Title: The multiplicities of Hilbert, "Kunz", Monsky, and a conjecture of Watanabe-Yoshida. 

Abstract: One of the most successful invariants of a local ring is its (Hilbert-Samuel) multiplicity. This integer plays a key role in intersection numbers for say complex curves. In the 1980s, P. Monsky noticed an error in a prior paper of Kunz, which allowed one to define a strange "Hilbert-Kunz" multiplicity which is a fruitful analogue of multiplicity only available for rings defined over the Galois field of p elements. This multiplicity has enjoyed an industry of study and has led to interesting conjectures, notably one by Watanabe and Yoshida, which characterize how small this multiplicity should be for singular settings. This colloquium will survey these ideas, discuss this conjecture, and explain a framework for deforming between these multiplicities as well as recent joint work with W. D. Taylor on the analogue in this deformation for hte Watanabe-Yoshida conjecture.