Mathematical Research Seminar - Archive
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Datum in ura / Date and time: 9.9.24
(15:00-16:00)
Predavalnica / Location: FAMNIT-MP1
Predavatelj / Lecturer: John Shareshian (Washington University)
Naslov / Title: Chromatic quasisymmetric functions
Vsebina / Abstract:
Richard Stanley initiated the study of the chromatic symmetric function X_G of a finite graph G. This symmetric function encodes more information about G than the well-studied chromatic polynomial, and has been studied closely. Michelle Wachs and I introduced a refined version of X_G. For most graphs G, this refined version is quasisymmetric but not necessarily symmetric. However, X_G is symmetric when G is an indifference graph. Chromatic quasisymmetric functions of indifference graphs are related to previously studied objects from algebraic geometry and representation theory of symmetric groups. After introducing chromatic symmetric and quasisymmetric functions, I will discuss these relations.
Datum in ura / Date and time: 5.9.24
(15:00-16:00)
Predavalnica / Location: FAMNIT-MP1
Predavatelj / Lecturer: Morgan Rodgers (University of Kaiserslautern-Landau)
Naslov / Title: Regular sets in finite polar spaces
Vsebina / Abstract:
A regular set or equitable bipartition in a (finite simple) graph is a set of vertices $Y$ such that there exist constants $a$ and $b$ for which each vertex in $Y$ has $a$ neighbors in $Y$, while each vertex not in $Y$ has $b$ neighbors in $Y$.
Such sets can only exist for graphs which are either biregular or regular. If $Y$ is a regular set, then $a-b$ is an eigenvalue of the adjacency matrix of the corresponding graph. We call $Y$ a regular set of an association scheme if $Y$ is a regular set for all the graphs in the association scheme.
Such sets can only exist for graphs which are either biregular or regular. If $Y$ is a regular set, then $a-b$ is an eigenvalue of the adjacency matrix of the corresponding graph. We call $Y$ a regular set of an association scheme if $Y$ is a regular set for all the graphs in the association scheme.
Regular sets appear in many geometric and combinatorial contexts, where examples in finite geometry include Cameron-Liebler classes, spreads, $m$-systems, and intriguing sets.
We will look at regular sets in the setting of the finite classical polar spaces, with a particular focus on Cameron-Liebler sets of generators. We will also describe a generalization of the concept of an $m$-ovoid to sets of isotropic subspaces having arbitrary (fixed) dimension. In some cases these generalizations of $m$-ovoids lead to constructions of non-trivial Cameron-Liebler sets of generators.
We will look at regular sets in the setting of the finite classical polar spaces, with a particular focus on Cameron-Liebler sets of generators. We will also describe a generalization of the concept of an $m$-ovoid to sets of isotropic subspaces having arbitrary (fixed) dimension. In some cases these generalizations of $m$-ovoids lead to constructions of non-trivial Cameron-Liebler sets of generators.
