Jean-Daniel Boissonnat: Building good triangulations.

Delaunay triangulations have been extensively used in many
applications and, in particular, in surface reconstruction and mesh
generation. Although they enjoy many nice properties, Delaunay 
triangulations have limitations: they are difficult to compute in high
dimensions, may contain flat simplices, and are strongly tied to Euclidean
spaces. The lectures will show how to overcome these limitations. Two 
main applications will be considered: the reconstruction of submanifolds
embedded in high dimensional spaces and the triangulation of Riemannian
manifolds.

1. Good Delaunay triangulations.
1.1. Delaunay triangulation of nets : combinatorial bounds, random 
subsets, algorithmic complexity.
1.2. Thick triangulations.
1.3. Stability and protection of Delaunay complexes.

2. Sampling theory for geometric objects

2.1. Sets of positive reach. Sampling conditions and quality of 
approximation.
2.2. Distance functions and homotopic shape reconstruction.
2.3. Reconstruction of submanifolds of $\R^d$.

3. Triangulation of manifolds

3.1. Anisotropic triangulations.
3.2. Delaunay triangulation of Riemannian manifolds.
3.3. Local criteria for homeomorphism.

Biograpical note.

Jean-Daniel Boissonnat is a research director at INRIA, the French Research Institute of Computer Science and Applied Mathematics and has been an invited professor at the Collège de France on the Chair of Informatics and Computational Sciences during the academic year 2016-2017. His research interests are in Computational Geometry and Topology, including geometric data structures, Voronoi diagrams, triangulations, randomized algorithms, robust computing, motion planning, shape reconstruction, mesh generation, topological data analysis. He published several books and more than 180 research papers. J-D. Boissonnat currently serves on the editorial board of the Journal of the ACM and of Discrete and Computational Geometry. He has been awarded an advanced grant from the European Research Council in 2014 to lead the GUDHI project (Geometry Understanding in Higher Dimensions).