DCUCD Discrete Mathematics Seminar Series

Speaker: Patrick Browne (TUS)

Title: Erdős–Ko–Rado type problems in root systems 

Abstract: Given a Lie algebra, two roots are said to be strongly orthogonal if neither their sum nor difference is a root. In this talk, we investigate sets of mutually strongly orthogonal roots. In particular, those such that any two such sets have the property that the difference between their sums can itself be expressed as the sum of a strongly orthogonal set of roots. We discuss this property and its relationship to Erdős–Ko–Rado type problems and finally discuss applications in terms of the existence of finite projective planes of certain orders. This is joint work with Qëndrim R. Gashi, University of Prishtina and  P Ó Catháin at DCU. 

Speaker: Lukas Klawuhn (Paderborn)

Title: Why everyone should know association schemes: An introduction to Delsarte Theory

Abstract: Interesting combinatorial structures can often be characterised as special subsets of association schemes. In his PhD thesis, Philippe Delsarte developed powerful linear programming techniques to prove non-existence and uniqueness results for such structures. Ideas of this type were fundamental in the work for which Marina Viazovska was awarded the Fields medal in 2022. This talk will begin with an overview of Delsarte theory. 

The association scheme of the symmetric group is well known. We will introduce it, and then study generalised permutations. They act on the set {1,2,...,n}, whose elements are coloured with one of r possible colours. We consider different notions of transitivity and interpret these algebraically in the appropriate association scheme . We will give existence results showing that there exist transitive sets of generalised permutations that are small compared to the size of the group. Many of these results extend results previously known for the symmetric group.

No particular knowledge of association schemes will be required to appreciate this talk. 

Speaker: Alena Ernst (Paderborn)

Title: Designs in finite general linear groups 

Abstract: It is known that the notion of a transitive subgroup of a permutation group G extends naturally to subsets of G.  It turns out that transitive subsets of the symmetric group give a combinatorial interpretation of the rather algebraic Delsarte T-designs in the corresponding association scheme. In this talk we give an overview of results known for the symmetric group and discuss a q-analog setting: we characterise Delsarte T-designs in the finite general linear group GL(n,q) in terms of subsets of GL(n,q) acting transitively on flag-like structures. 

Speaker: Daniel Hawtin (Rijeka)

Title:  Packings of spreads and designs in projective spaces

Abstract: A (line-)spread of a projective space is a set of lines of the space that induce a partition of the point set. One generalisation of this is the q-analogue of a design. A t-fold packing of a projective space is a collection P of spreads such that each line is contained in precisely t of the spreads in P. Similar concepts have been investigated for designs. We will discuss some of the history of t-fold packings, and their generalisations, and some applications in graph theory and coding theory. We also present some recent results regarding (q-1)-fold packings in odd dimensional projective spaces over GF(q), where q is a power of 2, and in infinite dimensional projective spaces.

Speaker: Andrew Fulcher (UCD)

Title: The cyclic flats of L-polymatroids

Abstract: In recent years, q-polymatroids have drawn interest because of their connection with rank-metric codes. For a special class of q-polymatroids called q-matroids, the fundamental notion of a cyclic flat has been developed as a way to identify the key structural features of a q-matroid. In this talk, we will see a generalisation of the definition of a cyclic flat that can apply to q-polymatroids, as well as a further generalisation, L-polymatroids. The cyclic flats of an L-polymatroid is essentially a reduction of the data of an L-polymatroid such that the L-polymatroid can be retrieved from its cyclic flats. As such, in matroid theory, cyclic flats have been used to characterise numerous invariants.

Speaker: Padraig Ó Catháin (DCU)

Title: Sequencing of Steiner Triple Systems

Abstract: A Steiner Triple System (STS) on the set {1, 2, ..., v} is a collection of blocks (subsets) of size three, such that each pair is contained in a unique block. Over the past few years, applications to problems in data storage suggested the problem of finding STSs such that no block is contained in a short interval of consecutive integers. With Daniel Horsley, we found the optimal bounds for this question (up to some log factors). In this talk, I'll give all the necessary background and explain why a naive application of the Lovasz Local Lemma fails for this problem, but a two-step version works surprisingly well.