Growth of Laplacian Eigenfunctions

Explore the intricacies of Laplacian eigenfunction growth in this 55-minute conference talk by Stefan Steinerberger, presented at the inaugural ZhengTong Chern-Weil Symposium in Mathematics. Delve into the classical problem of eigenfunction concentration on compact manifolds, tracing its study back to the 1950s. Examine the sharp growth rate attained on spheres and contrast it with the seemingly logarithmic growth on most manifolds. Discover a novel characterization of growth, revealing how eigenfunctions can exhibit 'spooky action at a distance' by significantly influencing far-away regions of the manifold. Observe this phenomenon in simple examples like S^1 and the unit square [0,1]^2. Consider the implications of the Berry random wave heuristic and explore the behavior of eigenfunctions on 'generic' manifolds where limited growth is expected due to the absence of long-range influences.