Many modern scientific, engineering, and social problems are amenable to analysis through the lens of collective behavior: many elements interact according to local rules to create complex phenomena. Efforts to directly model such systems are inhibited by their scale, nonlinear and stochastic behaviors, and the cost of obtaining even partial and noisy observations. However, high-level models, focusing on description at the level of qualia, have often succeeded where bottom-up models have not. Activity of individual neurons in the brain, for example, combine to encode representations of complex stimuli or drive behavior, and thus the state of the population of neurons can be usefully parameterized and modeled in terms of that stimulus or behavior. But, what can we do when we don't know or have access to such correlates?
In this talk I will discuss tools for characterizing and studying collective behaviors, built using techniques from computational, stochastic, and algebraic topology. I will discuss how to detect, or falsify hypotheses about, organizing principles using observations of a system. Then, I will discuss ongoing research aimed at rigorously assigning semantics to high-level activity in complex systems, as well as future plans to apply these tools to the question of how neural networks encode and learn nonlinear coordinate systems.