# Colloquium. Academic Year 19-20

Thursdays 4:10 pm in 5211 Stevenson Center, unless otherwise noted.

Tea at 3:33 pm in 1425 Stevenson Center

Colloquium Chair (2019-2020): Akram Aldroubi

## Orthogonal Fourier Analysis on Domains

Mihalis Kolountzakis, University of Crete

Location: Stevenson 5211

We all know how to do Fourier Analysis on an interval or on the entire real line. But what if our functions live on another subset of Euclidean space, let’s say on a regular hexagon in the plane? Can we use our beloved exponentials, functions of the form $e_\lambda(x) = {\rm exp}(2\pi i \lambda \cdot x)$ to analyze the functions defined on our domain? In other words, can we select a set of frequencies such that the corresponding exponentials form an *orthogonal basis* for L^2 of our domain? It turns out that the existence of such an orthogonal basis depends heavily on the domain. So the answer is yes, we *can* find an orthogonal basis of exponentials for the hexagon, but if we ask the same question for a disk, the answer turns out to be no. B. Fuglede conjectured in the 1970s that the existence of such an exponential basis is *equivalent* to the domain being able to *tile* space by translations (the hexagon, that we mentioned, indeed can tile, while the disk cannot). In this talk we will track this conjecture and the mathematics created by the attempts to settle it and its variants. We will see some of its rich connections to geometry, number theory and harmonic analysis and some of the spectacular recent successes in our efforts to understand exponential bases. Tea at 3:33 pm in Stevenson 1425. (Host: Akram Aldroubi)

## Decay Estimates for the Wave Equation on Manifolds

Jacob Sterbenz, University of California at San Diego

Location: Stevenson 5211

We’ll discuss long time decay estimates for the wave equation on a general class of asymptotically flat metrics. In the case of nontrapping metrics, when the operators are symmetric with slow time variation and a zero energy spectral condition, we’ll discuss local energy decay modulo finite dimensional dynamics. For a more general class of metrics, including black holes, we’ll discuss a general vector field method which takes local energy decay as an assumption. When used in combination, and also with the recent work of Dafermos, Rodnianski, and Shlapentokh-Rothman, these results establish a general asymptotic theory for both linear and null form equations on a wide variety of backgrounds. This is joint work with Jason Metcalfe, Jesus Oliver, Daniel Tataru. Tea at 3:33 pm in SC 1425. (Contact Person: Alex Powell)

## The Strong Cosmic Censorship Conjecture in General Relativity

Jonathan Luk, Stanford University

Location: Stevenson 5211

The Einstein equations in general relativity admit explicit black hole solutions which have the disturbing property that global uniqueness fails. As a way out, Penrose proposed the strong cosmic censorship conjecture, which says that this phenomenon of global non-uniqueness is non-generic. We will discuss this conjecture and some recent mathematical progress. This talk is based on joint works with Mihalis Dafermos, Sung-Jin Oh, Jan Sbierski and Yakov Shlapentokh-Rothman. Tea at 3:33 pm in Stevenson 1425. (Contact Person: Jared Speck)

## Around Cantor Uniqueness Theorem

Alexander Olevskii, Tel Aviv University

Location: Stevenson 5211

After a short introduction to Riemann’s uniqueness theory I will discuss the following problem, going back to early 60-s: Can a (non-trivial) trigonometric series

\sum c(n) e^inx , c( n) = o(1),

have a subsequence of partial sums which converges to zero everywhere? Joint work with Gady Kozma. Tea at 3:33 pm in SC 1425. (Host: Akram Aldroubi)