Appears in collection : Nexus Trimester - 2016 - Fundamental Inequalities and Lower Bounds Theme

Entropy inequalities (Shannon and non-Shannon) have been used to obtain bounds on the solutions to a number of problems. When the problems are restricted to the linear case, these bounds can sometimes be improved using linear rank inequalities such as the Ingleton inequality, which hold for entropy vectors coming from linear variables (also known as representable polymatroids) but not necessarily for all entropy vectors. This talk will describe methods for producing linear rank inequalities, or finding counterexamples to putative linear rank inequalities, comparing them to the methods for proving non-Shannon entropy inequalities, and show the differences between known results for the entropy region and for its linear analogue (actually analogues, since one has to divide into cases based on the characteristic of the underlying field).