Sectional theory of a morphism

• Grant number: 23/16525-7
• Support Opportunities: Scholarships abroad - Research Internship - Post-doctor
• Effective date (Start): June 01, 2024
• Effective date (End): May 31, 2025
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Geometry and Topology
• Principal Investigator: Daciberg Lima Gonçalves
• Grantee: Cesar Augusto Ipanaque Zapata
• Supervisor: Craig Christopher Westerland
• Host Institution: Instituto de Matemática e Estatística (IME). Universidade de São Paulo (USP). São Paulo , SP, Brazil
• Research place: University of Minnesota (U of M), United States
• Associated to the scholarship: 22/16695-7 - Sectional theory of a morphism, BP.PD

Abstract

A unified concept of sectional number have been recently introduced by the candidate. He define the $\mathcal{R}$-sectional number $\text{$\mathcal{R}$-sec}$ and the $\mathcal{R}$-mono sectional number $\text{$\mathcal{R}$-msec}$ of a morphism $f$ in a category $\mathcal{C}$ with covers where $\mathcal{R}$ is an equivalence relation over $\text{Mor}_{\mathcal{C}}(c,c^\prime)$ for each $c,c^\prime\in \mathcal{C}$. Usual sectional number and sectional category are recovered with the category of topological spaces and the trivial relation, and the homotopy relation, respectively. Recently, the candidate joint with Gonçalves and González present connections between sectional number, topological robotics, Borsuk-Ulam theory, Hopf invariants and fixed point theory. In addition Ellenberg, Venkatesh and Westerland apply stable homotopy theory to number theory. The present research project has as its goal the construction of new bridges between sectional theory and important areas in mathematics. Specifically, this project consists of the following important problems in the field of Geometry and Topology.(1) To developing new techniques for the sectional number related computations.(2) To study related problems with sectional number. For instance, sectional number and number theory.