Presented by: 
Thomas Lewiner (Pontifical Catholic University of Rio de Janeiro)
Date: 
Mon 2 Feb, 2:00 pm - 2:45 pm
Venue: 
Mansergh Shaw Building (45), room 204

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.