Partially ordered sets, also known as posets are interesting objects in combinatorics and computer science. Some related objects such as ideals, chains, antichains, linear extensions has been studied by computer scientists and combinatorists. I, in my graduate research, am interested to work on interesting, algorithmic problems on posets. I can categorize the problems that I want to work on in the following way:

The first category is "mixing problems": Markov chains are interesting for computer scientists because they sometimes make it possible to find an approximate solution for hard sampling problems. The problem of finding a fast Markov chain for a sampling problem is called a mixing problem.

We will talk about some mixing problems in this thesis as well as the problem "how we can make some already known Markov chains converge faster by adding some determinism to them". I will also talk about some diagnostic methods and whether it is possible to diagnose whether a Markov chain is mixed or not.

The second category is "homomesy": Homomesy is a Greek word meaning same average and it refers to the phenomenon of having some same statistic along orbits in a given set. This phenomenon have been introduced by J.Propp and T. Roby. I will discuss some instances when homomesy is observed and proven.