Geometry from algebra: The holographic emergence of spacetime in string theory

In the quest for the unification of gravity with quantum mechanics, a new idea has emerged called the holographic principle which states that gravitational physics in D spacetime dimensions can be described by a dual nongravitational theory in D − 1 dimensions. Thus Spacetime geometry is not fundamental but rather emerges holographically from a theory obeying the algebraic laws of quantum mechanics. In this thesis we explore the interplay between geometry and algebra implied by this principle in several new contexts. Using the techniques of general relativity, we extend the holographic principle to Gödel spacetimes, where we discover a possible holographic protection of chronology. Then we study matrix models which provide a simplified toy model of holography and apply these models to find new relations between gauge theories in six dimensions and integrable dynamical systems. Finally we view the geometry of M-theory orbifolds from the algebraic perspective of infinite dimensional Lie algebras and find simple conditions obeyed by the twisted sectors of these orbifolds.