Cities are now central to addressing global changes, ranging from climate change to economic resilience. There is a growing concern of how to measure and quantify urban phenomena, and one of the biggest challenges in quantifying different aspects of cities and creating meaningful indicators lie in our ability to extract relevant features that characterize the topological and spatial patterns of urban form. Many different models that can reproduce large-scale statistical properties observed in systems of streets have been proposed, from spatial random graphs to economical models of network growth. However, existing models fail to capture the diversity observed in street networks around the world. The increased availability of street network datasets and advancements in deep learning models present a new opportunity to create more accurate and flexible models of urban street networks, as well as capture important characteristics that could be used in downstream tasks.  We propose a simple approach called Convolutional-PCA (ConvPCA) for both creating low-dimensional representations of street networks that can be used for street network classification and other downstream tasks, as well as a generating new street networks that preserve visual and statistical similarity to observed street networks.

Link to the preprint

Configuration spaces of points in Euclidean space or on a manifold are well studied in algebraic topology. But what if the points have some positive thickness? This is a natural setting from the point of view of physics, since this the energy landscape of a hard-spheres system. Such systems are observed experimentally to go through phase transitions, but little is known mathematically.

In this talk, I will focus on two special cases where we have started to learn some things about the homology: (1) hard disks in an infinite strip, and (2) hard squares in a square or rectangle. We will discuss some theorems and conjectures, and also some computational results. We suggest definitions for "homological solid, liquid, and gas" regimes based on what we have learned so far.

This is joint work with Hannah Alpert, Ulrich Bauer, Robert MacPherson, and Kelly Spendlove.