The HADES seminar on Tuesday, February 14th will be at 3:30 pm in Room 740.

Speaker: Mitchell A. Taylor

Abstract: Let $(\mathrm{\Omega },\mathrm{\Sigma },\mu )$ be a measure space, and $1\le p\le \mathrm{\infty }$. A subspace $E\subseteq L_p(\mu )$ is said to do stable phase retrieval (SPR) if there exists a constant $C\ge 1$ such that for any $f,g\in E$ we have     $$\underset{|\lambda |=1}{inf}\Vert f-\lambda g\Vert \le C\Vert |f|-|g|\Vert .$$    In this case, if $|f|$ is known, then $f$ is uniquely determined up to an unavoidable global phase factor $\lambda$; moreover, the phase recovery map is $C$-Lipschitz. Phase retrieval appears in several applied circumstances, ranging from crystallography to quantum mechanics.

In this talk, I will present some elementary examples of subspaces of $L_p(\mu )$ which do stable phase retrieval, and discuss the structure of this class of subspaces. This is based on a joint work with M. Christ and B. Pineau, as well as a joint work with D. Freeman, B. Pineau and T. Oikhberg.