The lecture presented by Dr. Finat has begun with a short introduction to basic and most common concepts in Differential Geometry (manifold, fiber bundles, fields). Then, he has provided an overview of his contributions to different engineering areas related with Robotics, Computer Vision and Mechanics of Continous Media, where the MoBiVAP Group has been working for 20 years. The focus has been put on the need of having adaptive models able to self-adapt to evolving environmental conditions. This involves its interaction with other agents and/or the environment which requires efficient statistical models for estimating and predicting the changing conditions.
The goal of the talk is to show how Robotics, Computer Vision and Mechanics of Continuous Media provide meaningful benchmarks for Differential Geometry applications. However, the addition of events (modeled as singularities of fields) requires some extensions including D-modules. Invariants for linked graded structures suggest going further in the use of maps between corresponding differential filterings. The classical Differential Geometry is not enough for engineering, so it is necessary to extend it beyond Piecewise-smooth manifolds to semi-analytic cases by improving the currently available sampling and clustering methods.
The most outstanding challenges in the field are related to its extensions for Artificial Intelligence formulated in a tensor framework. This implies:
- a reformulation of tensor algebras to incorporate degenerate filterings,
- the management of degenerate cases with Complete Endomorphisms of Vector Spaces or, more generally, tensor algebras, and
- the alignment of tensor flows appearing in Differential Geometry.