The lack of national studies of the health effects of long-term exposure to ambient PM and its chemical components determining the PM toxicity represents a major evidence gap for the implementation of more effective air quality interventions. The US Environmental Protection Agency (EPA) is calling for research to explain heterogeneity in health responses to air pollutants that might be explained by the compositional differences in the pollution mixtures or sources or other factors.

We have developed Bayesian spatially varying coefficient regression models to estimate long-term effects of PM_2.5 on mortality while identifying the chemical composition that modifies the health effects. We will use spatio-temporal variation in health outcomes and exposure to estimate: 1) spatially varying health risks associated with long-term exposure to PM_2.5; and; 2) effect modification by PM_2.5 constituents. Our models will account for spatial misalignment of the data and uncertainty in the estimation of PM_2.5 chemical components.

We will apply our model to the Medicare Cohort Air Pollution Study (MCAPS) which includes 7.9 million Medicare enrollees followed for the period of 2000-2006 in the Eastern part of the US. We will use PM_2.5 data from 518 monitoring stations and PM_2.5 chemical components data from 241 monitors located in the Eastern region of the US. 

Modular curves as moduli spaces of elliptic curves with some additional structures provide an important tool for studying arithmetic of elliptic curves. In this talk we will explain how rational points of modular curves can be applied to the problem of determining the torsion subgroups and ranks of elliptic curves over number fields.