Asociacijske sheme, Avtomati in Incidenčne Strukture / Association Schemes, Automata and Incidence Structures

In my research, I was always fascinated by the many connections between mathematical topics. I experienced this during my Ph.D.
when I was working on problems arising from finite geometry and I came across the theory of association schemes. At that moment I
realized I could have used such a powerful theory to attack those problems. This motivated me to submit a project proposal on
association schemes.
Association schemes are algebraic-combinatorial structures, provided with a multifaceted nature: even if association have their roots
in statistical experiments, they have connections with error-correcting codes, combinatorial designs, graph theory, group theory and
finite geometry.
There are two well-known examples of association schemes: the Hamming schemes and the Johnson schemes, which are extensively
studied in various fields of mathematics.
These schemes belong to a very important class of association schemes, which are the core of my project: the schemes arising from
distance regular graphs.
In this project, I will look at objects and problems from other theories through the lens of association schemes.
More precisely, my proposal concerns the concepts of synchronization and separation for association schemes. These extend those
defined for permutation groups which in turn developed from the theory of synchronization in finite automata.
One of the more ambitious goals of my proposal is to extend further concepts from group theory into the language of association
schemes, starting from the case of schemes arising from distance-regular graphs.
Furthermore, some objectives regarding extended generalized quadrangles and connections with the theory of association schemes
are discussed.
All this implies several subproblems that are well-treated in my proposal, together with wide ranging projects which naturally
encompass it.