Invited Speakers:
Simeon Ball, Polytechnic University of Catalunya, Barcelona, Spain
Title: The geometry of error-correcting codes
Abstract: In this talk I will consider various error-correcting codes, including linear, additive and stabiliser codes. It is well known that if one considers the set of columns of a generator matrix of a linear code, then one can consider this set as a set of points in a finite projective space. The parameters of the code then translate over to properties of the point set. In this talk I will consider the geometry of various different types of codes and codes with certain nice properties, like Hermitian self-orthogonality. I will also discuss recent results on additive MDS codes.
Marston Conder, University of Auckland, New Zealand
Tile: Some helpful things I have learnt over my career
Abstract: This will be a highly non-standard talk! As it's a PhD summer school, I thought it could be helpful to offer some advice and insights to students from my last 40+ years in academia. These things will cover choice of research topics and projects, approaches to research problems, experimentation, looking for patterns, publication, participation in conferences, work-life balance, and a few others. I hope that many of them will be valuable.
Misha Muzychuk, Ben-Gurion University of the Negev, Israel
Title: On Jordan schemes
Abstract: In 2003 Peter Cameron introduced the concept of a Jordan scheme and asked whether there exist Jordan schemes which are not symmetrizations of coherent configurations (proper Jordan schemes).
In my talk I'll describe several infinite series of proper Jordan schemes and present first developments in the theory of Jordan schemes - a new class of algebraic-combinatorial objects.
Cheryl E Praeger, The University of Western Australia, Australia
Title: Edge-transitive Cayley graphs and graph quotients
Abstract: The edge-transitivity of a Cayley graph is most easily recognisable if the subgroup of `affine maps' preserving the graph structure is itself edge-transitive. These are the so-called normal edge-transitive Cayley graphs. Each of them determines a set of quotients which are themselves normal edge-transitive Cayley graphs, and which are built from a very restricted family of groups (direct products of simple groups). We address the questions: how much information about the original Cayley graph can we retrieve from this special set of quotients? Can we ever reconstruct the original Cayley graph from them: if so, then how?
Our answers to these questions involve a subgroup determined by the Cayley graph, which has similar properties to the Frattini subgroup of a finite group. I'll discuss this and give some examples. It raises many new questions about Cayley graphs.
Gabriel Verret, University of Auckland, New Zealand
Title: (k, t)-regular graphs
Abstract: A graph is called (k, t)-regular if it is k-regular and the induced subgraph on the neighbourhood of every vertex is t-regular. We are interested in the following question: For which pairs (k, t) does there exist a (k, t)-regular graph? This is a very simple yet interesting question about which little was known. I will discuss previous knowledge as well as some new results obtained with Marston Conder and Jeroen Schillewaert.
Jinxin Zhou, Beijing Jiaotong University, China
Title: On mixed dihedral groups and 2-arc-transitive normal covers of $K_{2^n,2^n}$
Abstract: In this talk, I will introduce the notation of a mixed dihedral group, which is a group $H$ with two disjoint subgroups $X$ and $Y$, each elementary abelian of order $2^n$, such that $H$ is generated by $X\cup Y$, and $H/H'\cong X\times Y$. We will give a graph theoretic characterization of this family of groups, and this is then used to investigate the $2$-arc-transitive normal covers of the `basic' graph $K_{2^n,2^n}$.
