Geometry processing has a long history of breakthrough developments that have guided design of 3D tools for computer vision, additive manufacturing, scientific computing, and other disciplines.  Research in this discipline has led to state-of-the-art approaches to a variety of problems, including the following:

• Acquisition and reconstruction
• Analysis and design for fabrication
• Architectural and industrial geometry
• Computational geometric design
• Computer-aided design and manufacturing
• Data-driven geometry processing
• Discrete differential geometry
• Exploration of shape collections
• Geometry and topology representations
• Geometry compression
• Geometric sorting, clustering, visualization
• Geometric deep learning
• Geometry processing applications
• Interactive shape design and editing
• Isogeometric analysis
• Machine learning in geometry
• Medical image analysis
• Mesh editing and deformation
• Meshing and remeshing
• Modeling for 3D printing
• Multiresolution modeling and subdivision 
• Procedural geometric modeling
• Processing of big geometric datasets
• Shape analysis and synthesis
• Simulation and animation
• Smoothing and denoising
• Surface and volume parameterization
• Topological data analysis

The Symposium on Geometry Processing conference annually showcases state-of-the-art research and tutorials in geometry processing; check out their Graduate School website for recorded talks introducing some of the big ideas.

Individual projects pursued week-to-week in our program will depend on the mentors who attend.  Given recent focuses in geometry processing research, a sampling of problems likely to be targeted during SGI includes the following:

• AI-driven shape representations:  While conventional ways to represent geometry on a computer include CAD models and triangle meshes, compatibility with emerging machine learning tools is inspiring new shape representations like neural fields and neural implicits.  We will develop methods that link the power of classical geometry processing algorithms to the flexibility and data-driven insight gained by incorporation of statistical methods.
• Geometry processing with uncertainty:  Motivated by computer graphics applications, classical geometry processing algorithms often assume smooth, manifold meshes as input, like those designed by artists and professional engineers.  We will aim to broaden the scope of geometry processing to computer vision and rapid prototyping by developing new algorithms and software that apply to messy, fuzzy, and/or incomplete signals coming from raw sensor data and novice users.
• High-throughput shape analysis:  In an effort to apply geometry processing to highly-detailed shapes as well as large datasets of models, new geometry processing algorithms have to take into account new models of parallel computation, with potential to advance not only classical applications in geometry but also PDE and simulation tools.
• Machine learning for geometry:  The vast expansion of machine learning research so far has had a fairly limited effect on geometry processing.  A primary reason for this disconnect is that popular tools for modern machine learning---specifically, convolutional neural networks---are not well-suited to the shape representations needed in many geometric applications (meshes, CAD models, and so on).  Many projects are focused on developing sensible learning techniques built from the ground up to ingest shapes rather than images or isolated data points.
• Higher-dimensional geometry processing:  The majority of geometry processing research projects focus on surfaces as two-dimensional objects embedded in 3D space.  New methods are being developed for volumes and even higher-dimensional manifolds of data points, drawing on intuition built in studying the lower-dimensional case.
• Unconventional applications of geometry processing: The exploratory projects in SGI allow our Fellows to pursue links between geometry processing and seemingly distant areas.  For instance, past projects have included geometry processing methods for astrophysics, modeling of complex biological systems, and combinatorics.