Quantum Field Theoretic and Combinatorial Perspectives on Intersection Numbers on the Moduli Space of Complex Curves: Explicit calculations in the Quartic Kontsevich Model & logarithmic concavity

The moduli space of complex curves, or Riemann surfaces, is endowed with a recursive structure, which has been extensively used in the investigation of the moduli space since its construction in the previous century. Algebro-geometric invariants of this moduli space, which are called intersection numbers, are related correlation functions of matrix models. In this thesis the Quartic Kontsevich Model as well as the LSZ model act as two examples for this. The origin of both matrix models, which in a particular limit are related to quantum field theoretic investigations, can be traced back to the quest of finding a unified physical theory of our universe by constructing a non-trivial interacting model on non-commutative space. It is, however, in the truncated matrix models, where most algebraic structures are evident. These are studied here applying the universal framework of topological recursion to explicitly compute expressions for the correlators of low topological type in terms of intersection numbers on the moduli space of curves. While the LSZ model, a matrix model of complex matrices, can be treated within the original topological recursion, its hermitian counterpart, the Quartic Kontsevich Model requires a generalization to what is called blobbed topological recur- sion due to its more involved loop equations. The additional data of this framework is provided for the correlators of the Quartic Kontsevich Model for low topological type. These explicit results are set to supply future research towards an understanding of the integrable structure of the Quartic Kontsevich Model (in the sense of the Japanese school) with useful data. This integrable structure is a consequence of the deep recursive nature of the inves- tigated models. It is, prominently, also shared by the Kontsevich model itself, the cubic analogue of the model studied here, which encodes the intersection numbers on the moduli space. In order to provide a different perspective on this, a relation between intersection numbers and a combinatorial approach to the lattice-point count in polytopes called Ehrhart theory is studied. A restricted setting allows to approach the natural question of the significance of those polytopes corresponding to intersec- tion number. By determining explicit data in the setting of Ehrhart theory this class of polytopes is characterized. Beyond the fact that this data has an interpretation in terms of a specific type of partitions, called order-consecutive partition sequences, in that process interesting patterns are observed. In particular logarithmic concavity is proved for the expansion data by different techniques, which reflect the significant and profound structure of the moduli space of curves.