The year was divided into six parts, each corresponding to a major area of research within discrete and computational geometry. Perhaps the single most important impact of the year has been the merging of several research communities which until then tended to ignore each other. For example, the long-established mostly European real-algebraic geometry community has now found a home in the younger but broader computational geometry community. Computational Geometers are increasingly concerned with nonlinear, higher-dimensional shapes, and for this reason, the work of researchers such as Roy, Neff, Canny, Renegar, Ierardi, Kozen (all of them DIMACS special year participants) has become more and more relevant to the community at large. Another example is the Austrian school of symbolic computation, under the leadership of Buchberger and his colleagues in Linz, which partly because of the year has attracted many followers stateside (e.g. Yap, Mishra, Hoffman, Lakshman). In Robotics, Schwarz, Canny and others have helped to bridge the gap between the implementation issues in automated assembly systems and their mathematical underpinnings. A remarkable confluence of interests was discovered among researchers who earlier thought of themselves as members of entirely different communities.
The entire area of randomized algorithms, spawned by the early work of Rabin, Clarkson, Haussler, Welzl, Reif, Sen has literally exploded. This avenue of research offers the perspective of algorithms that are both simple and efficient in practice. A whole DIMACS workshop was organized around this area of research which appears more and more promising. For example, Seidel has recently proposed a remarkably simple algorithm for linear programming in a fixed number of dimensions. Through the exchanges that it allowed, DIMACS was instrumental in bringing about this new focus on randomized algorithms.
The path breaking work of Gelfand (a DIMACS visitor during the year) and Rybnikov on the complexification of oriented matroids found an audience which promises to expand its impact significantly. Abhyankar and Bajaj are investigating computational asepects of algebraic geometry, with applications to computer graphics and design automation. Some of the top researchers in computer graphics participated in the special year and greatly enlightened the more theoretically-minded geometers in what constitute the main practical bottlenecks in graphics and what ought to be the next challenges for applied mathematicians in that area.
Index of DIMACS Special Year on Discrete and Computational Geometry
DIMACS Homepage