Magical improvement of spectral asymptotic estimates using averages
Speaker: Bob Strichartz, Cornell University
Location: Warren Weaver Hall 1302
Date: Tuesday, January 15, 2019, 11 a.m.
This will be a survey of recent work on improving estimates for the asymptotics of the eigenvalues of Laplacians in several different settings. For compact manifolds without boundary the classical result of Weyl is about a century old. Extension to manifolds with boundary are due to Ivri'i and include lower order terms. For example, if the manifold is the standard torus then the question is eqivalent to counting the number of lattice points inside a disk. The Weyl estimate is just the area of the disk, and the remainder is unbounded, but the rate of growth is still not known precisely. But if you average the remainder you get a decaying term that you can make very precise. I will describe examples where there are proofs of the results, and some where there is only experimental evidence, and I will discuss how experimental and theoretical results interact. One fun example is a case where experimental results yielded a certain constant to 7 decimal places of accuracy; then a google search gave the answer pi squared over 4; then we found a proof. (Much of this work has been joint with undergraduate students.) the talk will be on an elementary level so as to be accessible to a wide audience.