May 18-19, 2018

Groups in Galway has been running on an annual basis since 1978. The scope of the conference covers all areas of group theory, applications, and related fields. All who are interested are invited to attend.

- Rob Craigen (Manitoba)
- Ronan Egan (Rijeka)
- Bettina Eick (Braunschweig)
- Daniel Horsley (Monash)
- Eoin Long (Oxford)
- Maura Paterson (Birbeck)
- Götz Pfeiffer (Galway)
- John Sheekey (UCD)
- Ann Trenk (Wellesley)

Move the cursor over a title for the talk's abstract. Click a speaker's name for pdf of their talk if available.

09.45‒10.00 Opening remarks by Professor Martin L. Newell

10.00‒10.45
John Sheekey
Semifields and subspaces of matrices over finite fields
Finite *semifields* are nonassociative division algebras over a finite field. They have been studied since the early 1900s, with Dickson constructing the first nontrivial examples. The theory was further developed by Albert and Knuth in the 1960s.
Semifields correspond to various objects in both linear algebra and finite geometry. They correspond for example to a certain class of projective plane; certain types of *spreads* in a projective space $\mathrm{PG}(2n-1,q)$; nonsingular tensors in $\mathbb{F}_q^n\otimes \mathbb{F}_q^n\otimes \mathbb{F}_q^n$; and $n$-dimensional subspaces of $n\times n$ matrices in which every nonzero element is invertible. The last of these is of particular recent interest, as this is a special case of a *maximum rank distance (MRD) code*, a hot topic due to potential applications in network coding.
We will provide an overview for these connections, the known constructions and classifications, and present some open problems in the area.

10.45‒11.30 Coffee/tea

12.30‒13.15 Maura Paterson Reciprocally-weighted external difference families Let $G$ be a finite abelian group of order $n$. An $(n,k,\lambda)$ $m$-External Difference Family (EDF) is a collection of $m$ disjoint subsets of $G$ each of size $k$, with the property that each nonzero group element occurs precisely $\lambda$ times as a difference between group elements in two different subsets from the collection. A reciprocally-weight EDF (RWEDF) is a generalisation of an EDF in which the subsets may have different sizes, and the differences are counted with a weighting given by the reciprocal of the set sizes. These were proposed in connection with the study of weak algebraic manipulation detection codes. In this talk I will motivate the definition of RWEDFs before discussing recent results including the construction of infinite families of RWEDFs.

13.15‒14.30 Lunch

14.30‒15.15
Daniel Horsley
Symmetric coverings and the Bruck-Ryser-Chowla theorem
A $(v,k,\lambda)$-*block design* is a collection of $k$-element subsets, called *blocks*, of a $v$-set of *points* such that each pair of points appears together in exactly $\lambda$ blocks. A block design is *symmetric* if it has exactly as many blocks as points. The Bruck-Ryser-Chowla theorem famously establishes the nonexistence of various symmetric block designs, including projective planes.
In this talk I will discuss generalisations of the Bruck-Ryser-Chowla theorem to the setting of *coverings* where, instead of demanding that each pair of points appears together in *exactly* some fixed number of blocks, we only require that each pair of points appear together in *at least* some fixed number of blocks. The proofs of these generalisations exploit the structure of $XX^T$, where $X$ is the incidence matrix of a putative covering. In some cases it is sufficient to examine the determinants of these matrices. In others, arguments concerning rational congruence of matrices and making use of Hasse-Minkowski invariants are required.

16.15‒17.00 Coffee/tea

19.00 Conference Dinner: Il Vicolo Restaurant

10.00‒10.45 Ronan Egan Using groups to construct combinatorial structures and codes In this talk I will discuss methods of constructing combinatorial structures such as block designs from finite groups and using orbit matrices. I will also describe recent and ongoing work where generalized versions of orbit matrices are constructed from Hadamard matrices and weighing matrices using their permutation automorphism groups. This leads to the construction of self-orthogonal, self-dual, optimal and near optimal codes over finite fields.

10.45‒11.15 Coffee/tea

12.15‒13.00
Ann Trenk
The distinguishing chromatic number and NG-graphs
Albertson and Collins introduced the distinguishing number of a graph as the minimum number of colors needed to color the vertices so that the only automorphism of the graph which preserves colors is the identity. The *distinguishing chromatic number*, $\chi_D(G)$, is defined similarly, except that the coloring must also be proper, that is, adjacent vertices must get different colors. In this talk we discuss results about $\chi_D(G)$ including characterization theorems and connections to the Nordhaus-Gaddum inequality for graphs.

Galway is a small city. It takes about 15 minutes to walk from the Coach Station or Railway Station to the university (Google maps direction).

NUI Galway has a number of pay-and-display parking places for visitors. Cars parked in other spaces on the NUI Galway campus and not displaying a valid parking permit will be clamped.

- The Westwood House Hotel,
091 521442
- The House Hotel,
091 538900
- Bologna B&B, 091 523792
- Aneesha B&B, 091 524250
- Ashgrove House B&B, 091 581291
- Villanova B&B, 091 524849
- Coolavalla B&B, 091 522415
- Rosgal B&B, 091 524723
- De Sota B&B, 091 585064

See also Discover Ireland.

The organizers are Kevin Jennings, Tobias Rossmann, and Dane Flannery.

Groups in Galway 2018 is supported by

- The Office of the Registrar and Deputy President, NUI Galway
- The Irish Mathematical Society