Algebraic-topological methods for dynamical systems on differentiable and singular manifolds

• Grant number: 25/12435-9
• Support Opportunities: Regular Research Grants
• Start date: November 01, 2025
• End date: October 31, 2028
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Geometry and Topology
• Principal Investigator: Dahisy Valadão de Souza Lima
• Grantee: Dahisy Valadão de Souza Lima
• Host Institution: Centro de Matemática, Computação e Cognição (CMCC). Universidade Federal do ABC (UFABC). Santo André , SP, Brazil
• City of the host institution: Santo André
• Associated researchers: Ewerton Rocha Vieira ; Jean-Paul Michel Ildephonse Brasselet ; Maria Alice Bertolim ; Umberto Leone Hryniewicz

Abstract

This project aims to investigate problems in dynamical systems from a topological perspective, combining techniques from Conley Index Theory, Morse Theory, Spectral Sequences, and Intersection Homology. The proposed scientific problems will be addressed in two distinct settings: differentiable manifolds and singular spaces.In the differentiable setting, the project seeks to deepen the study of the dynamics and bifurcations associated with isolated critical points, aiming for precise topological characterizations of their stability via the homotopy Conley index and local Morse homology. It also proposes the development of a cancellation theory for isolated invariant sets based on the unfolding of spectral sequences, with the goal of obtaining minimal flows through successive cancellations.In the singular setting, we will explore two topological indices (Conley and Schwartz) on analytic singular varieties, investigating their interaction and correspondence with the Euler characteristic. Moreover, we propose to construct chain complexes based on dynamical data to compute the intersection homology with arbitrary perversities. Finally, we will study homotopical cancellations of GS singularities via spectral sequences, aiming to achieve minimal GS flows on singular spaces.The project seeks to contribute with new conceptual and computational tools for the analysis of dynamical systems in both smooth and singular settings. (AU)