This thesis develops statistical and algebraic methods for multivariate extremes. First, it studies sparse graphical models for Hüsler-Reiss distributions by introducing a precision matrix that encodes extremal conditional independence. It establishes graphical completion results for arbitrary connected graphs and uses them for consistent inference. Second, it develops stochastic representations of multivariate generalized Pareto distributions conditioned on general half-spaces. These representations yield new variogram and density formulas and show how the choice of half-space affects estimator bias and variance, with ensembles reducing variance. Third, it provides a common algebraic framework for covariance representations in extreme value theory and compositional data analysis using oblique projections, Moore-Penrose inverses, and variogram mappings. This framework clarifies links between parametrizations and supports the transfer of statistical methods between the two fields.