A dimer tree algebra A is the Jacobian algebra of a quiver Q (without loops and 2-cycles) with a canonical potential such that (i) every arrow lies in a chordless oriented cycle; and (ii) the dual graph is a tree. The category of non-projective syzygies over A is equivalent to the stable category of (maximal) Cohen Macaulay modules as well as to the singularity category of A. It is a triangulated 3-Calabi-Yau category.
We give a geometric model for the syzygy category for dimer tree algebras in terms of 2-diagonals in a polygon with checkerboard pattern.
This checkerboard pattern can be extended to a dimer model on the disk.

In this talk we explain a non-existence result for metrics of positive m-intermediate curvature (a notion of curvature reducing to positive Ricci curvature for m =  1, and positive scalar curvature for m = n-1) on closed orientable manifolds with topology $N^n = M^{n-m} x \mathbb{T}^m$ for $n \leq 7$. Our proof uses a slicing constructed by minimization of weighted areas, the associated stability inequality, and estimates on the gradients of the weights and the second fundamental form of the slices. This is joint work with Simon Brendle and Sven Hirsch.