Abstracts

Simone Dovetta: Variational methods for nonlinear Schrödinger equations on metric graphs

These lectures will provide an overview on the theory of normalized ground states for nonlinear Schrödinger equations on metric graphs, defined as global minimizers of the energy among functions with prescribed mass (i.e. the L² norm). In particular, we will focus on existence and non-existence results for ground states on general graphs. In recent years, this problem has been proved to be highly sensitive to the specific structure of the graph under exam and a wide phenomenology has by now been described. In the course, we will first illustrate a general existence argument for ground states, rephrasing in the context of metric graphs the typical framework of concentration-compactness. This general argument will be then applied to different families of graphs (graphs with half-lines, periodic graphs,...) to unravel how the topology and the metric of a graph affect the problem.

Roy Goodman: A consistent numerical approach to quantum graph computations

In principle, any mathematical question on a quantum graph should be amenable to numerical computations. The QGLAB project set out to make setting up and solving such problems simple, allowing users to easily construct and discretize quantum graphs and their Laplacian operators, solve linear and nonlinear problems, and visualize the results while working at a high level of abstraction.

In these lectures, I will introduce the basic numerical pieces used to build QGLAB. These include:

• Finite-difference and Chebyshev spectral discretizations.
• The block operator framework used to separate the enforcement of vertex conditions from the discretization of differential operators away from the vertices.
• Numerical continuation methods.
• Splitting methods for evolution methods containing both stiff and nonstiff terms.

I will apply these to the solution of several canonical problems on quantum graphs using QGLAB:

• Eigenvalue and Poisson problems.
• The computation of branches of stationary solutions to the nonlinear Schrödinger equation and their bifurcations.
• The evolution of solutions to PDE such as the heat, wave, and KPP equations.

If possible, I will hold a computational lab so that students may try solving their own quantum graph PDE problems.

For the course of Professor Goodman, to make the most of the hands-on sessions, we encourage participants to bring their own laptops with Matlab installed, if possible.

Delio Mugnolo: Spectral geometry of metric graphs and further branched spaces

Spectral geometry delves into the fascinating relationship between the shapes of domains or manifolds, and the patterns of the spectrum of associated Laplace(-Beltrami) operators. Traditionally, researchers have explored this connection by investigating certain properties like volume, diameter, or isoperimetric constant to establish spectral estimates.

While this exploration has commonly involved combinatorial graphs, which can be regarded as discretized manifolds, a newer focus has emerged on metric graphs. Metric graphs are like networks, consisting of connected intervals between points. Think of them as one-dimensional structures resembling simplified pathways. Over the last 25 years, the study of Laplacians associated with metric graphs has gained significant traction, considering these graphs as singular, or unique, types of spaces.

In these lectures, I aim to gently introduce the spectral geometry of metric graphs. Thanks to the inherent simplicity of metric graphs, being essentially one-dimensional, this area of study is richer compared to its higher-dimensional counterparts. I'll outline fundamental techniques, such as surgical and covering methods, that have been developed over the past decade to analyze the spectra of both metric and combinatorial graphs. Recent developments that involve non-standard quantities - like the torsional rigidity, the avoidance diameter and the mean distance of a metric graph - will also be reviewed.

I'll pay special attention to broadening these techniques to encompass additional operators, possibly nonlinear but homogeneous ones, including the intriguing realm of p-Laplacians and higher-order differential operators. If time permits, I'll touch upon potential connections with the parabolic theory of graphs.

Diego Noja: Time dependent NLS equation on metric graphs: Standing waves and their stability