This is a joint Discrete Mathematics / Algebraic Geometry seminar.  The tropical Laplacian is a symmetric square matrix associated with a balanced graph on a sphere, defined in a similar way to the Laplacian of an abstract graph. We will see by examples how the tropical Laplacian appears in the study of polytopes, matroids, and graphs. The speaker will pose many elementary questions on tropical Laplacians. A related but independent talk explaining the geometric origin of the questions will be given in the Tuesday (March 4) Algebraic Geometry Seminar.