I study applied algebraic geometry and tensor networks in statistics, computer science, and quantum information. Information-processing systems described as networks (e.g. Bayesian networks, quantum circuits) in seemingly disparate fields in fact have common mathematical foundations.
They are connected by variations on the graphical modeling language of tensor networks, or more generally monoidal categories with various additional properties. Basic questions about each type of information-processing system (such as what probability distributions or quantum states can be represented, or what word problems can be solved efficiently) quickly become interesting problems in shared algebraic geometry, representation theory, polyhedral geometry, and category theory.