Thesis advisor: Pankaj Mehta / Thesis advisor: Ziqiang Wang / Nature has revealed an astounding degree of phylogenetic and physiological diversity in natural environments -- especially in the microbial world. Microbial communities are incredibly diverse, ranging from 500-1000 species in human guts to over 1000 species in marine ecosystems. Historically, theoretical ecologists have devoted considerable effort to analyzing ecosystems consisting of a few species. However, analytical approaches and theoretical insights derived from small ecosystems consisting of a few species may not scale up to diverse ecosystems. Understanding such large complex ecosystems poses fundamental challenges to current theories and analytical approaches for modeling and understanding the microbial world. One promising approach for tackling this challenge that I develop in my thesis is to adapt and expand ideas from statistical mechanics to theoretical ecology. Statistical mechanics has helped us to understand how collective behaviors emerge from the interaction of many individual components. In this thesis, I present a unified theoretical framework for understanding complex ecosystems based on statistical mechanics, random matrix theories, and convex optimization. My thesis work has three key aspects: modeling, simulations, and theories. Modeling: Classical ecological models often focus on predator-prey relationships. However, this is not the norm in the microbial world. Unlike most macroscopic organisms, microbes relie on consuming and producing small organic molecules for energy and reproduction. In this thesis, we develop a new Microbial Consumer Resource Model that takes into account these types of metabolic cross-feeding interactions. We demonstrate that this model can qualitatively reproduce and explain statistical patterns observed in large survey data, including Earth Microbiome Project and the Human Microbiome Project. Simulations: Computational simulations are essential in theoretical ecology. Complex ecological models often involve ordinary differential equations (ODE) containing hundreds to thousands of interacting variables. Typical ODE solvers are based on numerical integration methods, which are both time and resource intensive. To overcome this bottleneck, we derived a surprising duality between constrained convex optimization and generalized consumer-resource models describing ecological dynamics. This allows us to develop a fast algorithm to solve the steady-state of complex ecological models. This improves computational performance by between 2-3 orders of magnitude compared to direct numerical integration of the corresponding ODEs. Theories:Few theoretical approaches allow for the analytic study of communities containing a large number of species. Recently, there has been considerable interest in the idea that ecosystems can be thought of as a type of disordered systems. This mapping suggests that understanding community coexistence patterns is actually a problem in "spin-glass'' physics. This has motivated physicists to use insights from spin glass theory to uncover the universal features of complex ecosystems. In this thesis, I use and extend the cavity method, originally developed in spin glass theories, to answer fundamental ecological questions regarding the stability, diversity, and robustness of ecosystems. I use the cavity method to derive new species backing bounds and uncover novel phase transitions to typicality. / Thesis (PhD) — Boston College, 2021. / Submitted to: Boston College. Graduate School of Arts and Sciences. / Discipline: Physics.