Directions in Infinite Graphs: topological, combinatorial and set-theoretical approaches

• Grant number: 24/01158-1
• Support Opportunities: Scholarships in Brazil - Doctorate
• Start date: June 01, 2024
• End date: February 29, 2028
• Field of knowledge: Physical Sciences and Mathematics - Mathematics
• Principal Investigator: Leandro Fiorini Aurichi
• Grantee: Lucas Silva Sinzato Real
• Host Institution: Instituto de Ciências Matemáticas e de Computação (ICMC). Universidade de São Paulo (USP). São Carlos , SP, Brazil
• Associated scholarship(s): 25/00669-5 - On topological and combinatorial properties of edge-ends, BE.EP.DR

Abstract

This Doctoral Scholarship Proposal is a natural extension of a previous FAPESP Master research project carried out by the applicant and his supervisor, which yielded one published paper and two preprints under elaboration. Through grant number 2021/13373-6, they worked with different branches of Infinite Graph Theory, a subarea of combinatorics still being developed abroad and with few representatives in Brazil. In this proposal, the student mentions the questions that arouse from its Masters, presenting also how they might be carried out in the Doctoral.In general lines, the problems found by the previous research are not exclusively related to graph-theoretical issues. For instance, positive results towards the unfriendly partition conjecture were obtained by transfinite algorithms and hierarchical constructions, suggesting that generalizations might be drawn with tools from set theory. On the other hand, the notion of ends in infinite graphs turned out to be appropriate when recovering classical theorems of finite ones, as broadly studied by the school in Hamburg.The current results in the literature on the subject point out the rich connection between this approach and general topology. Therefore, this Doctoral Scholarship Proposal aims to develop an interdisciplinary study of Infinite Graphs and to contribute with yet unexplored directions.