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Phone: (407) 823-6284;   Fax: (407) 823-6253;   MAP  207 2000-2001 MATHEMATICS COLLOQUIUM SERIES 01/19/01 Colloquium PROFESSOR PETER KUCHMENT DEPARTMENT OF MATHEMATICS & STATISTICS Wichita State University Differential Operators on Graphs as Models of Mesoscopic Systems Abstract: Differential (rather than difference) equations on graphs have arisen during the past decade as models of many important physical systems ranging from conjugated molecules to quantum wires, to photonic crystals. In these cases one often deals with propagation of electronic, electromagnetic, or acoustic waves in thin domains forming a graph-like structure. In the asymptotics of a very thin domain, one expects the wave propagation to be governed by a model on the resulting graph. The talk will contain a non-technical survey of this area of research. The particular topics that will be addressed include description of the situations when such differential or pseudo-differential models on graphs arise naturally, known and conjectured results on convergence to the graph models, and some conclusions one can derive from such models.