Algebra, Combinatorics, and Topology

Topology is the study of continuous functions between spaces, with broad latitude both for what qualifies as a space, and for which continuous functions are of interest. Of fundamental importance is the task of organizing spaces, treating two spaces as the same if, roughly speaking, each can be continuously deformed into the other. In this sense, the surface of a cube and the surface of a ball are the same, while neither is the same as the surface of a one holed doughnut.

Topology has strong ties to abstract algebra, notably the study of homology, cohomology, and homotopy theory.  Outside of mathematics itself, topology has real world applications in subjects such as biology, computer science, and robotics,  (e.g. knot theory, topological data analysis, and motion planning in a configuration space).