Discrete Differential Geometry - Helping Machines (and People) Think Clearly about Shape

Keenan Crane explains how discrete differential geometry translates continuous geometric concepts into algorithms that computers can process.

(For more information, see: keenan.is/here) The world around us is full of shapes: airplane wings and cell phones, brain tumors and rising loaves of bread, fossil records and freeform architectural surfaces. To a large extent, our ability to master these domains is limited by our capacity to design, process, and analyze geometry. But like much of mathematics, geometry makes liberal use of infinity -- a concept that is alien to machines with finite memory and limited precision. The driving force behind discrete differential geometry (DDG) is to develop a language that can be easily understood by a computer, yet still faithfully captures the way shape behaves in nature. A valuable consequence of constructing algorithmic descriptions is that real-world phenomena like "curvature" and "holonomy" (which traditionally demand expert terminology) can now be easily conveyed to anyone whose vocabulary includes words like "sum" and "triangle." In this talk I explore recent discoveries in the rapidly growing field of DDG, and demonstrate how a clear geometric perspective can lead to simpler, more efficient algorithms that are numerically robust and exhibit good scaling behavior. A somewhat remarkable fact is that a wide variety of seemingly dissimilar questions can be answered by computing solutions to a simple linear system known as a discrete Poisson equation. For instance: what's the shortest path from one point to another on a curved surface? How can you construct a flow with only the requested sources and sinks? And how does one manipulate surfaces without distorting important features like angles? These questions are deeply rooted in a number of classical and beautiful topics from physics and geometry such as heat flow, parallel transport, holomorphic functions, and the Dirac equation, which will all be explained in simple geometric terms. This lecture was given as part of the Everhart Lecture Series, which is a forum to encourage interdisciplinary interaction among graduate students and faculty, to share ideas about recent research developments, problems and controversies, and to recognize the exemplary presentation and research abilities of Caltech's graduate students. Lecturers discuss scientific topics at a level suitable for graduate students and faculty from all fields while addressing current research issues. Lecturers are selected to present their work based on each student's: Dynamic speaking skills, which capture the attention of and convey research material clearly to a diverse technical audience; Ability to communicate their research field's broader importance; and Impact on the scientific community through their research.