Title: The Geometry of Nodal Sets and Outlier Detection,
Abstract: Let $(M,g)$ be a compact manifold and let $-\Delta \phi_k = \lambda_k \phi_k$
be the sequence of Laplacian eigenfunctions. We present a curious new
phenomenon which, so far, we only managed to understand in a few highly
specialized cases: the family of functions $f_N:M \rightarrow \mathbb{R}_{\geq
0}$ $$ f_N(x) = \sum_{k \leq N}{ \frac{1}{\sqrt{\lambda_k}}
\frac{|\phi_k(x)|}{\|\phi_k\|_{L^{\infty}(M)}}}$$ seems strangely suited for
the detection of anomalous points on the manifold. It may be heuristically
interpreted as the sum over distances to the nearest nodal line and potentially
hints at a new phenomenon in spectral geometry. We give rigorous statements on
the unit square $[0,1]^2$ (where minima localize in $\mathbb{Q}^2$) and on
Paley graphs (where $f_N$ recovers the geometry of quadratic residues of the
underlying finite field $\mathbb{F}_p$). Numerical examples show that the
phenomenon seems to arise on fairly generic manifolds.