Abstract
In these first two sessions, we pose the general question of how to compare geometric objects and the spectra of their Laplacians.
This involves introducing various topologies, making it possible to tell when and how two objects "look alike", and to measure the "distance" between the spectra.
A non-exhaustive list of examples from recent literature in various fields of mathematics will be reviewed. We'll also begin to introduce the notion of convergence in the Benjamini-Schramm sense
Examples :
4) Convergence of metric spaces in the Gromov-Hausdorff sense, under the condition RCD(K, N)
5) Homogenization
6) Discrete torus: two scales, two limits.
References
- A. Bensoussan, J-.L. Lions, G. Papanicolaou, Asymptotic Analysis for Periodic Structures, North-Holland, Amsterdam, 1978.
- S. Kesavan, Homogenization of Elliptic Eigenvalue Problems: Part 1, Appl. Math. Optim. 5 153-167(1979)
- I. Benjamini and O. Schramm. Recurrence of distributional limits of finite planar graphs. Electron. J. Probab, 6:no. 23, 13 pp. (electronic), 2001.
- D. Aldous and R. Lyons. Processes on unimodular random networks. Electron. J. Probab, 12:no. 54, 1454-1508, 2007.
