Imagine a closed loop of string that looks knotted in space. How would you tell if you can wiggle it about to form an unknotted loop without cutting the string? How would a computer answer this question with absolute certainty? This is one of the many kinds of problems that we think about in computational geometry and topology.
The field of geometry has its roots in ancient times, and today geometry and topology are thriving fields of research that offer new insights into many different branches of mathematics. One of the most famous problems in topology is the Poincaré Conjecture, which was recently solved by Perelman, and for which he was awarded the Fields Medal and the first Clay Millennium Prize.
The topics we work on include innovative uses of familiar concepts such as distances, angles and surfaces, the manipulation of knots in three dimensions, the study of higher-dimensional spaces, and intrinsic geometric patterns that arise from algebra and arithmetic. These topics have important and sometimes surprising applications, covering fields such as microbiology, engineering, fluid flow, economics, and even the large-scale structure of the universe.