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The combinatorics of Coxeter groups has long been a rich area of study with important applications to representation theory and geometry. Many of the key ideas in this realm have natural analogues when we restrict our attention to involutions in Coxeter groups. Based on pioneering work of Richardson and Springer, we will survey many results translated through the lens of involution words, which are the natural analog of reduced words for involutions. Some highlights include a new insertion algorithm, an intuitive combinatorial interpretation of the Chinese monoid and applications to the geometry of spherical varieties. These results only scratch the surface, and many open problems remain! This is joint work with Eric Marberg and Brendan Pawlowski.

Note the special time!