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This article is a brief presentation of results surrounding the fundamentalgap. We begin by recalling Bakry-Emery geometry and demonstrate connectionsbetween eigenvalues of the Laplacian with the Dirichlet and Neumann boundaryconditions. We then show a connection between the fundamental gap andBakry-Emery geometry, concluding with a presentation of the key ideas inAndrews's and Clutterbuck's proof of the fundamental gap conjecture. Weconclude with a presentation of results for the fundamental gap of trianglesand simplices.

La g\'eom\'etrie de Bakry-\'Emery et l'\'ecart fondamental