Graphs and Patterns in Mathematics and Theoretical Physics

Suny At Stony Brook, Stony Brook NY

Investigators

Abstract

Abstract Award: DMS-0107455 Principal Investigator: Mikhail Lyubich The International Conference on Graphs and Patterns in Mathematics and Theoretical Physics at SUNY Stone Brook in 2001 will bring together researchers in several branches of mathematics and theoretical physics in which graphs play a central role, hoping to encourage fruitful interactions between these fields. This conference is a celebration of Dennis Sullivan's sixtieth birthday and addresses some of the topics of recent interest to Sullivan. Themes to be emphasized include (1) graphs and algebra, (2) graphs and discrete Riemannian geometry and discrete gauge theory, (3) graphs and bifurcation patterns in dynamical systems, (4) graphs and quantum field theory and topology. Minicourses on conference topics will be pointed towards graduate students and junior researchers. A graph in the sense used in the title of this conference is a data structure consisting of some number of "vertices," usually drawn as dots, joined by "edges," usually drawn as curves beginning at one vertex and ending at another. This kind of graph is a common combinatorial structure, encountered every day in problems of transportation and scheduling (consider a network of roads or telephone lines, for example), but useful in many less obvious ways. Some of the recent uses of graphs in geometry and physical theory grow out of Richard Feynman's celebrated idea of describing parts of quantum theory with graphs that represent possible lifetimes for a particle in space-time. Feynman's notion that a quantum theoretic quantity may represent a sum over all possible histories is reduced from a large, continuous calculation to a summation indexed by the different possible Feynman graphs. In a similar manner, modern geometers and physicists are exploring ways to work with the very large collections of all possible geometric structures on a particular space by reducing to manipulations of a family of graphs.