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.
Semantic foraging in memory is another nice example: concepts in memory can be represented spatially as locations in multidimensional space, and the route we travel in that 'space' has a lot in common with the optimal foraging movements animals adopt.
Dewalque, A. Brentano and the parts of the mental: a mereological approach to phenomenal intentionality. Phenomenology and the Cognitive Sciences, September 2013, Volume 12, Issue 3, pp 447-464
The book Theories of Meaningfulness of Louis Narens deals with Erlanger program, which is the connection between group theory and projective geometry.