I am generally interested in problems pertaining to
the interaction of systems, very broadly understood,
and the effects produced from such interactions.
The more fundamental a problem/question is, the more interest I can gather to entertain it.
My thesis' research concern.
The principal concern of my thesis work goes into understanding and uncovering interaction-related effects---termed, interactional effects---that arise from the interaction of systems.
We may describe the common situation of interest as small entities of systems coming together, interacting, and producing, as an aggregate, a behavior that would not have occured without interaction. Those situations are fundamental and appear in countless settings, some of which are contagion effects in societal systems, and cascading failures in infrastructures.
The goal of the research is to show that one can extract from a system the potential of it to generate such effects, and use those extracts to reconstruct, or characterize, the phenomena that emerge upon interaction.
My thesis work proposes a means to relate properties of an interconnected system to its separate component systems in the presence of cascade-like effects.
Building on a theory of interconnection reminiscent of the behavioral approach to systems theory, it introduces the notion of generativity, and its byproduct, generative effects. Cascade effects, enclosing contagion phenomena and cascading failures, are seen as instances of generative effects. The latter are precisely the instances where properties of interest are not preserved or behave very badly when systems interact.
The goal of the work is to overcome that obstruction. We show how to extract algebraic objects (e.g. vectors spaces) from the systems, that encode their generativity: their potential to generate new phenomena upon interaction. Those objects may then be used to link the properties of the interconnected system to its separate component systems. Such a link is executed through the use of exact sequences from commutative algebra.
The flavor and the tools.
My thesis work borrows mathematical tools and insight from a mix of systems theory, order/lattice theory, category theory, formal languages, commutative algebra, homological algebra, (algebraic) topology and geometry, and some combinatorics.