Numerical ranges and geometry in quantum information: Entanglement, uncertainty relations, phase transitions, and state interconversion

Studying the geometry of sets appearing in various problems of quantum information helps in understanding different parts of the theory. It is thus worthwhile to approach quantum mechanics from the angle of geometry -- this has already provided a multitude of interesting results. In this thesis I demonstrate results relevant to numerical ranges -- the sets of simultaneously attainable expectation values of several observables. In particular, I apply this notion in the problems related to uncertainty relations, entanglement detection, and determining bounds for the value of spectral gap. Apart from this, I present geometric structures helping with the question of state interconversion using channels commuting with a particular representation of a group.