Algorithmic Proofs of the Quillen-Suslin Theorem (Led by Donal O'Shea).
O'Shea's group consisted of Brian Johnson (Harvey Mudd), Margaret Hjalmarson (MHC), Ben Lee (Harvard), Laurel Reilly-Raska (MHC), and Carrie Snyder (Harvard). They investigated algorithmic proofs of the Quillen-Suslin theorem (also known as Serre's conjecture). This theorem asserts that any projective module over a polynomial ring is free. It is easy to show that the theorem implies that given any set of generators a1, a2, ..., ar of the polynomial ring k[x1, ..., xn], the syzygy submodule Syz (a1, ..., ar) of k[x1, ...,xn]r consisting of all ordered r-tuples (f1, ..., fr) such that f1a1 + ... frar = 0 must be free. It is less obvious, but true, that establishing this latter result would establish the Quillen-Suslin theorem.
The students worked on finding explicit bases for Syz (a1, ..., ar) in the case when the ai are in k[x,y]. Using Groebner basis methods, they were able to establish many partial results. In particular, they were able to classify the siuations in which finding a basis was most difficult. This turned out to be related to how many of the ai's were needed to write out a relation of the form c1a1 + crar = 1.
The students have written up their results.