December 2022 – Géométrie et topologie du calcul

Orateur : Mathieu Desbrun (website), GeomeriX team, Inria/Ecole Polytechnique

Titre : Exploiting the unreasonable effectiveness of geometry in computing

Résumé : While the fields of geometric design and numerical simulation have mostly evolved independently, we are now witnessing a convergence of thought: from isogeometric analysis, to geometric integrators and discrete exterior calculus, it has been repeatedly noted that the quality of computational tools ultimately boil down to properties of a fundamentally geometric or topological nature. This talk will describe a brief overview of our approach to computing through the lens of geometry to offer a versatile and efficient toolbox for a variety of applications, from shape processing to tangent vector field editing, and from fluid simulation to non-linear dimensionality reduction if time allows. We will point out how a strong grasp of classical differential geometry paired with a good understanding of the typical computational constraints in research and industry can bring forth novel theoretical and practical foundations for general-purpose computations. The importance of preserving differential geometric notions in the discrete setting will be a recurring theme throughout the talk to demonstrate the value of a geometric approach to computations.

Mathieu Desbrun, GeomeriX team, Inria/Ecole PolytechniqueBio: After obtaining a PhD in computer graphics in Grenoble, France, Desbrun joined Caltech as a postdoctoral fellow in 1998. He joined the CS department at the University of Southern California as an Assistant Professor in January 2000, where he remained for four years in charge of the GRAIL lab. He then became an Associate Professor at Caltech in the CS department in 2003, where he started the Applied Geometry lab and was awarded the ACM SIGGRAPH New Researcher award. He took on administrative duties after he became a full professor, becoming the founding chair of the Computing + Mathematical Sciences department and the director of the Information Science and Technology initiative from 2009 to 2015. More recently, he has been the Technical Papers Chair for the ACM SIGGRAPH 2018 conference, spent a sabbatical year at ShanghaiTech in the School of Information Science and Technology, and was elected as ACM Fellow in 2020. He is now working at LIX as both an advanced researcher at Inria Saclay, and a Professor at Ecole Polytechnique. He started the Geomerix research team with three local colleagues to focus on geometric numerics, covering data analysis, machine learning, and simulation.

Orateur : Aldo González Lorenzo (webpage), LIS

Titre :  A Heuristic for Short Homology Basis of Digital Objects

Résumé : Finding the minimum homology basis of a simplicial complex is a hard problem unless one only considers the first homology group. In this talk, we introduce a general heuristic for finding a short homology basis of any dimension for digital objects (that is, for their associated cubical complexes) with complexity $\mathcal{O}(m^3 + \beta_q \cdot n^3)$, where $m$ is the size of the bounding box of the object, $n$ is the size of the object and $\beta_q$ is the rank of its q-th homology group. Our heuristic makes use of the thickness-breadth balls, a tool for visualizing and locating holes in digital objects. We evaluate our algorithm with a data set of 3D digital objects and compare it with an adaptation of the best current algorithm for computing the minimum radius homology basis by Dey, Li and Wang.

Orateur : Yann-Situ, LIS

Titre : Étude d’objets homologiques computationnels

Résumé : En géométrie algorithmique, de nombreux travaux ont pour objectif d’analyser, comprendre ou classifier des formes. En raison de ses propriétés computationnelles, l’homologie constitue un descripteur topologique intéressant. En effet, il est possible de calculer un grand nombre d’informations homologiques à partir d’une représentation discrète d’objets (complexes de chaîne, complexes simpliciaux…). Il existe diverses méthodes de calcul homologique, fournissant des informations plus ou moins complexes et pertinentes. Nous introduirons ainsi plusieurs objets mathématiques associés à ces informations homologiques. Nous discuterons de leurs différences, leurs propriétés calculatoires, leurs relations d’un point de vue théorique et nous présenterons certaines pistes de recherche.