Tel Aviv University  -  Horowitz Seminar
 
Dear all,
 
The academic year is about to restart and the Horowitz seminar on
Probability, Ergodic Theory and Dynamical Systems renews its
activity.
We start with a special pre-semester seminar already this Monday! All
are welcome.
 
Speaker: Ram Band, University of Bristol
Title: Spectral Geometry on Graphs
 
Date: Monday, October 15
Time: 14:30
Place: Schreiber 309
 
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 sign changes of the
eigenfunction on tree graphs equals to the position of the
corresponding eigenvalue in the spectrum minus one. Is the reverse
true? If yes, one can tell a tree just by counting the number of its
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 number of sign changes.
Next, I will discuss the band spectrum obtained by varying the
magnetic phases on the graph. I will prove that the magnetic
band-to-gap ratio (quality of conductance) is a universal topological
quantity of a graph. This result highlights the spectral geometric
importance of this invariant and sheds a new light on previous works
about periodic potentials on graphs.
The talk contains content of a work in progress with Gregory
Berkolaiko.
All concepts will be explained and no previous knowledge of the topic
is required.
 
Best regards,

   Ron

Seminar webpage:
 <http://www.math.tau.ac.il/~peledron/Horowitz_seminar/Horowitz_seminar.html>
 
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Technion Math Net-2 (TECHMATH2)
Editor: Michael Cwikel   <techm@math.technion.ac.il> 
Announcement from: Ron Peled   <peledron@post.tau.ac.il>