I will discuss some common mathematical structures arising in information-processing networks in computer science, statistics and machine learning, and quantum information and many-body systems. These seemingly disparate fields are connected by variations on the graphical modeling language of tensor networks, or more generally monoidal categories with various additional properties. Tools from algebraic geometry, representation theory, and category theory have recently been applied to problems arising from such networks. 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, and category theory. The result has been new insights into problems ranging from recognizing images to classifying quantum phases of matter and interesting challenges in pure mathematics.