Georgia Institute of Technology College of Sciences

 One strategy for developing a proof of a claimed theorem is to start by understanding what a counter-example should look like.  In this talk, we will discuss a few recent results in harmonic analysis that utilize a quantitative version of this approach.  A key step is the solution of an inverse problem with the following flavor.  Let $T:X \to Y$ be a bounded linear operator and let $0 < a \leq \|T\|$.  What can we say about those functions $f \in X$ obeying the reverse inequality $\|Tf\|_Y \geq a\|f\|_X$?