices made were motivated by inherent interest in one problem or another. (I.e., I didn't have to twist too many arms: a lot of people naturally gravitated towards problems they'd thought up themselves.) The level of preparation has also been exceptionally high, as has the mastery of the subject matter. Though this semester's definitely had its ups and downs, I've got high hopes for a number of the students in this class as they move onward in their math careers. (At least three of them will be doing work with me this summer.)

For now, I must say adieu...I'm off to Abstract II, where I'll be hearing all about lattices, fuzzy groups, and the Sylow Theorems.