A short review of Edelsbrunner's achievements#

H. Maurer

Edelsbrunner has had a number of of path-breaking ideas in the general areas of computing and mathematics. He is too young to be a founding member of the area of Computational Geometry, but he left his mark my dominating the field with his work on algorithms and data structures in the early years of its development. He was instrumental to the field by reaching out to Discrete Geometry in the early 1990s and to practice and applications in the late 1990s. His landmark achievements in the area include

  • novel data structures, such as the interval tree and the first optimal as well as practical method for point location;
  • novel algorithms and algorithmic paradigms, including batched dynamic processing, topological sweeping, smooth analysis for sliver exudation;
  • novel bounds on combinatorial quantities in geometry, including repeated distances, ways to cut polyhedra and point sets, number of faces of envelopes and arrangements, etc.
  • novel geometric structures, including alpha shapes, skin surfaces, protein interfaces;
  • novel programming paradigms, such as the simulation of simplicity, which was a revolutionary general approach to coping with degeneracies in geometric data.

He is one of the founders of Computational Topology, perhaps the founder. This field uses methods from topology to study natural phenomena, including large scale mixing phenomena and data analysis questions. His single most important contribution to this field is the inception of the concept of Persistent Homology. This method uses algebraic topology to measure the scale of features in data such as continuous functions or point samples. Small scale is usually considered noise, and this method allows the elimination of the effect of noise without altering the data by removing the noise itself. This opens up new venues to a host of difficult modeling and design questions, but it also points the way to analyse multi-scale data from the sciences, in particular astronomy and biology.

The scientific activities of Edelsbrunner have been guided by questions from applications in a variety of areas of science, including Structural Molecular Biology, Systems Biology, Mechanical Engineering, and others. While motivated by applications, his methods have been designed from Pure Mathematical principles and equipped with fast algorithms. He contributed to the transfer to knowledge between fields with the development and distribution of software, but most importantly by founding Raindrop Geomagic, today Geomagic, a software company that works with three-dimensional scan data and provides sophisticated solutions in a variety of industries, including general manufacturing and medicine.

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