Keynote Speakers

Martin Rasmussen

Imperial College, London

An introduction to the qualitative theory of nonautonomous dynamical systems

The qualitative theory of dynamical systems, initiated by both Henri
Poincaré and Aleksandr Lyapunov in the early stages of the 20th century, has
been revolutionary and influenced the whole development of the dynamical
systems theory. The goal of the qualitative theory is to understand the
behavior of dynamical systems more from a geometrical and topological point
of view, rather than to search for explicit expressions of its solutions.

Recently, nonautonomous dynamical systems became very popular, because it is
often necessary to assume that the underlying rules which govern the
dynamics are time-dependent. Examples are given by nonautonomous velocity
fields in fluid dynamics, and also random dynamical systems and control
systems are special classes of nonautonomous systems.

This talk gives an introduction to the qualitative theory for nonautonomous
dynamical systems. Similarities and differences to the corresponding theory
for autonomous dynamical systems will be highlighted.

Thomas Lewiner

PUC-Rio, Rio de Janeiro

Looking for intuition behind discrete topologies

Topological properties of smooth scalar and vector fields on manifolds provide rich and robust information both on the field and the manifold. The early mathematical fundamentals in topology further enhance this information with intuitive interpretations, which help in designing efficient tools for analysis and visualization.

However, the continuous setting of traditional topology is not directly met by computational representations of scalar and vector fields. On the one hand, several applications rely on interpolating discrete data to recover a continuous representation, usually through multi-linear polynomials on grids. This approach conveys the mathematical intuition but the interpolation choice may contaminate the interpretation. On the other hand, combinatorial topology analysis of fields, such as Forman’s theory or Discrete Exterior Calculus, is well defined and very effective to compute, although it still needs its tools and concepts to be more intuitive.

In this talk, I will illustrate this apparent dichotomy through concrete examples from academic and industrial research, and develop some parallel aspects between interpolation approaches and Forman’s Morse theory, trying to get the best of both worlds.