  These lecture notes cover a brief introduction into some of the
algebro-geometric techniques used in the construction of BPS algebras. The
first section introduces the derived category of coherent sheaves as a useful
model of branes in toric Calabi-Yau three-folds. This model allows a rather
simple derivation of quiver quantum mechanics describing low-energy dynamics of
various brane systems. Vacua of such quantum mechanics can be identified with
the critical equivariant cohomology of the moduli space of quiver
representations. These are often counted by various crystal configurations.
Using correspondences in algebraic geometry, one can construct rich families of
affine-Yangian representations. We conclude with an exploration of different
algebraic structures naturally appearing in our story. The material was covered
in a 4-lecture mini-course within the Second PIMS Summer School on Algebraic
Geometry in High-Energy Physics. The text contains some new ideas, examples and
remarks that are going to be covered in detail in a joint work with Dylan
Butson.
