A striking phenomenon in probability theory is universality, where different probabilistic models produce the same large-scale or long-time limit. One example is the Kardar-Parisi-Zhang (KPZ) universality class, which contains a wide range of natural models, including growth processes modeling bacterial colonies, eigenvalues of random matrices, and traffic flow models originating from mRNA translation. Historically, mathematical proofs in the KPZ class have mostly relied on algebraic formulas. In this talk, I will introduce a general strategy of combining formulas with geometric and probabilistic analysis. I will focus on Exponential Last Passage Percolation, a pivotal model in the KPZ class, and discuss a few examples of results achieved using this perspective: correlation structure, statistics along geodesics, and behavior under perturbations. No prior knowledge of this topic will be assumed.