Given a number field K, the Hilbert class field of K is the maximal unramified abelian extension of K. Iterating this construction, one obtains a tower of fields known as the Hilbert class tower of K. The question of whether such towers can be infinite or must always stabilize after a finite number of steps is connected with a certain embedding problem and was finally resolved in the 1960s by Golod and Shafarevich. In the first part of the talk, I'll discuss this history in more detail. In the second part, I'll explain how one can sometimes obtain information indirectly about such towers using methods from computational group theory.

This talk is on the mapping properties of the Fourier transform between Banach function spaces. These are generalizations of Hausdorff-Young and Pitt's inequalities.

We provide several relations between weight functions, that guarantee the boundedness of the Fourier series coefficients, viewed as a map between weighted Lorentz spaces. As a useful machinery, we briefly introduce the quasi concave functions and generalize a number of known inequalities. Finally, we apply our results to Fourier inequalities in weighted Lebesgue spaces and Lorentz-Zygmund spaces.