Effective interoperation between multiple scientific disciplines is crucial to systems engineering. Can the study of interoperability—the working negotiations and hand-offs between theories and models—itself be made into a hard science? Hard sciences are based on mathematics, so this would require a mathematics of interoperability, a mathematics whose subject consists of the bridges and analogies that make data- and model-integration actually work. I propose that category theory serves this purpose exceptionally well.
In this talk, I will give evidence for the above claim, and without assuming the audience has seen any category theory before. I will focus on operads, which offer a framework for various forms of compositionality. In particular, I will discuss how operads model the interconnection of dynamical systems, provide a new method for solving systems of nonlinear equations, and explain how these two issues are connected category-theoretically. Finally, I'll explain how all this fits into a larger mathematical approach to interdisciplinarity.
David Spivak received a BS in mathematics from University of Maryland, College Park, and then a PhD in mathematics from UC Berkeley in 2007; his thesis was in algebraic topology. For the next three years he was a post doc in the math department at the University of Oregon. During this time his focus moved toward applications of category theory in science and engineering. Since 2010, he has continued this research as a post doc, and then research scientist, in the math department at MIT.