Graphs and Patterns in Mathematics and Theoretical Physics

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AbstractAward: DMS-0107455Principal Investigator: Mikhail LyubichThe International Conference on Graphs and Patterns inMathematics and Theoretical Physics at SUNY Stone Brook in 2001will bring together researchers in several branches ofmathematics and theoretical physics in which graphs play acentral role, hoping to encourage fruitful interactions betweenthese fields. This conference is a celebration of DennisSullivan's sixtieth birthday and addresses some of the topics ofrecent interest to Sullivan. Themes to be emphasized include (1)graphs and algebra, (2) graphs and discrete Riemannian geometryand discrete gauge theory, (3) graphs and bifurcation patterns indynamical systems, (4) graphs and quantum field theory andtopology. Minicourses on conference topics will be pointedtowards graduate students and junior researchers.A graph in the sense used in the title of this conference is adata structure consisting of some number of "vertices," usuallydrawn as dots, joined by "edges," usually drawn as curvesbeginning at one vertex and ending at another. This kind ofgraph is a common combinatorial structure, encountered every dayin problems of transportation and scheduling (consider a networkof roads or telephone lines, for example), but useful in manyless obvious ways. Some of the recent uses of graphs in geometryand physical theory grow out of Richard Feynman's celebrated ideaof describing parts of quantum theory with graphs that representpossible lifetimes for a particle in space-time. Feynman'snotion that a quantum theoretic quantity may represent a sum overall possible histories is reduced from a large, continuouscalculation to a summation indexed by the different possibleFeynman graphs. In a similar manner, modern geometers andphysicists are exploring ways to work with the very largecollections of all possible geometric structures on a particularspace by reducing to manipulations of a family of graphs.