WARTHOG — Cluster Algebras and Braid Varieties

For many (affine) algebraic varieties appearing in geometric representation theory and algebraic combinatorics, their coordinate algebra admits the structure of a cluster algebra. These algebraic varieties include (in order of increasing generality) the basic affine space, (double) Bruhat cells, positroid varieties, Richardson varieties, and braid varieties. Roughly speaking, this means that the corresponding algebraic variety comes with a collection of coordinate toric charts, with explicit coordinates and transformation rules between the charts that can be codified via quiver combinatorics. Conversely, the quiver and the coordinates can be read from combinatorial and geometric properties of the variety itself, though this is more of an art than a science. The goal of this workshop will be to understand the combinatorics and geometry underlying these cluster structures in the case of braid varieties, which include all of the cases described above and was established quite recently. The combinatorics can be encoded in two very different objects: weaves and 3D plabic graphs. Both appear in a variety of mathematical settings, including Soergel calculus and connections to high-energy physics. These rich combinatorial objects and their ties to other areas of mathematics inspire many questions about braid varieties, their cluster structures, and beyond.