We will review the relationship between Kirchoff's circuit laws, familiar from high school physics classes, and the Hodge Decomposition Theorem, familiar from somewhat more advanced courses. More precisely, we will re-cast Kirchoff's laws in the language of chain complexes and homology. Then we will prove that the resulting equations have an essentially unique solution using a simple finite-dimensional analogue of the Hodge Decomposition Theorem.
This talk can be understood as pure linear algebra, though its motivation comes from physics (electricity) and from differential geometry (Hodge theory). No prerequisites are assumed past linear algebra.