Intersection Homology and finite holomorphic maps

• Grant number: 24/07624-4
• Support Opportunities: Scholarships in Brazil - Doctorate
• Start date: October 01, 2024
• End date: July 31, 2028
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Geometry and Topology
• Principal Investigator: Nivaldo de Góes Grulha Júnior
• Grantee: Hana Marinho Lucena
• Host Institution: Instituto de Ciências Matemáticas e de Computação (ICMC). Universidade de São Paulo (USP). São Carlos , SP, Brazil
• Associated research grant: 19/21181-0 - New frontiers in Singularity Theory, AP.TEM

Abstract

In the field of topology, intersection homology, developed by Marc Goresky and Robert MacPherson in 1974, serves as a fundamental tool for investigating singular spaces, offering a complementary approach to singular homology. While singular homology applies to regular spaces, intersection homology is tailored to handle singularities by introducing "allowable" cycles that preserve the notion of intersection even in irregular spaces.Poincaré duality, a key concept in topology traditionally interpreted through intersection theory, establishes a fundamental relationship between the homology groups of a compact, oriented, and connected manifold and their dimensions. However, when the manifold exhibits singularities, such as multiple points, this interpretation becomes problematic.On the other hand, spectral sequences are a powerful mathematical tool used in various areas including algebraic topology, homological algebra, and algebraic geometry, to decompose complex objects into simpler elements and compute homology groups of topological spaces.Sets of multiple points of finite maps play a crucial role in various mathematical domains, from algebraic geometry to number theory. They naturally arise when considering the properties of map images between topological spaces and are essential for understanding the structure and behavior of these maps.The proposed scientific project aims to comprehensively and systematically explore sets of multiple points of finite maps, utilizing both intersection homology and spectral sequences. The objective is to develop methods for a detailed topological analysis of these sets, contributing to the advancement of mathematical knowledge across various fields.