Spatial Graphs and Their Application to Complex Molecular Structures

点击翻译按钮获取中文摘要

The broad goal of this research project is to use the tools of topology and geometry to help molecular biologists and chemists better understand the structure and behavior of DNA, proteins, and complex synthetic molecules. The topological model under study would help molecular biologists by simplifying the analysis of the site-specific recombination mechanism for closed circular DNA molecules. The investigator also aims to identify the forms of knots, links, and non-planar graphs that arise in proteins, and to model how these complex structures may have occurred. This information may offer valuable insights into protein folding mechanisms and degradation pathways. Synthetic organic molecules are normally too small to see with an electron microscope; when chemists synthesize a complex structure they use data from nuclear magnetic resonance (NMR) spectroscopy to provide evidence that the molecular structure has a particular form. Since these structures are large enough to be somewhat flexible, both topology and geometry have to be taken into account when comparing the symmetry properties of the NMR data to those of a physical model. The investigator is working with organic chemists to identify different types of symmetries exhibited by complex structures and to design new structures with interesting symmetry properties. In contrast with knots and links, whose topology depends exclusively on their embedding in the three dimensional sphere, the intrinsic structure of some graphs can affect the topological properties of every embedding of the graph in a given three dimensional manifold. For example, some graphs have the property that for any embedding G of the graph in a three-manifold M, there is no orientation reversing homeomorphism of the pair (M,G). Such a graph is said to be intrinsically chiral in M. The investigator will work on characterizing which graphs are intrinsically chiral in the three-sphere and in other three-dimensional manifolds, as well as determining other properties of embedded graphs which are independent of the particular embedding of the graph. The project draws on three-manifold results including Jaco-Shalen and Johannson characteristic decompositions, Mostow's rigidity theorem, Thurstons' hyperbolization theorem, and the classification of Seifert manifolds, as well as techniques from knot theory and the theory of tangles.