Research
Our overall scientific objective is to contribute both foundational and practical methods for data processing through a geometric perspective. We aim at producing modern computational tools for the furtherance of scientific exploration, allowing the analysis, processing, and simulation of a variety of data. The project-team’s focus on geometry offers a common language and tight intellectual cohesion, while promoting wide scientific applications due to the pervasiveness of topology and geometric aspects in topics from computer graphics, simulation, dynamical systems, and data science. Our research typically involves collaborations with mathematicians and computer scientists, and the outcomes are evaluated for their practical values with the help of domain experts. Our research program focuses on contributions in low- and high-dimensional geometry analysis and processing, and simulation of complex, multiphysics systems. In the long run, we wish to enable substantial interdisciplinary applications, such as accelerating scientific research through efficient digital prototyping, new geometry-derived insights for precision medicine, and discretizations for quantum and relativistic equations.
Research directions
While we intentionally leave the range of our mathematical foundations open so as not to
restrict our team-wide explorations, we concentrate our research on four concrete
themes, which we believe can be most significantly impacted by a geometric approach
to developing new numerical tools:
- Euclidean Shape Processing: From computer graphics to geometry processing and vision, the
analysis and manipulation of low-dimensional shapes (2D and 3D) is an important endeavor with applications covering a wide range of areas from entertainment and classical computer-aided design, to reverse engineering and biomedical engineering. Our project-team leads efforts in this competitive field, with key contributions in shape matching, geometric analysis, and discrete calculus on meshes. - Animation and Simulation: Traditional finite-element treatments of various physical models
have had tremendous success. Recently, a number of geometric integrators have upended the field, either through structure-preserving integration which offers improved statistical predictability by respecting the geometric properties of the exact flow of the differential equations, or by exploiting novel discretizations of state space. We aim at introducing novel integration methods for increasingly-complex multiphysics systems, as well as exploiting the use of learning methods to accelerate simulation. - Dynamical systems: We leverage the geometric nature of dynamical systems to investigate
and promote high-dimensional data analysis for dynamics. The study of dynamical systems from a limited number of observations of the state of a given system (for example, time series or a sparse set of trajectories) offers an opportunity to develop scalable computational tools to detect or characterize invariant sets, unusual features and coherent structures. - Data science: We also explore the underlying role of geometry in machine learning and statistical
analysis. This role has been put forward in the recent years, with the emergence of approaches such as geometric deep learning or topological data analysis, whose aim is to leverage the underlying geometry or topology of the data to enhance the performance, robustness, or explainability of the methods used for their analysis. We concentrate our efforts on topics related to explainable feature design, geometric feature learning, geometry-driven learning, and geometry for categorical and mixed data types.
