In this talk we will investigate extremal problems from 3 fields of analysis.
The first problem we consider is the Strohmer-Beaver Conjecture which is settled in the field of time-frequency analysis. It asks for the lattice of fixed redundancy which mininizes the condition number of the frame operator associated to the Gabor frame consisting of a Gaussian window and a lattice of given redundancy.
The second problem (Landau’s problem) is a packing problem for holomorphic functions. It asks for the largest (open) disc which can be found in the image of the unit disc under a holomorphic mapping whose derivative at the origin is fixed to be 1. The task is to find the function which minimizes the radius of the largest disc (Landau’s constant) found in this way. Most probably Rademacher found the correct solution in 1943, but the problem is still open.
The third problem concerns the heat kernel on a torus. We consider the minima of the heat kernel on a torus of fixed volume and ask for the torus which maximizes this minimum.
We will then look for connections of the mentioned problems and their potential solutions. Most likely all problems are equivalent.