Size bias and other distributional transforms play an important role in sampling problems.  They are also very useful tools in sharp Normal (and other distributional) approximation, giving slick Stein's method quantitative proofs of Central Limit Theorems.  Recently, Goldstein and Schmook discovered a connection between size bias and infinitely divisible distributions, yielding a new kind of Levy--Khintchine formula for positively supported distributions.

In this talk, I will discuss joint work with Goldstein exploring the free probability analogues of bias transforms, with applications to freely infinitely divisible distributions.  In most cases the classical results are transferable, but with a significant change in perspective required.  Among other things, this approach gives a probabilistic meaning to the (free) Levy--Khintchine measure, and a new result simply characterizing those distributions that are positively freely infinitely divisible.

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.