Los Angeles, along with many major cities, has a huge homeless problem. Right now there are over 50,000 homeless people in Los Angeles, and the problem is very complicated. There is little understanding as to the mechanisms that yield such high rates of homelessness and how individual characteristic traits influence the outcome of homeless individuals. For over two years now, I've been studying this problem from a number of different angles, which is leading to three rather different research problems: firstly by combining data such as median household income and various proxies for commerce that can be obtained for distinct geographic regions, along with annual homeless counts imputed by the Los Angeles Homeless Services Authority, it seems possible to predict, at least approximately, how the homeless population will evolve on small spatial scales known as census tracts. The main techniques used have included: topic modelling and artificial neural networks (yielding predictions of homeless population change that are much better than pure chance). Lastly, from a more theoretical perspective, I have been working to develop a partial differential equation that models the evolution of the homeless population density. The equation is a nonlocal, nonlinear reaction-advection-diffusion equation. I have been working to prove the well-posedness of the model, ensuring that as a mathematical model it yields physically sensible results (finite total populations, non-negativity of the population -- all the stuff that really should be true if the model equation can approximate reality in a meaningful way).
