Objects of everyday size and scale typically have something that you can call a shape. Footballs are spherical in most of the world, or prolate spheroids in the USA. Eggs are oviform, breasts are mammiform, and so on. Only, if you were to look closely, at an atomic level, at any macroscopic object, you'd see that the surface was not quite as solid as you thought, but an undulating border between atoms belonging to the object, and those not.
Down at the tiny level of atomic nuclei, the undulation is so intertwined with the nature of the nucleus, that the concept of shape breaks down. Many nuclei don't just sit there being a specific shape, but undulate so that they are a sometimes spherical, sometimes stretched spheres, sometimes squashed spheres, and occasionally other shapes. This is just quotidian quantum mechanics, but because it's counter-intuative, it's got its own name, and associated conferences. To be fair, it's as good a theme to use to understand any part of nuclear physics as any other, as long as one understands why we consider it a particular topic.
There were a lot of interesting talks, covering nuclear states that simultaneously exhibit multiple shapes, from both experimental and theoretical points of view. I gave a talk, and it invoked some mild disparagement from a fellow theorist, as theory talks always seem to do. It's too bad my interlocutor had to depart for a train before the end of the session I was in as we never managed to have a good chat about it.
As ever, I think about adding a figure to a blog post as I finish writing. Clearly, the term "shape coexistence" is used only amongst nuclear physicists, and not in other areas. This is perhaps not too surprising, as nuclei are the only quantum objects that are within their own size for a few orders of magnitude, so if that's the scale at which the effect happens, then there's nothing else for it to happen to. The picture is taken from a Cern Courier article about work by many of the people at the workshop, including its organiser, showing some of the different "co-existing" shapes in an isotope of lead.