Idea of seminar: make a meeting place for PhD students (and all others) from algebra, in particular a place where people doing noncommutative stuff can learn/teach people doing commutative algebra and algebraic geometry. The topics should include (non)commutative stuff, but no things requiring advanced machinery of geometry or noncommutative ring theory.
There was a summer graduate school Noncommutative Algebraic Geometry and Representation Theory, and an introductory workshop Noncommutative Algebraic Geometry and Representation Theory. The school consisted of four series of lectures:
"Noncommutative resolutions" (thanks for preparing it, Maciek!).
(partially done) Noncommutative resolutions are a technique to construct and study crepant resolutions of Gorenstein normal singularities. A Noncommutative Crepant Resolution of $X = \mbox{Spec } R$ is a ring $A$ (usually not commutative) that satisfies some properties (see Def. 2.5 from the notes). The point of this definition is that if $f :Y \to X$ is any projective birational map between Gorenstein normal varieties, $A$ is any ring that is derived equivalent to $Y$, then $f$ is a crepant resolution (in particular, Y is smooth!) if and only if $A$ is an NCCR (see Corollary 4.17). But this theory works best when dimension $\leq 3$; when this holds, then from an NCCR $A$ we can construct a crepant resolution $f: Y \to X$ such that $Y$ is derived equivalent to $A$ (see Theorem 5.17). We do this using quiver GIT quotients. When dimension $\geq 4$, then many theorems still are true, but there are singularities that have an NCCR, but do not have a crepant resolution and vice-versa.
Proposed schedule: