This talk will be expository and colloquium-style, with an emphasis on background and motivation.

Hessenberg varieties are subvarieties of the flag variety Flags(C^n), the study of which have rich interactions with symplectic geometry, representation theory, and equivariant topology, among other research areas. When the maximal torus T-action on Flags(C^n) restricts (wholly or in part) to a Hessenberg variety, they can be studied via familiar equivariant techniques such as GKM theory. Moreover, in some ways they exhibit behavior reminiscent of Schubert varieties and Schubert calculus, but computations also show that Hessenberg varieties behave in ways more complicated than in the purely Schubert setting. One particular avenue by which they have garnered recent, earnest, attention is the connection of Hessenberg varieties to the famous Stanley-Stembridge conjecture in combinatorics, where a certain S_n-action on the cohomology ring of regular nilpotent Hessenberg varieties plays a central role.

In this talk, I will give an non-exhaustive, highly biased overview of the subject, and point the audience to some of the exciting recent developments in the area.