These are lecture notes for a mini-course given at the Cornell Probability Summer School in July 2013. Topics include lozenge tilings of polygons and their representation theoretic interpretation, the $(q,t)$-deformation of those leading to the Macdonald processes, nearest neighbor dynamics on Macdonald processes, their limit to semi-discrete Brownian polymers, and large time asymptotic analysis of polymer’s partition function.
"Integrable probability: From representation theory to Macdonald processes." Probab. Surveys 11 1 - 58, 2014. https://doi.org/10.1214/13-PS225