This talk is a preview of a forthcoming book in the Applied Category Theory series of Cambridge University Press. The book uses basic category theory to describe all the central concepts and prove the main theorems of theoretical computer science. Category theory, which works with functions, processes, and structures, is uniquely qualified to present the fundamental results of theoretical computer science. We will meet some of the deepest ideas and theorems of modern computers and mathematics, e.g., Turing machines, unsolvable problems, the P=NP question, Kurt Gödel's incompleteness theorem, intractable problems, cryptographic protocols, Alan Turing's Halting problem, and much more. I will report on new things I learned about theoretical computer science and category theory while working on this project.

I’ll discuss a perspective on scientific computing centered around the modeling framework. Applied Category Theory gives us the tools to represent these modeling frameworks and build powerful software for representing, manipulating, and solving complex systems in science and engineering. This work builds on decorated cospans, and categorical logic for knowledge representation. I’ll end with some open questions involving dynamical systems.

In physics, wave phenomena are modeled by integrable systems of PDEs, the solutions of which are points in an infinite-dimensional Grassmannian. In this talk, I will show that the Grassmannian and certain other physically-relevant moduli spaces, time permitting, correspond to instances of a particular class of infinity toposes. I will conclude with some of the mathematical and physical implications.

I will give an intuitive introduction to representation and reasoning with hierarchical ontologies. Most of the talk will be done through a notes application, but I will also show a prototype graphical interface for creating ontologies.

This talk will give an informal and accessible introduction to the use of category theory for modeling logical (or ontological) information. Topics will include the relationship between categories and graphs, logical structures in a category and functorial semantics. If time allows, I will also say a bit about how these approaches generalize to other structures like monoidal categories and operads.