New, “big data” sources allow measurement of city characteristics and outcomevariables at higher collection frequencies and more granular geographic scales thanever before. However, big data will not solve large urban social science questionson its own. Big urban data has the most value for the study of cities when it allowsmeasurement of the previously opaque, or when it can be coupled with exogenous shocksto people or place. We describe a number of new urban data sources and illustrate howthey can be used to improve the study and function of cities. We rst show how GoogleStreet View images can be used to predict income in New York City, suggesting thatsimilar imagery data can be used to map wealth and poverty in previously unmeasuredareas of the developing world. We then discuss how survey techniques can be improved tobetter measure willingness to pay for urban amenities. Finally, we explain how Internetdata is being used to improve the quality of city services.

Bosonic topological insulators (BTI) in three dimensions are symmetry-protected topological phases (SPT) protected by time-reversal and boson number conservation symmetries. BTI in three dimensions were first proposed and classified by the group cohomology theory which suggests two distinct root states, each carrying a Z2 index. Soon after, surface anomalous topological orders were proposed to identify different root states of BTI, which even leads to a new BTI root state beyond the group cohomology classification. In this paper, we propose a universal physical mechanism via vortex-line condensation from a 3d superfluid to achieve all three root states. It naturally produces bulk topological quantum field theory (TQFT) description for each root state. Topologically ordered states on the surface are rigorously derived by placing TQFT on an open manifold, which allows us to explicitly demonstrate the bulk-boundary correspondence. Finally, we generalize the mechanism to ZN symmetries and discuss potential SPT phases beyond the group cohomology classification.

We study the heat kernel expansion of the Laplacian on n-forms defined on a subgraph of a directed complete graph. We derive two expressions for the subgraph heat kernel on 0-forms and compute the coefficients of the expansion. We also obtain the subgraph heat kernel of the Laplacian on 1-forms.