  In the last few years, the notion of symmetry has provided a powerful and
essential lens to view several optimization or sampling problems that arise in
areas such as theoretical computer science, statistics, machine learning,
quantum inference, and privacy. Here, we present two examples of nonconvex
problems in optimization and sampling where continuous symmetries play --
implicitly or explicitly -- a key role in the development of efficient
algorithms. These examples rely on deep and hidden connections between
nonconvex symmetric manifolds and convex polytopes, and are heavily
generalizable. To formulate and understand these generalizations, we then
present an introduction to Lie theory -- an indispensable mathematical toolkit
for capturing and working with continuous symmetries. We first present the
basics of Lie groups, Lie algebras, and the adjoint actions associated with
them, and we also mention the classification theorem for Lie algebras.
Subsequently, we present Kostant's convexity theorem and show how it allows us
to reduce linear optimization problems over orbits of Lie groups to linear
optimization problems over polytopes. Finally, we present the Harish-Chandra
and the Harish-Chandra--Itzykson--Zuber (HCIZ) formulas, which convert
partition functions (integrals) over Lie groups into sums over the
corresponding (discrete) Weyl groups, enabling efficient sampling algorithms.
