Spectral geometry studies the relationship between geometric shapes and their spectra, which are the eigenvalues of certain geometric differential operators such as the Laplacian. Its first eigenvalue already demonstrates fascinating connections between analysis and geometry: It is mainly controlled by the Ricci curvature and allows to estimate an isoperimetric constant.
A question that kickstarted the field in the 1960s was famously phrased as “Can one hear the shape of a drum?” – is it possible to reconstruct a Riemannian manifold from its spectrum? Or with a different twist: Given a differentiable manifold M, is there a metric on M such that the Laplacian has a prescribed spectrum? What if we only want to prescribe finitely many eigenvalues?
In this seminar, we will explore these questions and more, after quickly introducing the basics of analysis on manifolds and tools such as the heat kernel.