Abstract
This minicourse will review the basic notions of the theory of cluster algebras and associated varieties, loosely following our book draft (https://people.math.harvard.edu/~williams/book.html). Examples and computations will be emphasized throughout.
We will start by briefly discussing two key motivations for the introduction of cluster algebras: total positivity and canonical bases. After introducing the machinery of mutations of quivers and skew-symmetrizable matrices, we will define cluster algebras, formulate their key properties, and present the finite type classification.
Many cluster algebras arise in the context of representation theory, invariant theory, etc., in the form of coordinate rings of affine algebraic varieties. Typical examples include varieties of matrices of various kinds, Grassmannians, and basic affine spaces. We will discuss as many of these examples as time permits, together with relevant general theory.