Can you hear the fundamental frequency of a drum using probability?

In Mark Kac's famous 1966 paper, he asked ``Can you hear the shape of a drum?'' The precise question being, if you heard the full list of overtones and frequencies while you were blindfolded, would you be able to tell the shape of the drumhead $D\subset\mathbb{R}^{2}$ in some mathematical way? The problem I will primarily speak about is in regards to how the fundamental frequency of a drum and probability theory are related. This connection will be through an inequality involving the fundamental frequency of a drum with drumhead $D\subset\mathbb{R}^{d}$ and the maximum expected lifetime of Brownian motion started inside a domain $D\subset\mathbb{R}^{d}$. Brownian motion is a mathematical model for the random movement of a particle which was originally observed by Robert Brown in 1827 while looking at pollen grains through a microscope. We improve on the constants of a known inequality and prove a new asymptotically sharp inequality involving the moments of the expected lifetime of Brownian motion. We discuss conjectures about the sharp inequality and present our partial results about the extremal domains and sharp constants over a nice class of domains. This talk is based on joint work with Rodrigo Bañuelos and Jing Wang.