Abstract
In these first two sessions, we pose the general question of how to compare geometric objects and the spectra of their Laplacians.
This requires the introduction of various topologies, making it possible to tell when and in what way two objects may "resemble" each other, as well as to measure the "distance" between spectra.
A non-exhaustive list of examples from recent literature in various fields of mathematics will be reviewed.
Examples:
1) and 2): Rectangles and cones
3) Thickened metric graphs (after Exner & Post)
4) Convergences of metric spaces in the Gromov-Hausdorff sense, under the condition RCD(K, N) (after Cheeger-Colding, Gigli-Mondino-Savaré)
References
- S. Beckus and J. Bellissard, Continuity of the Spectrum of a Field of Self-Adjoint Operators, Ann. Henri Poincaré 17 (2016), 3425-3442
- P. Exner, O. Post: Convergence of spectra of graph-like thin manifolds, J. Geom. Phys. 54 (2005), 77-115.
- O. Post and S. Zimmer, Generalised norm resolvent convergence: comparison of different concepts, J. Spectr. Theory 12 (2022), 1459-1506
- J. Cheeger and T. Colding, On the Structure of Spaces with Ricci Curvature Bounded Below. Ill J. DIFFERENTIAL GEOMETRY 54 (2000) 37-74
- N. Gigli, A. Mondino, and G. Savaré : Convergence of pointed non-compact metric measure spaces and stability of Ricci curvature bounds and heat flows, Proc. Lond. Math. Soc. (3), 2015
- N Gigli: De Giorgi and Gromov working together, https://arxiv.org/abs/2306.14604
