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

Genome biology approaches have made enormous contributions to our understanding of biological rhythms, particularly in identifying outputs of the clock, including RNAs, proteins, and metabolites, whose abundance oscillates throughout the day. These methods hold significant promise for future discovery, particularly when combined with computational modeling. However, genome-scale experiments are costly and laborious, yielding "big data" that are conceptually and statistically difficult to analyze. There is no obvious consensus regarding design or analysis. Here we discuss the relevant technical considerations to generate reproducible, statistically sound, and broadly useful genome-scale data. Rather than suggest a set of rigid rules, we aim to codify principles by which investigators, reviewers, and readers of the primary literature can evaluate the suitability of different experimental designs for measuring different aspects of biological rhythms. We introduce CircaInSilico, a web-based application for generating synthetic genome biology data to benchmark statistical methods for studying biological rhythms. Finally, we discuss several unmet analytical needs, including applications to clinical medicine, and suggest productive avenues to address them.