Mixed Integer Linear Programming (MILP) is an important computational problem and is often solved on large scale inputs across academia and industry. Sophisticated software packages exist for this purpose, and the algorithmic paradigm used by these packages is well-documented. These algorithms, which are slow and inefficient according to the paradigm of worst-case analysis, are nevertheless very effective in practice. This mismatch between theory and observation holds for nearly every algorithmic component used in MILP software and poses a challenge for the study of algorithmic complexity. The objective of this project is to develop new mathematical tools to understand the performance of these algorithms. The specific subject of this project will be the simplex method for Linear Programming (LP), a state-of-the-art algorithm which is essential in every successful MILP solver. A number of theoretical models have previously been proposed to explain why the simplex method is efficient in practice, including average-case analysis, smoothed analysis, and parameterized analyses. Despite this concerted effort, existing approaches have a number of key limitations. The project’s objective is to develop new mathematical approaches that address these limitations. The desired outcome of this line of work is a mathematical framework for explaining the performance of the simplex method, and whose claims can be tested in computational experiments on real-world data. The project’s main output will be in theorems about the geometry and combinatorics of LP problems and convex polyhedra. Where applicable, computational experiments will be performed.