Computing through the Lens of Geometry

While numerical methods are often treated as low-level exercises in floating-point operations and index bookkeeping, looking at computation through the lens of geometry is frequently the key to predictive modeling. This talk offers a fast-paced survey of how respecting the geometry underlying various shape modeling and physical simulation tasks results in efficient, versatile tools for both research and industry. We begin by exploring how discrete connections provide a rigorous framework for both the grooming of virtual characters and high-dimensional data analytics. We then pivot to the geometric foundations of calculus, illustrating how mimicking the structure of exterior calculus ensures that computational operators remain faithful to their fundamental continuous definitions. The discussion shifts to the geometric nature of mechanics, where we show that preserving physical constraints—such as symmetry and conservation laws—is an effective way to ensure long-term simulation stability. We conclude with a candid look at the current state of machine learning for geometry, weighing the undeniable flexibility of neural methods against the risks of losing the structural guarantees provided by classical geometric methods. Throughout this talk, the recurring theme is that drawing inspiration from differential geometry in the discrete setting is rarely just about mathematical purity; it is a practical necessity for building a robust and general-purpose computational toolbox.

Design in the Age of AI and Spatial Computing

As the boundaries between the digital and physical worlds blur, we face a profound opportunity to reimagine how we design the world around us. While advanced manufacturing, artificial intelligence, and spatial computing offer unprecedented potential for architecture, engineering, and art, their impact is often limited by a lack of design tools that can seamlessly bridge human creativity with physical realizability. In this talk, I will explore the transformation of design workflows from traditional CAD tools toward intelligent design systems. I will discuss how optimization-based design and tailored data-driven models enable novel approaches for interactive shape exploration and beyond, demonstrating their applicability to challenges ranging from intricate microstructures to high-performance building facades. A central theme is the control problem: the inherent tension between the probabilistic nature of modern generative AI and the high precision and editability required for professional engineering. I will conclude by reflecting on the evolving role of algorithms as creative partners. I will share a vision for a future where technology provides the "digital superpowers" that complement rather than replace human intuition, enabling us to build a more sustainable, functional, and resilient world.

Implicit Modeling: From Dreams to Industry and Back to Dreams

Italian mathematician Alessandro Ricci laid foundational groundwork for implicit surface blending in his 1973 Ph.D. dissertation, which pioneered the use of continuous scalar fields and Boolean operations, such as union and intersection, to define complex solid shapes. Since Ricci's early work, implicit modeling has moved from being a fringe academic concept to a widely accepted and rapidly growing technique in the Computer-Aided Design (CAD) industry. While it has not replaced traditional CAD methods, it has become the industry standard for specific, highly complex engineering and advanced manufacturing workflows.

In 1985 my brother, Geoff Wyvill, visited me on sabbatical and we worked on a polygonizer for what would be known as implicit surfaces and modeling. After publication, Alessandro Ricci sent me a copy of his 1973 paper, opening up a window into what would become the direction of my research for most of my career. The trend at the time was to pursue mesh representations, whereas my contributions to computer graphics are in the general area of modelling techniques that do not use polygons. Moreover, there was a period of around 30 years when researchers interested in implicit modeling were considered unimportant by an anonymous ACM SIGGRAPH reviewer, or even to be "wasting valuable computer time," as David Parnas and Edsger Dijkstra commented to me at a talk on implicit modeling I gave at Memorial University in 1990.

My talk focuses on the development of implicit modelling and an encouragement to young researchers to stick with their original ideas even if, especially if, these ideas do not match mainstream research. I also delve into a few projects in the realm of computers in the arts, which leave the human as the artist without replacing a person with AI. My recent work on visualizing timelines for authors of novels with complex plots is similarly directed at facilitating the author as an artist rather than allowing AI to interfere in the creative process.

Symmetry, Steadiness, and other Revelations

The fundamental steps of inventing a solution are the revelations about the true nature of the problem or about an unexpected approach worth exploring. I am conscious of how these revelations grow in my head, even though I do not use words or images when I think. In this presentation, I attempt to share examples of such revelations that led to interesting solutions to problems of defining and computing averages and interpolations of points, curves, similarities, or tilings. Many of these revelations were inspired by two guiding principles: symmetry and steadiness.

Shape Modeling for Engineering Design and Manufacturing

Shape modeling is a fundamental computational technique in engineering design and manufacturing. Unlike visual shape modeling, engineering-oriented shape modeling must address geometric precision, topological robustness, design intent, and manufacturability altogether, making it both mathematically interesting and practically significant. In this talk, I will discuss our research on high-performance geometric computing, intuitive surface modeling, and robust solid modeling, and show how they help streamline design-to-manufacturing workflows. I will also share our recent efforts toward intelligent CAD, where geometric algorithms are integrated with data-driven strategies, engineering constraints, and domain knowledge. These explorations, hopefully, will support the development of next-generation CAD/CAM.

Toward Compact and Structured Representations of Detailed 3D Shapes

Detailed 3D shapes are fundamental to computer graphics, visual computing, and AI-driven 3D understanding, but their increasing geometric complexity poses significant challenges for rendering, reconstruction, and learning. Dense, unstructured representations often preserve surface detail but can be inefficient to store, difficult to process, and poorly suited for scalable computation. This talk presents my research vision toward compact and structured representations of detailed 3D shapes. The goal is to develop representations that retain high-frequency geometric fidelity while exposing useful structure for efficient rendering, accurate reconstruction, and effective learning from 3D data. By organizing complex geometry into compact and computationally meaningful forms, such representations can bridge classical geometric processing and modern data-driven methods, enabling future systems to represent, analyze, and synthesize detailed 3D shapes more efficiently and robustly.

Shaping natural landscapes through simulation

Many of the static macro-structures in the world around us are the frozen remnants of continuous, deep-time physical processes. Landscapes, in particular, emerge from the interplay between tectonic uplift and erosion. In this talk, we discuss different methods that leverage physical laws to simulate and predict the morphogenesis of large-scale terrains. A major challenge in this domain is accounting for the impact of various surface flows (rivers, sediments, debris flows, ...) which evolve at time scales orders of magnitude smaller than those of macro-scale mountain erosion. To address this, we explore the concepts of hydrology networks and flow paths: two computational proxies we developed to efficiently capture the long-term impact of these fast-evolving flows. We demonstrate the benefits of these methods from both an application and a methodological standpoint. In terms of application, they enable the formation of diverse geological structures. Methodologically, they allow for the formulation of novel mountain generation algorithms, for instance based on analytical solutions to the erosion law, which contributes to bridging the gap between physically-based simulation and procedural generation.

Scribbles to Structures: The Geometry of Creative Intent

Human expression, whether through over-drawn sketches, sparse points, or mid-air VR gestures, leaves behind an incomplete, ambiguous trail of data that fundamentally conflicts with the mathematically precise representations required by computers. In this talk, I reflect on my early-career journey dedicated to resolving this conflict by building discrete geometric frameworks that infer clean, reliable shapes from such imperfect inputs. I will trace my work across a trajectory unified by the use of discrete Voronoi-Delaunay dualities and abstract shape representations to extract intended geometry from chaos. Starting with my early work on 2D sketch processing, I show how these discrete principles extend to interactively decode structures like pixel grids and stippled art, and conclude with my recent advances in surfacing sparse 3D VR sketches.