Quantum graphs are an important example of metric measure spaces: they consist
of finite collections of intervals glued at their endpoints (vertices) in a graph-like
fashion. Under appropriate transmission conditions in the vertices, the heat
equation is well-posed and may enjoy additional properties: most notably,
conservation of mass. It turns out that the solutions converge to equilibrium at a
rate that is given by the first non-trivial eigenvalue of the relevant realization of the
Laplacian. It has been known since the 1980s that this quantity mirrors the
connectivity and further combinatorial features of metric graphs. After offering an
invitation to the lively topic of spectral geometry of quantum graphs, I will present
a new quantity that has recently gained popularity: the torsional rigidity of a
quantum graph, a quantity — routinely considered for Dirichlet Laplacians on
bounded planar domains — whose definition goes back to Pólya. Its interesting
geometric properties as well as its interplay with spectral theory have been
discovered very recently: I will present an overview of the main result in this field.
This is joint work with Marvin Plümer.
Partial Differential Equations @DipMat 