Rumi Salazar is a third-year student at the University of New South Wales studying a double major in mathematics and physics. He has a wide range of potential research interests in academia and is on a path to discover a more narrow focus. Outside of academia, he is passionate about learning and gaining a deeper understanding of all facets of the world through reading and listening. As well as this, his hobbies include engaging in social outings with his friends and the occasional gaming.
The Shape of a Drum
In 1966, Mark Kac asked whether one could hear the shape of a drum, thereby popularising a problem of Hermann Weyl. This question was really about whether the spectrum of the Dirichlet Laplacian on a domain in space determined the domain (up to rigid motions). There are a number of positive results: one can hear the area of the domain, the length of its boundary, and the number of holes, as well as negative results, due to Caroline Gordon et al. Recently, Zhiqin Lu and Julie Rowlett showed that the spectrum determines whether a domain has corners or not. However, their argument does not say how many corners there are. Thus, our objective is to discover if it is possible to determine this. If not, we aim to find a suitable bound on this number. We also aim to see whether either the presence or the number of cusps can be determined using the same (or similar) method.