Random growth models with half space geometry

Random growth models in 1+1 dimension capture the behavior of interfaces evolving in the presence of noise. These models are expected to exhibit universal behavior including intriguing occurrences of random matrix distributions, but we are still far from proving such results even in relatively simple models. A key development which has led to recent progress is the discovery of exact formulas for certain models with a rich algebraic structure. I will discuss some of these results, with a focus on models where a single boundary wall is present, as well as applications to other areas of probability.