The homotopy theory of toric objects

The homotopy theory of topological spaces is an active and extremely difficult area of research which has a plethora of application not just within pure mathematics but in real world problems at well. Toric objects in question come with a rich underlying combinatorial structure that surprisingly allows for more explicit study of their homotopy theory as it serves as a concrete, trackable bookkeeping information of topological transformations. Utilising the combinatorial structure, using the tools of algebraic topology, geometry, and commutative algebra, we plan to study the homotopy theory of toric objects.