We present an algorithm for the simplification of density maps in 3D. We assume that an input function specified at vertices of a triangulation and linearly interpolated within the tetrahedra. The fundamental operation in the simplification procedure is a topology preserving edge contraction. We describe simple and local conditions that can be used to check if an edge contraction preserves topology. These conditions are derived from the generic link conditions for 3-complexes described by Dey et. al. Besides providing a good approximation of the density map, the algorithm also aims to generate a mesh with good quality elements. We achieve this additional goal using a novel extension of the quadric error metric. We visualize isosurfaces extracted from the simplified density maps and use them along with an approximation error computation to evaluate the algorithm. We also develop validation tools to test the correctness of the simplified model and perform statistical measurements on the critical points of the density map to study the relationship between geometric and topological simplification.

We introduce local and global comparison measures for a collection of $k \leq d$ real-valued smooth functions on a common $d$-dimensional Riemannian manifold. For $k = d = 2$ we relate the measures to the set of critical points of one function restricted to the level sets of the other. The definition of the measures extends to piecewise linear functions for which they are easy to compute. The computation of the measures forms the centerpiece of a software tool which we use to study scientific datasets.

Given a Morse function $f$ over a 2-manifold with or without boundary, the Reeb graph is obtained by contracting the connected components of the level sets to points. We prove tight upper and lower bounds on thenumber of loops in the Reeb graph that depend on the genus, the number of boundary components,and whether or not the 2-manifold is orientable. We also give an algorithm that constructs the Reeb graph in time O($n \log n$), where $n$ is the number of edges in the triangulation used to represent the 2-manifold and the Morse function.

Here we present the results of the NSF-funded Workshop on Topology, which met on June 11 and 12 in Miami Beach,Florida. This report identifies important problems involving both computation and topology