Moment methods in extremal geometry

In this thesis we develop techniques for solving problems in extremal geometry. We give an infinite dimensional generalization of moment techniques from polynomial optimization. We use this to construct semidefinite programming hierarchies for approximating optimal packing densities and ground state energies of particle systems. For this we...

The Eigenvalue Method for Extremal Problems on Infinite Vertex-Transitive Graphs

This thesis is about maximum independent set and chromatic number problems on certain kinds of infinite graphs. A typical example comes from the Witsenhausen problem: For $n \geq 2$, let $S^{n-1} := \{ x \in \R^n : \|x\|_2 =1 \}$ be the unit sphere in $\R^n$, and let $G=(V,E)$ be the graph with $V = S^{n-1}$, in which two points in $S^{n-1}$ are...