Morgan Rogers

Abstract monoids

The notion of monoid requires relatively little categorical structure to define. That means that we can study monoids in all sorts of contexts, and try to understand how they are transformed by functors between these contexts. It also means that there are all sorts of ways to build actions of monoids, from the simplest kind (set-monoids acting on sets, as in my early work) to those appearing in representation theory (set-monoids acting on vector spaces), topology (topological monoids acting on topological spaces) or ergodic theory (measurable monoids acting ergodically on measure spaces), to more exotic monoids (indexed monads). My long term research goals revolve around lifting tools from conventional domains in which monoids and groups act to be able to apply those tools across contexts.

Complexity theory

Monoid actions appear in computability and complexity theory at a very abstract level: we can define a generic model of computation to consist of a space (of configurations), which might carry a topological or measure structure, equipped with a monoid action expressing the effect of the operations which the model is capable of performing. Programs, possibly subject to some constraints, form structures over these actions, and so complexity classes emerge as invariants. My current research program, which is developing in the cracks between collaborations, is to try to use toposes of monoid actions as a context for computing and comparing these actions alongside other monoid actions.

Masters research projects (stages)

The following are projects I am proposing. The former falls squarely into (categorical) logic, the latter is an algebra project. Both have scope to lead into PhD projects.
De Morgan toposes. English description; French description.
Quantifying Burnside semigroups and rigs. English description.