In this talk, we will introduce Lie theory -- an indispensable mathematical toolkit for capturing symmetries in continuous spaces -- and show how to utilize it in the design of optimization and sampling algorithms under symmetry. We first present the basics of Lie groups, Lie algebras, and the adjoint actions associated to them. Subsequently, we will 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 will present the Harish-Chandra--Itzykson--Zuber (HCIZ) formula which converts partition functions (integrals) over Lie groups into sums over the corresponding (discrete) Weyl groups, enabling efficient sampling algorithms. Time permitting, we will mention the classification theorem for Lie groups that establishes the connection between continuous and discrete groups.

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