Title: Inequalities between length and multiplicity

Abstract: This talk will review some classical inequalities that relate the colength and the multiplicity of an ideal in a Noetherian local ring, especially one due to Lech. Then we will consider a conjectured inequality, and talk about some recent progress on it. This part of the talk will be work in progress with M. Mustata, S. Tagaki, and K. Watanabe.