Description
There is a natural way to associate a linear operator to graphs and
surfaces, the so called Laplace operator. In the case of a
graph, this operator is simply a symmetric matrix reflecting the
combinatorial structure of the graph. In this project we will be
mainly concerned with compact surfaces and finite graphs, for which
the Laplace operators has a discrete/finite set of eigenvalues. The
set of these values is called the spectrum of the Laplace
operator. In the case of a surface with a Riemannian metric, these
eigenvalues are almost impossible to compute explicitly, but there are
many exciting results about relations between them and
geometric/topological properties of the surface. The study of these
connections is a topic of a special research area called Spectral
Geometry. The ultimate question is whether the surface/graph is
completely determined by the set of eigenvalues. This question has a
negative answer, and different surfaces/graphs with the same set
of eigenvalues are called isospectral. A nice general
construction to produce those isospectral examples was introduced in
1984 by Toshikazu Sunada. In spite of this negative result, a great
amount of the geometry and topology of a graph/surface is still
determined by the spectrum. Here is are some examples highlighting
connections between the Laplace spectrum and geometry and topology.
- There is an isoperimetric constant which measures the ratio
between the boundary and interior when cutting graphs/surfaces into
two pieces. In the case of graphs, this constant is also called the
edge expansion rate, and it reflects in some sense the
connectedness of the graph. Cheeger's inequality is a
fundamental inequality which relates this isoperimetric constant with
the lowest non-trivial eigenvalue of the Laplace operator. In the case
of graphs, this leads to the highly active area of expander graphs and
Ramanujan graphs, which have connections to many areas of
mathematics like group theory, random walks and number theory and even
in theoretical computer science (robust communication networks and
even construction of good error correcting codes).
- For hyperbolic surfaces (i.e., surfaces of constant curvature -1),
the eigenvalues below 1/4 are called small eigenvalues. It was
long conjectured that a closed hyperbolic surface of fixed genus g
cannot have more than 2g-2 small eigenvalues. (Students with
topological background will notice that this number is, up to sign,
the Euler characteristic of the surface.) This conjecture was proved
in 2009 by Jean-Pierre Otal and Eulalio Rosas.
- The so-called Weyl asymptotic shows that the Laplace
eigenvalues determine the total area of a closed surface with
arbitrary Riemannian metric. In the particular case of hyperbolic
surfaces, the Theorem of Huber states that the eigenvalues also
determine the lengths of all the closed geodesics of the surface. This
is also one of the many applications of the famous Selberg Trace
Formula.
Many other exciting results could be added to this list. After
acquiring the basic concepts, the students in this project will branch
out into more specific topics linking different areas of mathematics
in a beautiful way.
Prerequisites
Analysis III/IV and Differential Geometry III are
indispensable. Riemannian Geometry IV and Topology III are useful but
not necessarily required.
Resources
The following classical references are excellent introductions into
the project and cover a wide variety of aspects
- Pierre H. Berard: Spectral Geometry: Direct and Inverse
Problems, Springer-Verlag, 1986
- Peter
Buser: Geometry and Spectra of Compact Riemann Surfaces,
Birkhäuser, 1992
- Isaac
Chavel: Eigenvalues in Riemannian Geometry, Academic Press,
1984
- Fan R. K. Chung: Spectral Graph Theory, American
Mathematical Society Publication, 1997
- Shlomo Hoory, Nathan Linial, Avi Wigderson: Expander graphs
and their applications, Bulletin of the AMS 43, 2006
- Giuliana
Davidoff, Peter Sarnak, Alain Valette: Elementary Number
Theory, Group Theory, and Ramanujan Graphs, Cambridge University
Press, 2003
After specialisation, the students will read and study original
research articles related to their topics.
email: N Peyerimhoff
or O Post
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