Title: Mathematical and practical lessons learned from mentoring undergraduate students
Abstract: I will discuss some recent work that was completed this past summer as part of the Columbia mathematics undergraduate summer research program, available at arXiv:2508.05603. In this project, we prove probabilistic invariances for a discrete directed polymer model by uncovering algebraic structures present in the system. On the probabilistic side, invariances of this flavor go back to the 1950s work of Burke in queuing theory. At the end of the talk, I will discuss some practical lessons learned from advising students in a summer undergraduate research program. This work was done jointly with undergraduate students Eva Engel, Benjamin Jasper Kra-Caskey, Oleksandr Lazorenko, Caio Hermano Maia de Olivera, Ivan Wong, and Ryan Xu, as well as a graduate TA, Xinyi Zhang.