Automorphism Groups of K3 Surfaces and 𝔸¹-Multiplicities of Tropical Quartic Curves

This thesis investigates two distinct but computationally intensive problems in algebraic and tropical geometry. The first part addresses the automation of generating sets for automorphism groups of K3 surfaces, along with their associated rational curves and fundamental domains. By analyzing a computational framework in detail, this work clarifies the theoretical foundations and enables systematic verification of these groups, highlighting the complexity of their structure. Existing methods from the literature were independently assessed to ensure the soundness and reliability of the underlying approach rather than its numerical implementation. This combined focus on methodological scrutiny and automation provides a more robust basis for further study of K3 surface automorphisms. Ultimately, the investigation confirms the theoretical robustness of the framework, while demonstrating that practical computational feasibility remains a nontrivial challenge. The second part of this thesis focuses on computing 𝔸¹-multiplicities for the bitangent classes of smooth generic tropical quartic curves and leveraging these computations to analyze the structure of the associated secondary fan. By systematically determining these multiplicities, the study quantifies the proportion of the secondary fan corresponding to the invariant 14ℍ, providing a refined perspective on the geometric distribution of bitangent shapes. I employ a hierarchical Bayesian framework for this analysis, which not only integrates prior geometric knowledge but also models variability at multiple levels, both globally across the entire fan and locally within specific secondary cones and their subcones. This hierarchical structure allows for more nuanced inference, especially in the presence of sparse or unevenly distributed data, and enables us to borrow statistical strength across related regions of the secondary fan. As a result, the analysis yields more stable posterior estimates of multiplicity frequencies and offers deeper insight into the geometry of the parametric space for smooth tropical quartics.