In this talk, I present several examples illustrating how math can be used to prove global dynamics from assumed local dynamics between individuals. The examples are primarily drawn from my own work and that of collaborators.
The first example concerns large-scale transitions of thought from one extreme viewpoint to another within a close-knit community or society. In the minimalistic model that we study, we find surprising behavior. For example, increasing the resolve of moderate thinkers to remain moderate can actually make a population more vulnerable to extremist thinking. A second project studies the relationship between the spread of epidemics and the structure of the underlying social network. We show that we can learn something about the topology of historical social networks simply by studying how diseases expanded across them. Finally, the third project that I discuss regards the general problem of desynchronizing consumption of a shared resource. I propose a general solution and prove that it does indeed achieve stable asynchrony.
I close with a speculative look toward psychological foundations for fields that depend on an understanding of human cognition and behavior.