A central goal of my research is to mathematically understand how intricate structures behave as their “complexities” tend to infinity. I have studied this question under various guises, including analyzing statistical mechanical systems (crystal growth and interacting particle systems) with many particles; random matrices of high dimension; and surfaces of large genus. My work develops algebraic (symmetric functions and representation theory), analytic (multi-scale analysis and elliptic regularity), and probabilistic (percolation and coupling) frameworks to analyze these systems, with the distinctive feature of often employing these three methodologies in tandem to combine their strengths. In this way, I have registered several advances in the above topics, including the most general proofs of boundary effects in surface (dimer and six-vertex) models; the first exhibition of extended states in heavy-tailed matrices; and proofs of conjectures enumerating flat surfaces of large genus.