This dissertation aims to probe four-dimensional spaces via the study of surfaces embedded in them, which involves representing these surfaces diagrammatically with the aid of trisections.

The "combinatorial" project studies Lagrangian surfaces in the complex projective plane, equipped with its standard trisection, by representing them with "triple grid diagrams," which can be thought of as specific shadow diagrams on the central surface. Diagrams satisfying one extra condition represent Lagrangian surfaces, thus capturing this geometric information combinatorially. Part of this project is ongoing joint work with David Gay and Peter Lambert-Cole.

The "group theoretic" project studies group trisections from a few perspectives. In joint work with Benjamin Ruppik, we provide examples of group trisections of all cyclic groups. Then in joint work with Robion Kirby, Michael Klug, Vincent Longo, and Benjamin Ruppik, we extend the definition of group trisections to smoothly knotted surfaces in 4-manifolds, and show how to recover a knotted surface from its group trisection. Stallings folding guides the proof.