Quantity and size: Auslander-type results in silting theory

A famous theorem of Auslander states that a finite dimensional
algebra is of finite representation type if and only if every
module is additively equivalent to a finite dimensional one. This
establishes a correlation between quantity (of indecomposable
finite dimensional modules) and size (of indecomposable modules).

We will discuss the ocurrence of an analogous correlation in
silting theory. Indeed, for a finite dimensional algebra A we
prove that

1) A is \tau-tilting finite if and only if every silting module
is additively equivalent to a finite dimensional one.

2) A is silting discrete if and only if every bounded silting
complex is additively equivalent to a compact one.

This is based on joint work with L. Angeleri Hügel and F. Marks
and on joint work with L. Angeleri Hügel and D. Pauksztello.