Mathematics Colloquium

Waves are ubiquitous in our daily life. Two best-known linear models are the free wave and free Schrödinger equations, whose simplest forms are very amenable to Fourier analysis. Still, a basic question—how large can a solution be, and where can it be large?—is surprisingly subtle and only partly understood, especially in higher dimensions. Over decades, it transpired that in order to answer this fundamental question, one often needs to understand whether and how much the solution can concentrate on important subsets of $\mathbb{R}^n$. I will discuss three kinds of such subsets (convex sets, semialgebraic sets and lattices) and their importance based on sample problems. Some of them have nice connections to nearby areas such as number theory, geometry and combinatorics.