Technion - Israel Institute of Technology Department of Mathematics ===================================== PDE AND APPLIED MATHEMATICS SEMINAR ===================================== DATE: Monday, 24.9.2012 SPEAKER: Rami Band (University of Bristol) TITLE: Spectral Geometry on Graphs PLACE: Room 814, Amado Mathematics Building, Technion TIME: 14:30 ABSTRACT: The talk's theme is the extraction of geometric information about graphs (metric or combinatorial) from the spectra of the graph's Schroedinger operators (continuous or discrete), and from the distribution of sign changes on the corresponding eigenfunctions. These include questions such as e.g., the ability to "hear the shape of the graph"; the extent to which the spectral sequence and the sequence of the number of sign changes (or number of nodal domains) complement or overlap each other; the derivation of topological information from the study of the response of the spectrum to variation of scalar or magnetic potentials on the graph, etc. In the present talk I shall illustrate this research effort by reviewing several results I obtained recently. The first example answers the question "Can one count a tree?" which appears in the following context: It is known that the number of nodal domains of the eigenfunction on tree graphs equals to the position of the corresponding eigenvalue in the spectrum (ordered as a non-decreasing sequence). The same is true for the number of sign changes. Is the reverse true? If yes, one can tell a tree just by counting the number of its nodal domains or sign changes. For the proof I shall introduce an auxiliary magnetic field and use a very recent result of Berkolaiko and Colin de Verdiere to connect the spectrum and the zero count. Similarly, by introducing an auxiliary potential on the vertices, I shall discuss the analogous problem for counting the nodal domains. Finally, I will consider the band spectrum obtained by varying the magnetic phases on the graph, and show that the magnetic band-to-gap ratio (quality of conductance) is a universal topological quantity of a graph. Based on joint works with Gregory Berkolaiko, Iosif Polterovich, Uzy Smilansky and Idan Oren. For further info: Yehuda Pinchover <pincho@techunix.technion.ac.il> For past and future Applied Math/PDE seminars see: <http://www.math.technion.ac.il/pde/seminar.html> NOTE: This announcement is being announced using the old Techmath system because of some remaining difficulty with the new system. But please continue trying to submit events via the new system. M.C. --------------------------------------------------------- Technion Math. Net (TECHMATH) Editor: Michael Cwikel <techm@math.technion.ac.il> Announcement from: Yehuda Pinchover <pincho@tx.technion.ac.il>