Applied Math Colloquium

This talk will discuss some recent work on the spectra of discrete Laplace operators coming from boundary maps in a simplicial complex. For two families of simplicial complexes, the chessboard complexes (studied by J. Friedman and P. Hanlon) and matroid complexes (studied by W. Kook, D. Stanton and myself), these spectra are known to be integral, and interpretations of the spectra have been given. Why these particular complexes should have integral spectra is still mysterious, as is the connection to spectra of Laplacians on Riemannian manifolds.

One corollary to the interpretation of the spectra for matroid complexes may be paraphrased as stating that one can "hear" the chromatic polynomial of a graph.

Every child learns early that brittle objects break easily. Yet on a bit of reflection the process should seem mysterious. Weak forces are applied over large length scales to an object, and spontaneously the object uses that energy to snap atomic bonds. It happens through propagation of a crack, a propagating structure with a singularity at its tip. George Irwin created an ingenious theory for how cracks propagate, posed entirely in the framework of continuum mechanics, which cleverly evades all questions about what happens very near the singular tip of the crack. However, there have long been some dynamical puzzles about how cracks moved. For example, they seemed to move at half the speed that theory predicts. I will argue that these puzzles are the result of trying to push continuum mechanics too hard, and will present analytical, numerical, and experimental evidence that one can understand crack motion rather thoroughly when continuum mechanics is replaced by the more realistic underlying discrete mechanics.

Abstract. Durrett and Levin (1994) proposed that the behavior of stochastic spatial models could be inferred from the mean field ordinary differential equation which results from assuming the system is "homogeneously mixing." Specifically the answer to the question is no if the ODE has a single attracting fixed point but yes if there is more than one stable fixed point or periodic orbits. We will present theorems, conjectures, and videotaped simulations in support of this picture.