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By Yaiza Canzani.

Section 7.5, Theorem 65: Faber-Krahn inequality for the first Dirichlet eigenvalue. Author name appears on the lecture-note title page; no year inferred from host directory.

Yaiza Canzani works in spectral geometry, studying how the geometry of a space controls eigenfunctions and eigenvalues of its Laplace operator. Her research account includes bounds for eigenfunction concentration, averages over submanifolds and spectral counting remainders.^[ref]

Her Analysis on Manifolds via the Laplacian by Yaiza Canzani develops the geometry behind this spectral language. On a circle, for example, the modes $1$, $\cos (k\theta )$ and $\sin (k\theta )$ have nonnegative Laplace eigenvalues $0$ and $k^2$. Their oscillation scale grows with $k$, making the link between shape, frequency and eigenvalue tangible. On a general Riemannian manifold, the corresponding questions concern how waves concentrate and how geometry changes their allowed modes.^[ref]