Analysis on Networks - Encounters of Mathematical Physics, Discrete Mathematics, and Theoretical Computer Science

• Applicant: Professor Dr. Delio Mugnolo
• Subject Area: Mathematics
• Term: since 2018
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 397230547

This project is the continuation of a research program supported by the DFG between September 2018 and August 2021, aimed to study several properties of the Laplacian operator on networks. In this proposed renewal phase, our research project will pick up three thematic clusters and further expand them. Our investigations will be motivated by mathematical objects, models, and concepts arising in theoretical physics and computer science. Our studies will be inspired by known results for higher dimensional domains, but conveniently exploit the 1D structure of metric graphs to deliver analytic results and explicit formulae. The following list of objectives forms the backbone of my research program:- Network topology and geometric aspects of diffusion equations. We will initiate a study of the heat kernels’ dependence on metric and topological features of networks.- Spectral and partition geometry of networks. We will systematically investigate the impact of combinatorial and metric quantities on Laplacian eigenvalues and spectral minimal energies.- Variational approaches to network partitioning and data clustering. We will study innovative approaches to partitioning and segmenting in networks based on heat kernels and eigenvalues.To pursue these goals, we will further sharpen the spectral geometric investigations of the first phase: we will develop a geometric theory of eigenfunctions, discuss the influence of network topology on their nodal domains, and study spectral implications of diverse metric and combinatorial invariants of possibly infinite networks. Systematically analyzing the properties of heat kernels depending on the network topology will be critical to our project. To this aim, we will initiate the systematic study of torsional rigidity on networks: this classical object will help us bridge the distance between spectral and parabolic investigations.

• DFG Programme: Research Grants