If (X,d) is a metric space, X is said to be Menger convex if for any distinct x and y in X, there is a z (not equal to x or y) such that d(x,z) + d(y,z) = d(x,y). In a complete and locally compact metric space, Menger convexity implies the metric space is in fact a length space. Furthermore, propositions regarding Menger convexity closely mirror that of standard convexity in a linear space. For this talk, a basic understanding of metric spaces is helpful, but not required. All other concepts (length space, convexity in a linear space, etc.) will be defined.

Rotation systems are structures that may be placed on graphs that help us determine the local orientation of edge ends at a vertex. Rotation systems may be thought of as algebraic representations of embeddings of graphs. We present some examples of the results we have observed about rotation systems, and their implications for graph duals, and abstract algebra. We may use rotation systems to study embeddings of graphs into surfaces of arbitrary genus. We also provide a discussion of how to identify equivalent rotation systems.

The "Higher Dimensional Continuum Dislocation Dynamics" (hdCDD) model can capture curved dislocations in a crystal under plastic deformation. Parameterized dislocation lines are mapped to a higher dimensional configuration space by considering the angle of the tangent to a curve with respect to the Burgers vector as an additional variable. This model and a solution approach using a Runge-Kutta Discontinuous Galerkin (RKDG) finite element method with an unusual basis will be discussed.