There are certain confluences involving graphs in physics,molecular biology, number theory, Teichmueller theory, dynamicalsystems, fluid dynamics, and 3-manifold invariants, which in turnreflect possibly deeper relations among their theoreticalfoundations which have yet to be developed.  The investigatorsorganize a workshop to consider mathematical structures withapplication to specific problems in fields increasingly connectedto mathematics, for example the global geometry of macromolecularfolding, including both proteins and RNA.  Recent mathematicalresults allow for a treatment of the full spatial foldingproblem: how do the physico-chemical properties of macromoleculesdetermine their spatial folding characteristics?  Roughly,folding can be modeled as a family of weighted arcs, each arccorresponding to a chemical bond, the weight the Boltzmannstrength of the bond.  The graphs formed from combining theseweighted ars give a space of weghted graphs representing allpossible foldings.  Can an appropriate energy functional be foundon such a space to study the gradient flow minimizing the totalenergy?  Several participants use this approach to understanddynamic folding.  Related spaces of graphs in biology are those ofphylogenetic trees formed by speciation.  Though the globalgeometry of these spaces is understood, the nature of the localgeometry is not known; some of the participants study thesequestions.  A recent discovery in string theory is the exoticstructure of an "operad" associated with spaces of graphs.  Someof the researchers at the workshop investigate these higher orderstructures, and their possible use in biology.     Graph theory is a branch of mathematics dealing with objectscalled graphs.  A graph consists of a set of points or verticesthat are connected by links or edges.  An ordinary map, forexample, is a graph whose vertices are towns and whose edges areroads.  Although graphs are abstract objects, they have been usedto represent a wide variety of real objects or processes.  Afamiliar use of graphs is to represent the evolutionaryrelationships between a group of organisms -- family trees.  It isa testimony to their great versatility and usefulness that graphshave been used in fields as diverse as biology, economics, andcomputer science.  In the field of biotechnology, there has beenan explosion in the use of graphs to represent the complexphysical structure of the basic proteins produced by DNA.  Thiswork has important applications in understanding genetic diseasesand in designing medicines to cure them.  Because methods based ongraph theory have been developed independently in many fields, itis extremely likely that sharing problems and results betweenfields will lead to advances.  The purpose of this workshop is tobring together mathematicians and biologists working on problemsinvolving graph theory to describe and discuss their work in aninterdisciplinary setting.