The magnitude function and spectral geometry
Analysis and Geometry Seminar
20th February 2020, 2:00 pm – 3:00 pm
Fry Building, 2.04
(joint with Magnus Goffeng and Nikoletta Louca) The notion of magnitude was introduced by Leinster as a generalization of the Euler characteristic of a finite category. Magnitude has since been extended to an invariant of compact metric spaces, but basic geometric questions remain open even for compact subsets X of R^n or a general Riemannian manifold with the induced metric d. In this talk we discuss the geometric significance of the magnitude function M(R)=mag((X,R*d)), X a smooth compact domain. In particular, in the semiclassical limit of large R, M admits an asymptotic expansion, and its leading terms relate to the volume, surface area and integral of the mean curvature. This proves an asymptotic variant of the convex magnitude conjecture by Leinster and Willerton, with corrected coefficients. Our approach connects the magnitude function with ideas from spectral geometry, scattering theory and the Atiyah-Singer index theorem.