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
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
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
Based on joint works with Gregory Berkolaiko, Iosif Polterovich, Uzy
Smilansky and Idan Oren.
For further info: Yehuda Pinchover <firstname.lastname@example.org>
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