Artur Schäfer

## Contact:

### University Address:

Mathematical Institute,

University of St. Andrews,

North Haugh,

St. Andrews,

Fife,

Scotland,

KY16 9SS

artur@mcs.st-and.ac.uk

## My Position

Currently, I am a teaching fellow at the University of Dundee and a PhD student at the University of St. Andrews jointly supervised by Prof. Dr. Nik Ruskuc and Prof. Dr. Peter J. Cameron.## Research Interests

In general, I am interested in algebra, group theory and representation theory, combinatorics, algebraic graph theory, semigroup theory and related things.

Currently, I am working on synchronizing permutation groups and semigroups, which relates to the research of graph endomorphisms. However, by investigating the graph endomorphisms of the Hamming graph I moved to describing Latin hypercuboids of class r.

##### Background on synchronizing groups

A semigroup of transformations of an n-element set is called synchronizing if it contains a transformation of rank 1 (size of its image). Interest in the topic stems originally from the 45-year-old (and still open) Černý conjecture, which asserts that a synchronizing semigroup S contains a rank 1 element which is a word of length at most (n-1)^2 in the generators of S.

It is known that a semigroup is non-synchronizing if and only if it is contained in the endomorphism monoid of a simple non-null graph. Graph endomorphisms have traditionally been important in constraint satisfaction problems, and some of the techniques developed there may be of use, here.

Recent research on synchronizing semigroups focuses on semigroups generated by a permutation group G and a singular transformation f. The group G is called synchronizing, if the semigroup generated by G and f is synchronizing for all singular transformations f. It is known that a synchronizing group is primitive, and a 2-transitive group is synchronizing, but neither of these implications reverse.

## Publications

### In Preparation

- Embeddings of Latin Hypercuboids of Class R, see abstract
- Extensions of Latin Hypercuboids of Class R and Evan's Conjecture
- Normalizing Semigroups and their Strong G-Decompositions, see abstract

### Papers

#### 2016

- Endomorphisms of Cuboidal Hamming Graphs, Latin Hypercuboids of Class r, and Mixed MDS Codes http://arxiv.org/abs/1602.05515, (2016), submitted
- Endomorphisms of The Hamming Graph and Related Graphs, http://arxiv.org/abs/1602.02186, (2016), submitted
- Generating Sets of the Kernel Graph and the Inverse Problem in Synchronization Theory, http://arxiv.org/abs/1601.04295, (2016), submitted

#### 2015

- (with J. Araujo, W. Bentz, P.J. Cameron, G. Royle) Primitive groups, graph endomorphisms and synchronization, http://arxiv.org/abs/1504.01629, (2015), submitted

#### 2014

- (with G. Nebe) A nilpotent non abelian group code, Algebra and Discrete
Mathematics,

Vol.18 Nr 2, pp. 268-273, (2014). pdf

### Other

- Masters Thesis: Two Sided and Abelian Group Ring Codes (2012) pdf
- Bachelor Thesis: Ueber p-Gruppen kleiner Ordnung (German) (2011)

## Curriculum Vitae

Education | ||
---|---|---|

- PhD in Pure Mathematics at St. Andrews University | (Apr. 2013 - Dec. 2015) | |

- MSc in Mathematics at RWTH Aachen University (2 year degree) | (Oct. 2011 - Oct. 2012) | |

- BSc in Mathematics at RWTH Aachen University | (Oct. 2008 - Jun. 2011) | |

- German Gymnasium | (Sep. 1999 - Jun. 2008) | |

Employment | ||

- Teaching Fellow at the University of Dundee | (Sep. 2015 - May 2016) | |

- Tutor at the University of St. Andrews | (Sep. 2013 - Aug. 2015) | |

- Tutor at RWTH Aachen University | (Oct. 2009 - Mar. 2013) | |

- Assessor at tasteMINT at RWTH Aachen University | (Jan. 2011 - Oct. 2012) |