Abstract: Data science represents some of the biggest opportunities and challenges to science today. However, many of the algorithms underlying machine learning are not well understood. In this talk I will discuss a number of ways that mathematical analysis can help in understanding these algorithms. In particular, I will discuss how a broad range of tools from modern analysis (including differential equations, variational analysis and probability) can be used to understand crucial questions in machine learning relating to optimization routines, overfitting and reinforcement learning. This represents joint work with a number of statisticians, engineers, and mathematicians. Most of this talk is designed to be accessible to undergraduates.