Midlands Graduate School 2017

9 - 13 April, 2017

Graphical Linear Algebra

Pawel Sobocinski

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 concepts such as controllability in a compositional way.

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