Of all the categories, "mathematicians working in a different field" is probably the hardest to write a proper research summary for. I will try my best.

Homotopy theory originates in algebraic topology, the study of topological spaces (mostly the nice ones: think manifolds) through algebraic invariants (e.g. vector spaces). In fact, there are several routes that can lead you (and that have, historically, led people) from algebraic topology to homotopy theory.