I’m Matteo Capucci, I’m currently enrolled at the University of Strathclyde as a PhD student in the amazing MSP group. I’m working in applied category theory, focusing on categorical cybernetics and applied topos theory.
- Categorical foundations of cybernetics
Cybernetic systems self-adapt through the observations they make of the ‘environment’ which interacts with them. In games, this brings players to play equilibria. In machine learning, it makes models learn from a dataset.
Category theory can put the mathematical treatment of these systems on strong and flexible foundations. We can then use string diagrams to describe systems compositionally, and categorical logic to impose guarantees on their behaviour.
1. Generalised open learners by Bruno Gavranović,
2. Games with players by myself,
3. Compositional Active Inference by Toby St. Clere Smithe.
- Relativization of stochastic calculus
Some theories are unnaturally complicated when described in the language of sets. Using the languages of other topoi can help to make them look simpler, and therefore to be simpler to work with. In a sense, relativization is the search for the natural habitat of a mathematical theory, where it can be ‘its true self’ and thrive. Stochastic calculus is one such a theory: topoi of sheaves over suitably defined sites make the theory tame and natural-looking. My current goal is to construct Ito’s and Stratonovich’s integrals in this way.
1. How topos theory can help commutative algebra by Ingo Blechschmidt,
2. My name is stochastic calculus but everybody calls me calculus by myself,
3. Topos theory and measurability by Asgar Jamneshan.
- Applied sheaf theory
Sheaf theory is a wonderful abstraction to deal with local/global data. Can we use it to better understand compositional behaviour? This is an open ended question and a long-term research topic. A crucial step into answering this question is: how do we bridge the cartesian-monoidal gap in order to use sheaves on categories of systems?
1. Systems, Generativity and Interactional Effects by Elie Adam,
2. Lax functors describe emergent effects by Jules Hedges,
3. Fantastic sheaves and where to find them by myself