We will overview some conjectures on the mixed Hodge structure of character varieties in the framework of non-abelian Hodge theory on a Riemann surface. Then we introduce and study toric analogues of these spaces, in particular we prove that the toric character variety retracts to its core, the zero fiber of the toric Hitchin map, that its cohomology is Hodge-Tate and satisfies curious Hard Lefschetz, as well as the purity conjecture. We will indicate how these shed light on the $P=W$ conjecture in the toric case as well as for general character varieties. This is based on joint work with Nick Proudfoot.[-]

Let $X$ be an algebraic subvariety in $(\mathbb{C}^*)^n$. According to the good compactifification theorem there is a complete toric variety $M \supset (\mathbb{C}^*)^n$ such that the closure of $X$ in $M$ does not intersect orbits in $M$ of codimension bigger than dim$_\mathbb{C} X$. All proofs of this theorem I met in literature are rather involved.
The ring of conditions of $(\mathbb{C}^*)^n$ was introduced by De Concini and Procesi in 1980-th. It is a version of intersection theory for algebraic cycles in $(\mathbb{C}^*)^n$. Its construction is based on the good compactification theorem. Recently two nice geometric descriptions of this ring were found. Tropical geometry provides the first description. The second one can be formulated in terms of volume function on the cone of convex polyhedra with integral vertices in $\mathbb{R}^n$. These descriptions are unified by the theory of toric varieties.
I am going to discuss these descriptions of the ring of conditions and to present a new version of the good compactification theorem. This version is stronger that the usual one and its proof is elementary.[-]

In joint work with Konstanze Rietsch (arXiv:1712.00447), we use the $\mathcal{X}$-cluster structure on the Grassmannian and the combinatorics of plabic graphs to associate a Newton-Okounkov body to each $\mathcal{X}$-cluster. This gives, for each $\mathcal{X}$-cluster, a toric degeneration of the Grassmannian. We also describe the Newton-Okounkov bodies quite explicitly: we show that their facets can be read off from $\mathcal{A}$-cluster expansions of the superpotential. And we give a combinatorial formula for the lattice points of the Newton-Okounkov bodies, which has a surprising interpretation in terms of quantum Schubert calculus.[-]

Reid's recipe is an equivalent of the McKay correspondence in dimension three. It marks interior line segments and lattice points in the fan of the G-Hilbert scheme (a specific crepant resolution of $\mathbb{C}^{3} / G$ for $G \subset S L(3, \mathbb{C})$ ) with characters of irreducible representations of $G$. Our goal is to generalise this by marking the toric fan of a crepant resolution of any affine Gorenstein singularity, in a way that is compatible with both the G-Hilbert case and its categorical counterpart known as Derived Reid's Recipe. To achieve this, we foray into the combinatorial land of quiver moduli spaces and dimer models. This is joint work with Alastair Craw and Jesus Tapia Amador.[-]

Splice type surface singularities were introduced by Neumann and Wahl as a generalization of the class of Pham-Brieskorn-Hamm complete intersections of dimension two. Their construction depends on a weighted graph with no loopscalled a splice diagram. In this talk, I will report on joint work with Patrick Popescu-Pampu and Dmitry Stepanov (arXiv: 2108.05912) that sheds new light on these singularities via tropical methods, reproving some of Neumann and Wahl'searlier results on these singularities, and showings that splice type surface singularities are Newton non-degenerate in the sense of Khovanskii.[-]