Spectral theory on combinatorial and quantum graphs

par Evans Harrell ( Georgia Institute of Technology- Atlanta-USA)

ce cours sera donné en anglais. Slides : harrellkairouancimpa2016-1.pdf (1.84 Mo), harrellkairouancimpa2016-2.pdf (1.01 Mo), harrellkairouancimpa2016-3.pdf (1.33 Mo), harrellkairouancimpa2016-4.pdf (1.4 Mo)

Abstract: Many mathematical models in applied science, from microelectronic circuits to social networks, to biological systems, are based on graphs, that is, discrete sets of vertices with connections. In a combinatorial graph the only information needed is which vertices are connected to which, while in a quantum graph, the edge between vertices carries a differential operator of Sturm-Liouville type. I will describe some of the models that lead to graphs and focus on the problem of learning about their structure through the eigenvalues of operators on the graph. After developing the standard tools, which are similar to those that are used in understanding problems of vibrating membranes and quantum mechanics, I will develop some novel tools of spectral analysis and provide a selection of open research problems.