Homological and Homotopical methods in dynamical systems

• Grant number: 13/15119-3
• Support Opportunities: Scholarships abroad - Research Internship - Doctorate
• Start date: October 15, 2013
• End date: October 14, 2014
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Geometry and Topology
• Principal Investigator: Ketty Abaroa de Rezende
• Grantee: Ewerton Rocha Vieira
• Supervisor: John M. Franks
• Host Institution: Instituto de Matemática, Estatística e Computação Científica (IMECC). Universidade Estadual de Campinas (UNICAMP). Campinas , SP, Brazil
• Institution abroad: Northwestern University, Evanston, United States
• Associated to the scholarship: 10/19230-8 - Transition Matrix Theory, BP.DR

Abstract

The goal of this project is twofold. Firstly, we will investigate chain complex descriptions of dynamical systems and study underlying algebraic and topological properties in order to obtain dynamical informations. This is the case in the well known Morse-Witten chain complex. In this setting, the differential of this complex is given by a connection matrix. Hence, we make use of Conley index theory, more specifically the theory of connection and transition matrices in order to obtain dynamical informations.At present we have concentrated our efforts in studying the transition matrix theory. There are currently in the literature, three different constructions of transition matrices: singular, topological and algebraic. All are defined differently and having different properties. We are concluding an article, with Prof. Franzosa which presents a generalized topological transition matrix which are used to detect codimension one bifurcation of connecting orbits in a Morse decomposition of an isolated invariant set. Inspired by these results we are working together with Prof. Franzosa on a generalization of the transition matrix with a view towards unifying the theory as a whole. We will study more intensely the algebraic informations provided by the spectral sequence and in particular the transition matrices that arise in the Sweeping Method in terms of its dynamical significance. Prof. Franks has done extensive work with this underlying theme, i.e, relating back and forth, a dynamical situation to a chain complex setting and to a homological algebraic description. The invariants obtained in each context shed light on the others.The second purpose of this project is more open ended. In our weekly meetings with Prof. Franks, we hope to be able to broaden and deepen our knowledge of this approach to dynamical systems for which he is renowned in this field of research. Learning not only modern techniques but coming in contact with the cutting edge research at the forefront of this field will be, without a doubt, an invaluable experience. This opportunity to work with Prof. Franks is inestimable to gain more insight on homological and homotopical methods applied to the study of dynamical systems and group actions on manifolds. (AU)