The concept of topological persistence defined and computed in recent work by Edelsbrunner et al. opens the door to the construction of hierarchies of progressively simpler Morse complexes. Such hierarchies are related to density modification methods common in the reconstruction of protein structures from electron densities. In this project, we study the topological, combinatorial, and computational properties of Morse complexes defined over triangulated 2- and 3-manifolds, and of the hierarchies of such complexes.

For 2-manifolds, our theoretical understanding is fairly complete, and we pursue the implementation of an algorithm that combines the methods of topological and geometric simplification. The resulting software is expected to freely allow for graded local simplification based on viewpoints or foci of interest. The use of topological simplification controls the degradation of the topological structure as expressed by the critical points and their dependences.

For 3-manifolds, we are in the process of formulating and simultaneously implementing the algorithm for constructing and topologically simplifying Morse complexes. The difficulties we face in three dimensions are significantly more severe than in two dimensions. The complexes have more complicated topological structure, they are more difficult to construct, and they are more cumbersome to visualize. Nevertheless, we expect that the software will be at least as useful as the two-dimensional one. We plan to experiment with using readily visualized substructures, such as the unstable 1-manifolds to identify the basic structure of electron densities, etc., to enhance reconstructions from inherently three-dimensional data.