**Completeness problems for complex exponentials**

I will talk about two well-known problems of harmonic analysis related to completeness of complex exponentials in L^{2}-spaces. The first problem was solved by Beurling and Malliavin in 1960s. I will discuss its modern generalizations, obtained via the use of Toeplitz operators, and further open problems. This part of the talk is based on joint work with Nikolai Makarov.

In the second part of the talk I will discuss the so-called type problem, which stood open until recently. Starting with a brief history of the problem, I will present recent results and

new examples.

- N. Makarov and A. Poltoratski: Beurling-Malliavin Theory for Toeplitz Kernels,
*Invent. Math.*, Vol. 180, Issue 3 (2010), 443-480. - A. Poltoratski: A problem on completeness of exponentials,
*Annals of Math.*, Volume 178 (2013), 983-1016 - A. Poltoratski: Toeplitz Approach to Problems of the Uncertainty Principle, Conference Board of the Mathematical Sciences (CBMS) series, AMS/NSF, 2015.