# Topology and Homotopy Theory

Topology studies properties of spaces that are invariant under continuous deformation, like stretching or contracting but not breaking. Topology has deep connections to numerous areas of mathematics: in algebraic topology, algebraic structures are used to tell spaces apart; in differential topology, one deals with spaces that carry additional smooth structure; while in topological combinatorics one studies combinatorial objects, such as graphs, by associating spaces

to them.

Faculty members in the Department of Mathematics represent all active research areas within topology, including algebraic, low-dimensional, combinatorial, applied, and differential topology.

An area of focus in the Department of Mathematics is homotopy theory. Historically a branch of algebraic topology, homotopy theory is now its own discipline with deep connections to other areas, including algebraic geometry, number theory, category theory, and theoretical computer science, among others. Several of our faculty members work in homotopy theory.