Analysis Seminar

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.

Synopsis:

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.