Complex geometry is a central research field in current mathematics that also forms part of the backbone of many subjects in theoretical physics, for example quantum mechanics and string theory. During the last 50 years, use of the Einstein Field Equations in complex geometry has become more and more popular. The main reason for this is that in complex geometry the Einstein Field Equations constitute an intersection point between several different areas in mathematics, Geometry, Algebra and Probability, providing surprising links between concepts that were not previously known to be related.
This thesis contains four papers contributing to this field. The first paper explores a recently discovered connection to probability and statistical mechanics in which the objects of interest can be described as clouds of large numbers of interacting particles. The second paper proves that solutions to an equation related to the Einstein Field Equations minimise a certain energy type quantity, which can be formulated in terms of the theory of Optimal Transport. The motivation for this work is twofold. On the one hand, it paves the way for an interpretation in terms of statistical mechanics for these equations. On the other hand, it is a preparation to adress a question about certain types of degenerating geometric objects, asked independently by several mathematicians in the early 2000’s, motivated by the notion of mirror symmetry in string theory. In the third paper, the discovery of a new type of geometric objects is presented, which puts Einstein Field Equations into a more general setting. This paper also provides a connection to the algebraic point of view in geometry, similar to a link which has been studied by many mathematicians during the last 20 years that relates the Einstein Field Equations to the algebraic point of view in geometry. In the fourth paper we show that in many situations this connection can be expressed in surprisingly concrete terms using the geometry of polytopes.