In the 2024-2025 academic year, I served as a graduate mentor for the Queens Experience in Discrete Mathematics REU. I worked with Luis Hernandez (York College) and Sean Ku (NYU, now UCLA) under the guidance of Dr. Radosław Wojciechowski (York College, CUNY GC) and Dr. Jun Masamune (Tohoku University).
Our project characterized form uniqueness on weakly spherically symmetric graphs. A weakly spherically symmetric graph \(X\) is a locally finite connected infinite graph, with a finite “center” set \(O\), and which is spherically symmetric about \(O\) with respect to edge weight and measure. The canonical examples would be trees or antitrees with uniform edge weight and measure, but they can get a lot more complicated. The measure of a weakly spherically symmetric graph may be finite or infinite, and there are no restrictions on combinatorial degree.
Form uniqueness refers to uniqueness of energy forms associated to \(X\). We can define a minimal energy form \(Q^{(D)}\) and a maximal energy form \(Q^{(N)}\). Here, minimal and maximal refer to the domains of these operators–the domain of \(Q^{(D)}\) is the space of functions which decay to 0, and the domain of \(Q^{(N)}\) is the space of all finite-energy functions in \(\ell^2\). On a given graph, if these domains are equal, the maximal and minimal forms agree and we say the graph satisfies form uniqueness.