Hessenberg varieties are subvarieties of flag varieties invented in the early 1990s by de Mari and Shayman.  De Mari and Shayman were motivated by questions in applied linear algebra, but, very quickly, people realized that Hessenberg varieties are very interesting objects of study from the point of view of algebraic groups.  

I got interested in Hessenberg varieties because of their connection to questions in combinatorics, in particular, the Stanley-Stembridge conjecture.  I'll explain this conjecture, now a theorem due to Hikita, and I will explain some of my work with Tim Chow, which resolved a conjecture of Shareshian and Wachs connecting Hessenberg varieties directly to Stanley-Stembridge. (I'll also try to say a few words about Hikita's work and the very exciting state the field is in now.)  Then I'll explain joint work with Escobar, Hong, Lee, Lee, Mellit and Sommers on the moduli of Hessenberg varieties.