## Graphical Linear Algebra

Lecturer: Paweł Sobociński (University of Southampton). About the lecturer: Pawel Sobocinski obtained his PhD from BRICS, University of Aarhus, in 2004. After postdocs at the Universities Pisa, Paris VII and Cambridge, he has been on the faculty of University of Southampton since 2007. His research has concentrated on concurrency theory, graph rewriting, compositional modelling and verification, and applications of category theory in computer science. He is a member of the editorial board of Mathematical Structures in Computer Science, IFIP WG 1.3 on Foundations of System Specification and IFIP WG 2.2 on Formal Description of Programming Concepts. Since 2015 he has been writing the Graphical Linear Algebra blog, aimed at a general audience.

Course summary: In recent years there has been an explosion in the use of string diagrams - a graphical syntax for the arrows of higher dimensional categories - in various fields, including computer science, physics and engineering. Symmetric monoidal categories have been especially prevalent and are now being used as a mathematical domain for the compositional analysis of the topology and computations of Petri nets, electrical circuits and signal flow graphs, quantum circuits and quantum information, amongst many other applications.

This course will consist of an introduction to symmetric monoidal categories and their diagrammatic syntax, and focus on one particular application: elementary finite dimensional linear algebra. We will rediscover the natural numbers, integers, rational numbers, matrices and linear spaces as certain diagrams, arising through the interaction of two very important mathematical structures: bimonoids and Frobenius monoids. By building up our diagrammatic lexicon, we will arrive at diagrams that amount to a compositional algebra of signal flow graphs, a foundation model of computation of electric engineering and control theory. Indeed, we will see that the theory allows us to approach classical concepts such as controllability in a compositional way.

Slides to the lecture can be found here.

Assignment: You can find the problem set here. Please send the solutions to the address provided on the problem sheet by Friday, February 10. To get grate N you need to solve N-1 problems, for N=3,4,5.