Surprising shapes of free resolutions

Boij and Soederberg offered an important shift in perspective when they suggested that graded free resolutions over a polynomial ring were more easily understood when viewed "up to scalar multiple." The proof of their conjectures by Eisenbud and Schreyer confirmed an unexpectedly simple polyhedral structure on these free resolutions.
We discuss this result and its analogue for minimal free resolutions over local rings. Over a regular local ring and their hypersurface rings, we classify the possible shapes of such resolutions. This illustrates the existence of free resolutions whose Betti numbers behave in surprisingly pathological ways. This is joint work with Daniel Erman, Manoj Kummini, and Steven V Sam.