Grothendieck Inequalities, Nonlocal Games and Optimization

Motivated by applications in quantum information theory and optimization we introduce new variants of a celebrated inequality known as Grothendieck's Inequality. In quantum information theory we apply these mathematical tools to study of one of the most surprising and counter-intuitive predictions of Quantum Mechanics: entanglement. In optimization we use them to determine the precision of efficient approximation algorithms for geometric problems that arise naturally from the study of entanglement and from models of interacting particles considered in classical statistical physics. In this thesis we study entanglement by using nonlocal games. A nonlocal game involves two or more players who are not allowed to communicate with each other, but do interact with an extra party usually referred to as the referee. At the start of the game the referee asks each of the players a question, upon which they each reply to him with some answer. Then, the referee decides if the players win or lose based only on the questions he asked and the answers he received. The players know in advance what set of answers would cause them to win, which of course is their objective. The catch is that they only know the question that was aimed directly at them and not any of the other players' questions. The players thus don't