Algebraic theory of graphs and neural networks

• Grant number: 22/01501-2
• Support Opportunities: Scholarships in Brazil - Scientific Initiation
• Start date: April 01, 2022
• End date: March 31, 2023
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Geometry and Topology
• Principal Investigator: Farid Tari
• Grantee: Thiago Verissimo Leal Gonçalves
• Host Institution: Instituto de Ciências Matemáticas e de Computação (ICMC). Universidade de São Paulo (USP). São Carlos , SP, Brazil
• Associated research grant: 19/07316-0 - Singularity theory and its applications to differential geometry, differential equations and computer vision, AP.TEM

Abstract

This project is embedded in a major proposal that includes applications to graph theory to coupled dynamical systems. It provides an introduction to the algebraic theory of graphs applied to dynamical systems, a topic that is rapidly expanding in results and applications during the last twenty years. The studies are directed to the recognition of graphs as a tool in the investigation of networks of coupled dynamical systems, which we call networks. A graph is a representation of symmetrical or asymmetrical relationships between discrete objects and, in this sense, a network is defined as a graph in which vertices and edges have attributes. The perspectives of this project are to learn about the spectral theory from linear algebra applied to the study of graphs and also to learn the tools for the realization of a vector field of a network as an admissible vector field to the graph associated with this network. One case study will be considered: a neuronal synapsis with excitatory and inhibitory couplings. The continuation as an extended high-level second undergraduate project is promising along this line and has potential for the publication of original results. (AU)