# Seminar Spectral Geometry, SoSe 2023

Prof. Dr. Joachim Lohkamp, Dr. Matthias Kemper

Seminar for BSc and MSc Mathematics.

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.

If you want to participate, join the **preliminary meeting on Thursday, April 13 15:30 in room 304 ⅔** („Lichthof” on the third floor), Einsteinstr. 62, or contact Matthias Kemper.

## Literature

I. Chavel: *Eigenvalues in Riemannian Geometry*. Academic Press (1984).

M. Berger: *A Panoramic View of Riemannian Geometry*. Springer (2002).

P. Buser: A note on the isoperimetric constant. *Ann. scient. E.N.S.* **15** (1982), 213–230.

Y. Colin de Verdière: Spectre de variétés Riemanniennes et spectre de graphes. *Proc. Intern. Cong. Math. Berkeley* (1986), 522–530.

Y. Colin de Verdière: Construction de laplaciens dont une partie finie du spectre est donnée. *Ann. scient. E.N.S.* **20** (1987), 599–615.