Hessenberg varieties are subvarieties of the full flag variety. In this talk, I will concentrate on Lie type A. I will talk about a flat degeneration of a regular semisimple Hessenberg variety to a regular nilpotent Hessenberg variety, and I will explain how we can use this flat family to compute some Newton-Okounkov bodies of the Peterson variety of dimension 2. Along the way, we will also see that any regular nilpotent Hessenberg variety is a local complete intersection; this is a generalization of a result in Erik Insko’s PhD thesis. This is a joint work with Lauren DeDieu, Federico Galetto, and Megumi Harada.

Counting linear extensions of a partial order (linear orders compatible with the partial order) is a classical and computationally difficult problem in enumeration and computer science. A family of partial orders that are prevalent in enumerative and algebraic combinatorics come from Young diagrams of partitions and skew partitions. Their linear extensions are called standard Young tableaux. The celebrated hook-length formula of Frame, Robinson and Thrall from 1954 gives a product formula for the number of standard Young tableaux of partition shape. No such product formula exists for skew partitions.

In 2014, Naruse announced a formula for skew shapes as a positive sum of products of hook-lengths using ”excited diagrams” of Ikeda-Naruse, Kreiman, Knutson-Miller-Yong in the context of equivariant cohomology. We prove Naruse’s formula algebraically and combinatorially in several different ways. Also, we show how excited diagrams give asymptotic results and product formulas for the enumeration of certain families of skew tableaux. Lastly, we give analogues of Naruse's formula in the context of equivariant K-theory.

This is joint work with Igor Pak and Greta Panova.

Cancelled due to road conditions.

We begin with a brief introduction to Grobner theory and tropical geometry. Grobner bases are one the most fundamental tools in computational algebra. Tropical geometry can be described as a piecewise linear version of algebra/algebraic geometry and comes from looking at a variety from "infinity". It has many applications in different areas such as phylogenetics and optimization. We then talk about new results (joint with Chris Manon) about far extending Grobner theory concepts and doing algorithmic computations in general algebras equipped with valuations (in particular coordinate rings of varieties). In particular, this makes a direct connection between tropical geometry and recently emerged theory of Newton-Okounkov bodies. A central problem in this web of ideas is "degenerating" a given variety to a toric variety. There are many connections with other areas such as applied algebra, symplectic geometry (Hamiltonian systems) and representation theory (reductive group actions).