# Tim Boykett

Photo from Alex Davies, Austrian sugar overdose at Ottensheim main square.

At the university I am involved in the Algebra group based around Prof Guenter Pilz, who are principally involved in researches in the field of nearrings. My interests have to do with the relations between theoretical computer science and abstract algebra.

Publications and Resources

Mathematical Family Trees

Current projects include:

• I am working as an assistant professor at the Kepler university for a semester, so I have decided to keep track of the developments and try not to get too lost by having a blog. So if you are bored and want to read some badly formatted live mathematical thoughts, come over. If you want some contest, try a real mathematical blog like Peter Cameron's here.

• Polynomials over lattices. When are all order preserving functions on a lattice polynomial? This is known as the order-polynomial completeness problem. Furthermore, if not all functions are polynomials in a given lattice, how can we tell whether a given function is polynomial?

• Group Automata. If the state set of an automaton is given group structure and the mappings of the input alphabet have certain relations to that group, what can we say about the automata from a group theoretical or nearring theoretical perspective? In general we are interested in specific forms of the Cerny Conjecture.

• Lattice and order arithmetic. Can we use permutation group tools to investigate the size of free distributive lattices?

• An investigation of representations of one dimensional cellular automata, based upon their language structures and defects in that structure. This project has been funded by the Linzer Innovationstopf in 2004.

• Strong homomorphisms of labelled graphs. In particular ways to determine whether two labelled graphs generate the same biinfinite language. Preliminary results with Yao-Kun Wu in Shanghai point to some positive possibilities.

• Pregeometry derived from certain graphs: Cahill and colleagues have determined a class of graphs and postulate that they have a strong 3-sphere global structure. I am working upon developing visualisations of these as a part of a LinzExport project.

• Difference families from fixed point free automorphism groups of groups.

• Difference families on nonassociative algebras (e.g. loops and quasigroups). It appears that we obtain nothing new by this approach. In fact we can say that all difference family structures, without requiring associativity of addition, can be obtained from groups.

• Central groupoids: obtaining exhaustive lists of such objects using fast and efficient combinatorial algorithms. See paper in Publications.

• Vector spaces with 0-1 basis. Can we characterise the matrices of which such spaces are the NullSpaces? This is related closely to the next area iof investigation.

• Conservation laws in cellular automata. Is the space of conserved quantities of a reversible (or even irreversible CA) easy to describe? Results indicate that there is a connection to the congruence structure of the associated binary algebra. See publication with Kari and Taati.

• Languages of CA. In particular the fixed point languages of reversible CA. This seems to be important in obtaining computational structure in CA, see the proof of the universality of rule 110.

So what does all this mean? Here's a list of keywords that might begin to explain what it is that I do.

• Applied Abstract Algebra and Theoretical Computer Science, using algebraic knowledge to analyse computational problems and structures.

• A situationist form of theatre, a theatre without actors, only facilitators.

• Biomechanics, an approach to ourselves as whole bodies, anti-dualistic.

• Music should be either fully automated or fully improvised.

• Reversible Computation; to structure computation to make it more comprehendable.

Email tim.boykett (at) algebra.uni-linz.ac.at

Also heavily involved in Time's Up, a laboratory for the composition of experimental situations.

There are some music projects that are on hold: IZLAZ and High Speed Lady Died.

Updated: 17 October 2008

Mathematicians are machines that turn coffee into theorems