Human cognitive and social systems are perhaps the final frontier for mathematical scientific theory.  While well-known methods of statistical physics and scientific computation are useful as entry points to a fast growing body of data, critical formal innovations are also necessary that describe these systems in their own terms.

Cities, in particular, provide a rich, novel and increasingly empirically available set of problems where open-ended adaptation at different scales builds large-scale socioeconomic networks in interaction with infrastructural systems embedded in space and time.

In this talk, I will describe the emerging mathematics of cities.  The crucial starting element deals with the quantification of the general properties of urban areas, which become apparent through scaling analysis and associated statistics. Based on a set of regularities that I will demonstrate empirically, I then build a mean-field theory that derives the scaling of many socioeconomic, infrastructural and physical properties of cities and reveals the basic trade-offs involved in these systems.

I will then demonstrate how the detailed fabric of cities can be understood through a process of spatial selection and show how the complexity of explanations at the local level (groups, neighborhoods) can be quantified in units of information relative to more coarse-grained descriptions, in a way analogous to renormalization group transformations in statistical physics.

I will end with some general (speculative) thoughts on the convergence between methods of statistical physics, the mathematics of selection and basic aspects of human social behavior and cognition that may provide a path to a more integrated quantitative understanding of complex adaptive systems.