The complexity of the human brain gives the illusion that brain activity is intrinsically high-dimensional. Nonlinear dimensionality-reduction methods such as uniform manifold approximation and t-distributed stochastic neighbor embedding have been used for high-throughput biomedical data. However, they have not been used extensively for brain activity data such as those from functional magnetic resonance imaging (fMRI), primarily due to their inability to maintain dynamic structure. Here we introduce a nonlinear manifold learning method for time-series data-including those from fMRI-called temporal potential of heat-diffusion for affinity-based transition embedding (T-PHATE). In addition to recovering a low-dimensional intrinsic manifold geometry from time-series data, T-PHATE exploits the data's autocorrelative structure to faithfully denoise and unveil dynamic trajectories. We empirically validate T-PHATE on three fMRI datasets, showing that it greatly improves data visualization, classification, and segmentation of the data relative to several other state-of-the-art dimensionality-reduction benchmarks. These improvements suggest many potential applications of T-PHATE to other high-dimensional datasets of temporally diffuse processes.