In this talk, I will begin by introducing graphons, which are infinite-size limits of adjacency matrices of sequences of growing graphs. I will then define graph reaction-diffusion (RD) equations, which are systems of differential equations that are defined on the nodes of a graph. For a sequence of growing graphs that converges to a graphon, the solutions of the sequence of graph RD equations also converge. The limiting solution solves a nonlocal differential equation that we call a graphon RD equation. Furthermore, the graph RD equation is related to a stochastic birth-death process on graphs. I will show that this birth-death process converges to the graphon RD equation via a hydrodynamic limit.

In 1958, Hirzebruch produced a generating function for the Hodge numbers of a smooth hypersurface Z in P^n, and in 1969, Griffiths produced the residue map from the space of polynomials to differential forms. If a group G acts linearly on Z, the Griffiths residue map is G-equivariant. This map allows us to describe the primitive cohomology of Z in terms of graded pieces of a particular ring — the Griffiths ring. In this talk we generalize Griffiths' construction to smooth hypersurfaces in Grassmannians Gr(k,n), assuming some mild divisibility constraints on k,n, and the degree of the hypersurface.

Non-linear Random Matrix Theory (RMT) has recently emerged as a powerful paradigm for the theoretical understanding of deep learning theory. Throughout recent works, a universality principle, the \textit{Gaussian Equivalence Theorem} (GET), has become an indispensable tool allowing for the behavior of complex nonlinear neural networks to be understood through tractable linear kernel models. This thesis contributes to this emerging field, first by using the GET universality principle to derive a novel scaling law in Neural Tangent Kernel (NTK) regression, and second by studying the implications of this idealized linear equivalence on a high-dimensional nonlinearly separable dataset.