There is a strong interplay between combinatorics and algebraic geometry that has recently led to significant advances in both disciplines. This project focuses on the development of combinatorial and computational tools in the study of algebraic varieties, and applying these techniques to questions about matroids. It aims to (i) develop new tools using divisor theory and toric degenerations for attacking two long-standing and extremely important open problems in combinatorics and algebra, namely the conjectures of Stanley and Postnikov-Shapiro, and (ii) devise new algorithms for computational problems in divisor theory with concrete real-world applications in system reliability (industrial and risk engineering) and the analysis of neural networks. Here is one of the applications of the problems studied in this project: Network Reliability is an important and well-studied problem appearing in various contexts, including railway management, economics, epidemiology, and redundancy elimination in electronic systems and electrical power grids. To find the reliability of a network, it is critical that we have efficient algorithms. However, the problem is known to be NP-hard. On the other hand, the problem has an algebraic formulation that can be exploited to obtain faster algorithms for many real-world instances. This project will study the algebraic formulation and exploit it to characterize families of networks in which reliability can be computed efficiently.