Explore the connections between cluster varieties, Landau-Ginzburg potentials, and projective representations in this 45-minute lecture by Béa de Laporte from the University of Cologne. Delve into the world of partial compactifications of Fock-Goncharov's cluster varieties, including flag varieties significant in algebraic group representation theory. Examine how these compactifications lead to Landau-Ginzburg potentials on dual cluster varieties, and how their tropicalizations define polyhedral cones parametrizing the theta basis. Discover a new interpretation of these potentials as F-polynomials of projective representations of Jacobian algebras. Follow the lecture's progression through introductory concepts, motivations, natural spaces, dual cluster varieties, examples, and results, culminating in a theorem and its proof. Gain insights into this collaborative work with Daniel Labardini-Fragoso, presented at the Institut des Hautes Etudes Scientifiques (IHES).