Information Theory in Network Coding (Nan), and Algebraic Approaches to the Belgian Chocolate Problem (Charles)

Information Theory in Network Coding
Ting-Ting Nan
Network coding has been used in many applications. However, one of the basic problems, finding the coding capacity of most networks, is still unsolved. The entropy region is central to computing network coding capacities, but it is not well characterized for n > 3 random variables.
Each point in the entropy region for n=4 has an Ingleton score. This score is bounded above, but its supremum is still unknown. The Four-Atom Conjecture, which said that s < 0.089373, was disproved by Matus and Csirmaz in 2013. In this talk, we introduce a systematic approach that obtains larger values of s than Matus and Csirmaz and investigate the true value of its supremum. Algebraic Approaches to the Belgian Chocolate Problem Zach Charles Machinery and other physical systems are often designed to have built-in feedback that regulates their behavior. We are often concerned with when the resulting feedback loop is stable. This leads to the issue of representing these systems mathematically and what it means for the system to be stable. We will present a famous open problem concerning the stabilization of such systems. The problem asks for which values of a process parameter $\delta$ we can stabilize a specific feedback loop. In contrast to previous methods that used optimization and search techniques, we will discuss recent algebraic methods that have led to largest known $\delta$ for which the system can be stabilized.