Confirmed Invited Speakers:
Title: Eigenvalue bounds for the independence number of graph powers and their application to coding theory
Abstract: Spectral graph theory looks at the connection between the eigenvalues of a matrix associated with a graph and the corresponding structures of a graph. In this talk we will show how spectral methods provide a handy tool for obtaining results concerning the structure of graphs, and also how these results are a powerful tool to be used in other fields. In particular, we will derive sharp eigenvalue bounds on an NP-hard distance-type graph parameter: the k-independence number of a graph (or equivalently, the independence number of a graph power). We will see how to use polynomials and mixed integer linear programming in order to optimize such bounds. Finally, we will illustrate some applications of the new eigenvalue bounds to coding theory.
Title: New directions for direction sets
Abstract: Let q be a power of a prime p, and let f be a function from GF(q)^m to GF(q). The graph of f is the set of points in the affine space AG(m+1,q) of the form (x,f(x)), where x ranges over GF(q)^m. The directions determined by the graph of f are the points at infinity corresponding to the slopes of lines connecting pairs of points on the graph.
In this talk, we will show how properties of the set of determined directions yield information about the function. We will explain how the graphs of functions and their associated direction sets can be used to construct various extremal objects in finite geometry, for instance, the smallest point sets in the plane that meet every line in at least t points, or the smallest multisets in space that meet each line in a number of points divisible by p. We will also discuss applications related to an Erdős-Ko-Rado type problem.
Title: Factored lifts of graphs and their spectra: The Abelian and non-Abelian cases
Abstract: In this talk, we introduce the concept of factored lift, associated with a combined
voltage graph, as a generalization of the lift graph (or voltage graph). We present a new method for computing all the eigenvalues and eigenspaces of factored lifts in two cases, when the underlying group is Abelian and non-Abelian.
Title: Quasirandomness through lenses of combinatorial limits
Abstract: A combinatorial object is said to be quasirandom if it resembles a random object in a certain robust sense. The notion of quasirandom graphs, which was developed in the work of Rödl, Thomason, Chung, Graham and Wilson in 1980s, is particularly robust as several different properties of truly random graphs, e.g., subgraph density, edge distribution and spectral properties, are satisfied by a large graph if and only if one of them is.
We will present classical and recent results on quasirandomness of different combinatorial objects, in particular, graphs, directed graphs, permutations, hypergraphs and Latin squares. We will cast the results using the language of the theory of combinatorial limits, which we introduce during the talk, and demonstrate how analytic methods provided by the theory of combinatorial limits can be used to obtain results concerning quasirandom combinatorial objects.
Title: The linear dimension of permutation groups
Abstract: From a linear group G acting on a vector space V we obtain transitive permutation representations of G, the orbits on vectors. One may ask to what extent these permutation representations determine the linear representation. Conversely, given a transitive permutation group G, we may ask which linear representations of G yield an orbit of G on vectors that is equivalent to the permutation action of G. The `linear dimension' of a permutation group G is the smallest dimension of such a linear representation. One can see this as the most efficient way to linearise the action of G. This was defined recently by D’Alconzo and Di Scala and is inspired by cryptographic questions. In this talk I will give some overview of the recent results and explain how this leads to fundamental and difficult questions in the representation theory of finite groups.
