Project Details
Abstract
Wider research context. The main objects of this Erwin Schrödinger project are two different classes of Laplacian operators on graphs: discrete Laplacians on discrete graphs and quantum graphs on metric graphs. Our main interest is the spectral theory of these graph Laplacians. We concentrate on infinite graphs (i.e. the graph has infinitely many edges and vertices) and questions appearing in this context.
Objectives. The project goal is to develop several new directions in the spectral theory of graph Laplacians emerging from recent results in this field. The overall aim is to achieve a significant step in spectral geometry on graphs and understand the interplay of geometry of graphs with spectral properties of graph Laplacians on several new levels.
We plan to work on four main topics: first of all, we investigate connections between these two operator classes, aiming at a spectral equivalence of discrete Laplacians and quantum graphs. Secondly, we will search for geometric Hardy inequalities on graphs by modern notions in graph geometry. Thirdly, we will make significant progress in describing self-adjoint extensions of graph Laplacians in terms of suitable boundary notions and boundary conditions for infinite graphs. Finally, we address questions emerging from the role of graph Laplacians in algebraic geometry of graphs.
Approach. The main theme of our approach is a uniform treatment of discrete Laplacians and quantum graphs. The range of project topics is rather wide and this requires a variety of different methods from spectral theory, functional analysis and geometry of graphs.
Level of innovation. Our goal is to work in several new research areas of spectral theory of graph Laplacians and treat problems which arise from recent developments in the field.
Primary researchers involved. The primary researchers involved are the project applicant Noema Nicolussi and the hosting researchers Omid Amini (École polytechnique), Matthias Keller (University of Potsdam) and Gerald Teschl (University of Vienna).
Objectives. The project goal is to develop several new directions in the spectral theory of graph Laplacians emerging from recent results in this field. The overall aim is to achieve a significant step in spectral geometry on graphs and understand the interplay of geometry of graphs with spectral properties of graph Laplacians on several new levels.
We plan to work on four main topics: first of all, we investigate connections between these two operator classes, aiming at a spectral equivalence of discrete Laplacians and quantum graphs. Secondly, we will search for geometric Hardy inequalities on graphs by modern notions in graph geometry. Thirdly, we will make significant progress in describing self-adjoint extensions of graph Laplacians in terms of suitable boundary notions and boundary conditions for infinite graphs. Finally, we address questions emerging from the role of graph Laplacians in algebraic geometry of graphs.
Approach. The main theme of our approach is a uniform treatment of discrete Laplacians and quantum graphs. The range of project topics is rather wide and this requires a variety of different methods from spectral theory, functional analysis and geometry of graphs.
Level of innovation. Our goal is to work in several new research areas of spectral theory of graph Laplacians and treat problems which arise from recent developments in the field.
Primary researchers involved. The primary researchers involved are the project applicant Noema Nicolussi and the hosting researchers Omid Amini (École polytechnique), Matthias Keller (University of Potsdam) and Gerald Teschl (University of Vienna).
| Status | Finished |
|---|---|
| Effective start/end date | 1/10/21 → 30/09/23 |
Collaborative partners
- University of Vienna (lead)
- École Polytechnique
Keywords
- spectral graph theory
- analysis on graphs
- discrete Laplacians
- quantum graphs
- metric graphs
- Graph Laplacians