Constant Vector Curvature in 3 Dimensions: A Complete Description

A relatively new area of interest in differential geometry involves determining if a model space has the properties of constant vector curvature or constant sectional curvature. The natural setting in which to begin studying these properties is in 3-dimensional space. This paper in particular examines these properties in the Lorentzian setting, where all Ricci operators take on one of four Jordan-Normal forms. We determine that three of these four model space families (Ricci operators) possess the property of constant vector curvature, and that under an orthonormal basis, only the diagonalizable family has constant sectional curvature, and that is only when the Ricci Operator has precisely one eigenvalue. By examining these families together, we draw some interesting and unifying conclusions that may be useful for exploring these properties in higher dimensions. Presenter(s): Gabriel Lopez Major: Mathematics Faculty Mentor: Dr. Corinne Johnson Title: Modeling Self-Assembled DNA Nanotubes Abstract: Emerging laboratory techniques have been developed using the Watson-Crick complementarity properties of DNA strands to achieve self-assembly of graphical complexes. One recent focus in DNA nanotechnology is the formation of nanotubes, which we model with a two-dimensional lattice that wraps around to form a tube. The vertices of the lattice graph represent k-armed branched junction molecules, called tiles. Using concepts from graph theory, we seek to determine the minimum number of tile and bond-edge types necessary to create a desired self-assembled complex. Results are known for certain infinite classes of graphs, but are yet to be found for several other classes. Specifically, results are unknown for lattice graphs which motivates our study of triangle and hexagonal lattice graphs. While some laboratory settings allow for the possibility of the formation of smaller complexes using the same set of tiles, we examine the problem under the restriction that no smaller complete complex may be formed.