My previous research has established a game-semantic model of higher-order computation and extended it to a combinatorial foundation of (almost) all mathematics.
In this talk, I report some new directions of my ongoing research, which applies the unique features of the game semantics for logic and computation to algebra, topology and geometry.

One direction is to classify a broad range of structures and theorems by complexity classes and apply the latter as invariants to the former, where unlike classical models of computation (e.g., Turing machines) the generality of the game semantics makes it possible to assign computational complexity to (almost) all structures and theorems.

Another direction is based on the observation that every game comes with a canonical topology similar to yet much more general than Scott topology. By the replacement of finiteness defining open sets in this topology with computability, topology and computability become the two sides of the same coin in the game-semantic foundation, and the two sides are naturally refined into geometry and computational complexity, respectively, by caring quantities, for which again the generality of the game semantics is crucial because classical models of computation can handle only countable spaces.
The second direction is then to explore this link between computational and geometric concepts and by combining it with the first direction to study structures and theorems through topology and geometry.