WG4 - Variational methods on graphs and networks
Overview
Proceeding from classical variational methods for the analysis of Laplace-type operators, the aim of this Working Group is twofold: on the one hand, classic spectral geometry applies to elliptic operators generating semigroups that are Markovian (i.e., they describe the time evolution of expectation with respect to a Markov process); the plan is to investigate possible generalizations to generators of semigroups that have such a probabilistic interpretation only after a suitable transient; nonlinear contexts will also be studied. On the other hand, insights that have recently allowed the extension of gradient systems with respect to Wasserstein metric in spaces of measures will be extended to nonlinear dynamical systems on further discrete structures. A particular challenge is that the implementation of boundary conditions, which is a central issue in the theory of networks, is not yet well understood in the context of flows on metric spaces.
Tasks
- Develop variational methods for nonlinear dynamical systems on graphs (in collaboration with W3).
- Prove suitable isoperimetric inequalities for contexts that lack maximum principles.
- Exploit spectral geometric information to settle questions related to the long-time behaviour of the dynamical system on graphs.
- In collaboration with WG5 develop appropriate numerical methods for applications.
