Sectional theory of a morphism

• Grant number: 22/16695-7
• Support Opportunities: Scholarships in Brazil - Post-Doctoral
• Start date: June 01, 2023
• End date: April 30, 2026
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Geometry and Topology
• Principal Investigator: Daciberg Lima Gonçalves
• Grantee: Cesar Augusto Ipanaque Zapata
• Host Institution: Instituto de Matemática e Estatística (IME). Universidade de São Paulo (USP). São Paulo , SP, Brazil
• Associated research grant: 16/24707-4 - Algebraic, geometric and differential topology, AP.TEM
• Associated scholarship(s): 23/16525-7 - Sectional theory of a morphism, BE.EP.PD

Abstract

The present research project has as its goal the construction of foundations of sectional theory in category theory and build bridges between important areas in mathematics. Specifically, this project consists of the following important problems in the field of Geometry and Topology.(1) We define and study a generalization in the category theory setting of the Schwarz genus (it is also known as the sectional category) of a fibration. Indeed, for an arbitrary morphism $f:X\to Y$ in a category $\mathcal{C}$, we define a numerical invariant, the sectional number of $f$, in terms of covers of the target $Y$.(2) To study the sectional number of (some) morphisms which appear in robotics. (3) To study the sectional number of a homomorphism of groups. It generalizes the covering number of a group. (4) To study the sectional number of a functor. (5) To study the related problems with sectional number. (6) To study concrete applications of the sectional number in industry and technology. For example, we present a solution for the motion planning problem for a multi-robot system of firefighting and rescue unmanned ground vehicles (UGVs). In the air transport sector, we will present a solution to the problem of aircraft navigation. On the other hand, we will present a study of the complexity and design of solutions to some problems generated by the COVID pandemic, for example, economic and social problems.