Splines are a fundamental tool across applied mathematics and analysis, used in areas such as computer graphics, engineering models, and data interpolation. Our research considers a more abstract idea of splines in which we work with an algebraic-combinatorial generalization of splines on an edge-labeled graph dual to a triangulation. A spline on a graph is a way of labeling the vertices such that if two vertices share an edge, then their vertex-labels differ by a multiple of the edge-label. As part of a longstanding open problem sometimes called the “upper-bound conjecture,“ we are developing approaches to find a basis (and/or the dimension) of the space of splines when using polynomial labels of degree at most 2. We will introduce the basic mathematical objects involved in this research and build up to some approaches we’ve been working on to answer this question.
by Hope Pungello '21