Historically, a fundamental paradigm in systems neuroscience for elucidating brain function has been to identify "single-neuron receptive fields." From this point of view, a functional description of a neural system is equated with a list of single-cell receptive fields, i.e., a description of the trial-averaged activity of each cell, conditioned on one or more sensory or behavioral variables. However, in light of the on-going proliferation of exciting multi-neuronal recording techniques, there has been a push from the theoretical and computational communities to develop techniques and ideas which can leverage population recordings to provide insightful new descriptions of brain function. In this seminar, I will describe two synergistic research projects (one by myself and Ryan Low, the other by Rishi Chaudhuri and Ila Fiete), organized around the idea of studying neural circuits in terms of the "intrinsic geometry" of their population activity. Specifically, I will give a detailed description of the manifold learning paradigm from machine learning and data science, and how we have applied novel techniques of this kind to infer properties of "cognitive maps" represented in the neural activity recorded from the hippocampus and thalamus of rats navigating in real and abstract spaces (as described in the primer by Ila Fiete). These techniques provide a language for characterizing neural population activity in a concise and unsupervised manner, and more importantly, we will demonstrate their potential for discovering novel properties of neural systems.