How has geometry been applied in cognitive science? What areas of geometry are used in psychology/cognitive science/neuroscience? Are the applications of a sophisticated nature or superficial?
Here are a few off the top of my head from neuroscience:


neural activity may primarily exist on low dimensional attractors.
reconstructing PET signal origins from emitted gamma rays
It's widely believed our brains are gyrencephalic (wrinkly) to maximize surface area.
Various distance metrics (Euclidean, Mahalanobis) are common tools for clustering data, for example spike sorting.
Neuron morphology (shape) follows function.
Microgeometry of objects is important for tactile texture perception.
Bat echolocation and fish electrosensation are limited in range due to the inverse square law.