### Paweł Sobociński

School of Electronics and Computer Science

University of Southampton

United Kingdom

## Graphical linear algebra

#### Abstract

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.

#### Course materials

- P. Sobocinski. Graphical linear algebra. Slides from the EWSCS 2017 course.
- Videos from the lectures (unedited, large files).
- P. Sobocinski. Graphical linear algebra blog. [blog on Pawel's website]
- F. Bonchi, P. Sobocinski, F. Zanasi. The calculus of signal flow
diagrams I: linear relations on streams.
*Inf. and Comput.*, v. 252, pp. 2-29, 2017. [doi link] - F. Bonchi, P. Sobocinski, F. Zanasi. Full abstraction for signal
flow graphs. In
*Proc. of 42nd Ann. ACM SIGPLAN-SIGACT Symp. on Principles of Programming Languages, POPL 2015*, pp. 515-526, ACM, 2015. [doi link] - F. Bonchi, P. Sobocinski, F. Zanasi. Lawvere theories as composed
PROPs. In I. Hasuo, ed.,
*Proc. of 13th Int. Wksh. on Coalgebraic Methods in Computer Science, CMCS 2016*, v. 9608 of*Lect. Notes in Comput. Sci.*, pp. 11-32. Springer, 2016. [doi link] - B. Fong, P. Rapisarda, P. Sobocinski. A categorical approach to
open and interconnected dynamical systems. In
*Proc. of 31st Ann. ACM/IEEE Symp. on Logic in Computer Science, LICS 2016*, pp. 495-504, 2016. [doi link] - F. Bonchi, P. Sobocinski, F. Zanasi. Interacting Hopf
algebras.
*J. of Pure and Appl. Alg.*, v. 221, n. 1, pp. 144-184, 2017. [doi link]

Last changed **
March 19, 2017 0:25 Europe/Helsinki (GMT +02:00)**
by
local organizers, ewscs17(at)cs.ioc.ee

EWSCS'17 page:
http://cs.ioc.ee/ewscs/2017/