**Trace Diagrams and their Applications** , Olin College Research Seminar, Feb. 5, 2010.

**Abstract:** Trace diagrams are graphs (with some additional structure) whose edges may be labeled by matrices. Every trace diagram may also be interpreted as a multilinear function. Because of this, many results of linear algebra may be proven in a completely rigorous way by reasoning with diagrams; the resulting proofs are often more elegant than those done using traditional notation. This talk will demonstrate a few short proofs of classical linear algebra using the diagrams. We will also show how they apply to much deeper questions, in particular to the understanding of geometric structures on surfaces.