Algorithms, Computational Geometry and Topology

Herbert Edelsbrunner

The core of Herbert Edelsbrunner's research is a combination of mathematics and computer science, always driven by relevant questions in applications.  During a past shift from geometry to topology (which are related subjects without clear separation), the group noticed an increase in relevant application questions we could address. These include questions in scientific visualization, structural molecular biology, systems biology, but also geometry processing, medical imaging, and orthodontics. The common theme is the importance of shape and the recognition, matching, and classification of shape. Topology is the area within mathematics whose methods most directly speak to that need. Algorithms and computer software are needed to make mathematical insights useful in applications, which is the motivation to study in topology and also geometry from a computational point of view.

Contact
Herbert Edelsbrunner
Institute of Science and Technology Austria (IST Austria)
Am Campus 1
A – 3400 Klosterneuburg

Phone: +43 (0)2243 9000-3301
E-mail: herbert.edelsbrunner@remove-this.ist.ac.at

Herbert Edelsbrunner's website (with CV & Publication list)

Assistant
Elisabeth Hacker

Phone: +43 (0)2243 9000-1015
E-mail: elisabeth.hacker@remove-this.ist.ac.at

Team

  • Ulrich Bauer, Postdoc
  • Michael Belkin, Visiting Professor
  • Stefan Huber, Postdoc
  • Mabel Iglesias-Ham, PhD Student
  • Salman Parsa, Student Intern
  • Florian Pausinger, PhD Student
  • Pawel Pilarczyk, Postdoc
  • Jan Reininghaus, Postdoc
  • Qichao Que, Student Intern
  • Olga Symonova, Postdoc
  • Yusu Wang, Visiting Professor

Current Projects

- International Delaunay Laboratory in Yaroslavl
The Mega project on Discrete and Computational Geometry funded by the Russian Government and headed by Herbert Edelsbrunner establishes a research laboratory within the Yaroslavl State University. The research focus are mathematical and computational topics in geometry as well as topology, including questions about sphere arrangements, Voronoi and Delaunay tessellations, geometric reconstruction, and persistent homology. We develop courses on these topics in Yaroslavl, hold summer schools and conferences, and pursue applications, including to geographic information systems and medical image analysis.

- Genome wide analysis of root traits

This NSF funded project, led by Philip Benfey, aims at relating the phenotypes of agricultural root systems with their genotypes. The work in our group focuses on capturing and describing phenotypes. The challenging steps are the reconstruction of the shape of a root system from 2D images, the development of geometric and topological descriptors that characterize the intrinsic and extrinsic shape of roots, and the exploration of changes through growth.

- Software

  • The group developed the 3D Alpha Shapes software some 20 years ago, and see it now as the origin of a number of different developments, in industry as well as in academic research. The software takes as input a finite set of points with x-, y-, and z-coordinates, and produces a multi-scale description of the shape the points sample.
  • A productive branch of development took alpha shapes into the world of molecular structures, in particular proteins and nucleic acids. The alpha shapes provide the mathematical and computational foundation for the fastest and most accurate volume, area, and derivative computations we have today. 
  • Another development leads from alpha shapes to persistent homology, which is the most important innovation that has yet emerged from the young field of computational topology. Indeed, the pervasive concept of a filtration was first perceived as a nested sequence of alpha shapes.
  • In industry, the Geomagic Wrap software was motivated by the Alpha Shape software but needed a major new idea to lead to reliable surface reconstruction, a problem that arises in many areas of manufacturing and medical modeling. That new idea is a discrete geometric flow on a simplicial complex, and idea that combines geometric and topological aspects and is eminently computable.

Selected Publications

  • Edelsbrunner H, Harer JL. 2010. Computational Topology. An Introduction. American Mathematical Society, Providence, Rhode Island.
  • Edelsbrunner H. 2001. Geometry and Topology for Mesh Generation. Cambridge University Press, Cambridge, England.
  • Edelsbrunner H. 1987. Algorithms in Combinatorial Geometry. Springer-Verlag, Heidelberg, Germany.

Career

2009 Professor, IST Austria
2007–2008 Visiting Professor, Berlin Mathematical School, Germany
2007 Visiting Professor, Ecole Normale Superieur, Paris, France
2006 Moore Distinguished Scholar, Caltech, Pasadena, USA
2004– Professor for Mathematics, Duke University, Durham, USA
2002 Visiting Professor, Lawrence Livermore National Laboratory, USA
1999– Professor for Computer Science, Duke University, Durham, USA
1996– Founder, Principal, and Director, Raindrop Geomagic
1994–1995 Visiting Professor, Hong Kong University of Science and Technology
1985–1999 Assistant, Associate, and Full Professor, University of Illinois, Urbana-Champaign, USA
1981–1985 Assistant, Graz University of Technology, Austria
1982 PhD, Graz University of Technology, Austria

Selected Distinctions

2014 European Association for Theoretical Computer Science (EATCS) Fellowship
2011 Corresponding Member, Austrian Academy of Sciences
2009 Member, Academia Europaea
2008 Member, German Academy of Science (Leopoldina)
2006 Honorary Doctorate, Graz University of Technology
2005 Member, American Academy of Arts and Sciences
1991 Alan T. Waterman Award, National Science Foundation

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