# Wednesday, 7 December 2016: NODES Seminar, Durham

General meeting of all members of NODES and their PhD students.

## Location:

Pemberton Building, room PG21

## Timetable:

14.00-14.45 Max Gadouleau (Durham, CS): An introduction to the theory of Finite Dynamical Systems

14.45-15.30 Tea/Coffee break

15.30-16.15 Evgenios Kakariadis (Newcastle, Maths): Operator algebras associated with subshifts

Max Gadouleau: An introduction to the theory of Finite Dynamical System

Abstract: We are interested in complex networks of interacting entities (such as genes, neurons, persons, computers, etc.), where each entity has a finitely valued state and a function which updates the value of the state. Since entities influence each other, this local update function depends on the states of some of the entities. Such a network is called a Finite Dynamical System (FDS). The main problem when studying an FDS is to determine its dynamics given a limited knowledge of it; for instance, we may only know the interaction graph, i.e. which entities influence each other. In this talk, we will review some of the seminal results in the theory of FDSs and focus on the maximisation of images and periodic points for a given interaction graph.

Evgenios Kakariadis: Operator algebras associated with subshifts

Abstract: A factorial language is characterized by a set of allowable words on d symbols and encodes the allowable operations an automaton performs.

In the late 1990’s Matsumoto constructed a C*-algebra associated to a factorial language, deriving initially his motivation from the work of Cuntz and Krieger. These C*-algebras were then studied in depth in a series of papers.

In a recent work with Shalit we take another look at this context and study factorial languages in terms of classification programmes on nonselfadjoint operator algebras and Arveson’s Programme on the C*-envelope. We investigate two types of nonselfadjoint operator algebras and we show that they completely classify languages: (a) up to the same allowable words, and (b) up to local conjugacy of the quantized dynamics.

In the case of sofic languages the quantized dynamics are encoded through the follower set graph and the two types of nonslefadjoint operator algebras offer classification: (a) up to label isomorphisms, and (b) unlabeled isomorphism. In addition we discover that the C*-algebra fitting Arveson’s Programme is the quotient by the generalized compacts, rather than taking unconditionally all compacts as Matsumoto does. Actually there is a nice dichotomy that depends on the structure of the language.

Nevertheless in the process we accomplish more in different directions. This happens as our case study is carried in the intersection of C*-correspondences, subproduct systems, dynamical systems and subshifts. In this talk we will give the basic steps of our results with some comments on their proofs.

This is joint work with Shalit and with Barrett.

Snapshots of the talks

Coffee break