I see three major divisions for a Petri net initiative: math, science and computation.
Starting from their definition, Petri nets can be treated as purely mathematical entities. On the one hand, there is the classical mathematics which studies them a a special kind of Markov process, looks at the rate equations and the structure of the solutions, etc. And then there are more modern, abstract extensions of this math into category theory.
In the scientific branch, we study the application of these models to actual empirical systems. As part of this, we there investigate the extent to which the models capture the actual phenomena.
The computational branch can be partitioned into algorithms and development.