MSI Weekly Bulletin - Week starting Monday 6 November, 2006

We consider the problem of exhaustively generating classes of combinatorial objects without isomorphic objects appearing. The two main approaches, orderly generation and canonical augmentation, will be explained. Examples will be given from graph theory and geometry.

The Bablyonians did algebra in 1800BC. We've learned a little bit since then, and I'd like to share with you some parts of modern mathematics that I find amazing. Algebra provides us with the tools to study geometries inaccessible to our other senses. Come along and learn about some of these tools: - Singular homology and simplices - Differential forms and deRham cohomology - Sheaves Pictures included! It's all good voodoo.

Abstract: The current primality test in use for Mersenne primes continues to be the Lucas-Lehmer test, invented by Lucas in 1876 and proved by Lehmer in 1935. In this paper, a practical approach to an elliptic curve test of Gross for Mersenne primes is presented and analyzed. One of most important advantages of the test is that, unlike the Lucas-Lehmer test which requires O(p) arithmetic operations in order to determine whether or not 2^p-1 is prime, it only needs O(lambda) arithmetic operations, with lambda << p. Hence it is more efficient than the Lucas-Lehmer test, but is still as simple, elegant and practical

I will discuss joint work with Julie Clutterbuck on oscillation estimates for parabolic equations, and applications of this to obtain estimates on low eigenvalues of the Laplacian. In the class of compact Riemannian manifolds with a given diameter bound and a given lower bound on Ricci curvature, we deduce the sharp lower bound on the first nontrivial eigenvalue. Finally, by combining the method with some new sharp estimates on log-concavity of the first eigenfunction (also proved using the heat equation), we prove an old conjecture on the spectral gap for the Dirichlet problem on a convex domain in Euclidean space.