Speaker:Robert Curl   

Title: A Pedestrian's Encounter with Specialized Polyhedra

Abstract: In the Fall of 1985, we stumbled into a new three dimensional world of carbon chemistry based upon polyhedra consisting of twelve five-membered rings and two or more six-membered rings of carbon atoms. The real discovery was that these actually form when carbon vapor condenses under the right conditions. This lead us into a very old polyhedral structure, the truncated icosahedron, some quite old mathematics created by Descartes and Euler, and some rather old results of group theory. We did no graph theory. Others notably the theoretical group at Texas A&M Galveston reaped a rich harvest of new mathematical results. Some efforts, which turned out to be adventures, in teaching these concepts to middle school children will be described. The later development of single walled carbon nanotubes provided yet another set of mathematical applications for them and others.

Speaker: Douglas J. Klein

Title: Fullerenes & Beyond

Abstract: In 1985 Kroto, Heath, O'Brien, Curl, & Smalley detected a special C60 molecule for which they proposed a "uniquely elegant" structure: a truncated icosahedron which as a molecule they called "buckminsterfullerene". Later this C60 structure was verified, and the material obtained in macroscopic quantities. Also several other carbon-based tri-valent hexagonal-ring-motif "bucky" structures have been experimentally realized: further bucky-polyhedra, chains of inter-connected bucky-hedra, bucky-onions, bucky-tori, bucky-cones, semi-infinite bucky-networks, and especially bucky-tubes. Thence there is a large rapidly growing area of novel elemental carbon structures, beyond the classical cases of graphite & diamond.

             Besides a bit of history, a review is made of "bucky"-structure mathematical systemics, including work from Galveston. Some simple mathematical rationalization of chemically plausible structures is noted, especially as viewed in the context of graph theory. Note is made of the relevance of dimer coverings, of adjacency-matrix eigenspectra, of the "isolated pentagon rule", of "combinatorial curvature", of Euler-Poincare characteristics, & more. A range of plausible carbon nano-structures beyond those experimentally realized are noted, e.g., structures which are scrolled, seamed, different genuses, or negatively curved.

Speaker:Glenn Tesler

Title: The Fragile Breakage versus Random Breakage Models of Chromosome Evolution

Co-authors: Qian Peng and Pavel Pevzner
                     University of California,
                     San Diego Department of Computer Science and Engineering

Abstract:For many years, studies of chromosome evolution were dominated by the random breakage theory, which implies that there are no rearrangement hot spots in the human genome. In 2003, Pevzner and Tesler argued against the random breakage model and proposed an alternative "fragile breakage'' model of chromosome evolution. In 2004, Sankoff and Trinh argued against the fragile breakage model and raised doubts that Pevzner and Tesler provided any evidence of rearrangement hot spots.

            We investigate whether Sankoff and Trinh indeed revealed a flaw in the Pevzner and Tesler arguments. We show that Sankoff and Trinh's synteny block generation algorithm is flawed and that their parameters do not reflect the realities of the comparative genomic architecture of human and mouse. We further argue that if Sankoff and Trinh had fixed these problems, their arguments in support of the random breakage model would disappear.

Speaker:Vijaya Ramachandran

Title: The Diameter of Sparse Random Graphs

Abstract: We derive an expression of the form : c ln n + o(ln n) for the diameter of a sparse random graph with a specified degree sequence. The result holds a.a.s., assuming certain convergence and supercriticality conditions are met. The classes of random graphs for which this result applies include the classical random graph model G_{n,p} when p=(1+b)/n for any positive constant b, and `power-law' graphs when the number of edges is linear in the number of vertices.

              The proof is constructive and yields a method for computing the constant c. Our methods also establish that almost all pairs of vertices that are connected by a path in such a random graph have almost the same shortest path distance.

              Among the technical contributions of this paper are tools for bounding the size of breadth-first search neighborhoods in sparse random graphs, and a characterization of the structure of a typical longest simple path in such graphs. (This is joint work with Dan Fernholz.)

Speaker: Terry A. McKee

Title: Thinking outside of the graph

Abstract: A variety of well-studied graph classes can be profitably viewed using representations that are themselves graphs, with nodes that correspond to specific sorts of subgraphs of the original graphs. This talk will focus on tree representations, generalizing the classical clique-tree intersection graph representations of chordal graphs. These ideas are related to applications of chordal graphs to statistics (decomposable loglinear models) and matrices (sparse-inverse determinantal formulas) that might conceivably have analogs for other graph classes; we'll look at one related application to probability (Bonferroni-type bounds).