Spectral sequences for Morse-Bott and Morse-Novikov flows study

• Grant number: 14/11943-6
• Support Opportunities: Scholarships in Brazil - Post-Doctoral
• Start date: September 01, 2014
• End date: March 31, 2018
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Geometry and Topology
• Principal Investigator: Ketty Abaroa de Rezende
• Grantee: Dahisy Valadão de Souza Lima
• Host Institution: Instituto de Matemática, Estatística e Computação Científica (IMECC). Universidade Estadual de Campinas (UNICAMP). Campinas , SP, Brazil
• Associated research grant: 12/18780-0 - Geometry of control systems, dynamical and stochastics systems, AP.TEM
• Associated scholarship(s): 15/10930-0 - Symplectic dynamics and spectral sequences, BE.EP.PD

Abstract

In this project , we propose to develop Conley's theory in the context of Morse-Bott and Morse-Novikov flows, associated to Morse-Bott and Morse circular functions, respectively. More specifically, we intend to use the algebraic theory of spectral sequences together with connection matrices to study continuation properties of these flows. We want to associate the topology of the manifold to the trajectories connecting critical manifolds (singularities, respectively) of a Morse-Bott flow (Morse-Novikov flow, respectively). We will consider a Morse-Bott chain complex (Novikov chain complex, respectively) such that its differential contains information on these trajectories. We will study the connection matrices and transition matrices obtained by the sweeping method adapted to this situation, in order to understand dynamical behaviors associated with each stage of the process. The ultimate goal is to understand the bifurcations that can occur in the steps associated to transition matrices when a one-parameter family of flows is considered.

Scientific publications

(4)

(References retrieved automatically from Web of Science and SciELO through information on FAPESP grants and their corresponding numbers as mentioned in the publications by the authors)

BERTOLIM, MARIA A.; LIMA, DAHISY V. S.; MELLO, MARGARIDA P.; DE REZENDE, KETTY A.; DA SILVEIRA, MARIANA R.. Algebraic and dynamical cancellations associated to spectral sequence. EUROPEAN JOURNAL OF MATHEMATICS, v. 3, n. 2, p. 387-428, JUN 2017. (12/18780-0, 14/11943-6)

LIMA, D. V. S.; RAMINELLI, S. A.; DE REZENDE, K. A.. HOMOTOPICAL CANCELLATION THEORY FOR GUTIERREZ-SOTOMAYOR SINGULAR FLOWS. JOURNAL OF SINGULARITIES, v. 23, p. 33-91, 2021. (14/11943-6, 18/13481-0, 15/10930-0, 16/24707-4)

LIMA, D. V. S.; MANZOLI NETO, O.; DE REZENDE, K. A.. On handle theory for Morse-Bott critical manifolds. Geometriae Dedicata, v. 202, n. 1, p. 265-309, OCT 2019. (12/18780-0, 16/24707-4, 14/11943-6)

LIMA, DAHISY V. DE S.; MANZOLI NETO, OZIRIDE; DE REZENDE, KETTY A.; DA SILVEIRA, MARIANA R.. CANCELLATIONS FOR CIRCLE-VALUED MORSE FUNCTIONS VIA SPECTRAL SEQUENCES. TOPOLOGICAL METHODS IN NONLINEAR ANALYSIS, v. 51, n. 1, p. 259-311, MAR 2018. (15/10930-0, 12/18780-0, 14/11943-6)