Abstract
Isoperimetric estimates stretch back for thousands of years in geometry, and for more than a hundred years in harmonic analysis and mathematical physics. We will touch on some of these highlights before describing recent progress that uses rotational symmetry to prove sharp upper bounds on sums of eigenvalues of the Laplacian. For example, we prove in 2 dimensions that the scale-normalized eigenvalue sum \[ (\lambda_1 + \cdots + \lambda_n) \frac{A^3}{I} \] (where $A$ denotes area and $I$ is moment of inertia about the centroid) is maximized among triangles by the equilateral triangle, for each $n \geq 1$. This theorem, which is due to the author and B. A. Siudeja, generalizes a result of Pólya for the fundamental tone.
Numerous related problems will be discussed, such as the inverse spectral and spectral gap problems for triangular domains.
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