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Computational Geometry and Topology


                                                               

 

Abstract:

A. Delaunay Complexes  
The Delaunay triangulation and its dual Voronoi diagram are among the most fundamental concepts in computational geometry.  After defining them, we present some of their pertinent properties, and we discuss their generalization to weighted point data.  
 
B. Discrete Morse Theory  
The radius function on the Delaunay triangulation fits a recent generalization of Forman's discrete Morse theory. We describe this algebraic framework and apply it to elucidate important properties of the Delaunay triangulation.  
 
C. Persistent Homology  
Among the many faces of this extension of the classical theory of homology groups is the connection to discrete Morse theory, which it extends by pairing not necessarily incident simplices and derive meaning from this pairing.

Lecturer - Herbert Edelsbrunner

Herbert Edelsbrunner is Professor at the Institute of Science and Technology Austria. He graduated from the Graz University of Technology, Austria, in 1982, he was faculty at the University of Illinois at Urbana-Champaign from 1985 through 1999, and Arts and Sciences Professor at Duke University from 1999 to 2012. He co-founded Geomagic in 1996, a software company in the field of Digital Shape Sampling and Processing.

His research areas are algorithms, computational geometry, computational topology, data analysis, and applications to biology. He has published four textbooks in the general area of computational geometry and topology. In 1991, he received the Alan T. Waterman Award from the US National Science Foundation. In 2006, he received an honorary degree from the Graz University of Technology. He is a member of the American Academy of Arts and Sciences, the Germany Academy of Sciences (the Leopoldina), the Academia Europaea, and the Austrian Academy of Sciences.

Tutor - Mabel Iglesias Ham

Mabel Iglesias Ham received the MSc. in Computer Science from University of Havana, Cuba, in 2005. From 2005 to 2011 she was a Research Associate in the Pattern Recognition Department at CENATAV, Cuba. In 2008 and 2009 she was visiting the PRIP group at TU Vienna as Research Assistant under the supervision of Prof. Walter Kropatsch. Since 2012 she has been a PhD student at IST Austria in the group of Algorithms, Computational Geometry and Topology under the supervision of Prof. Herbert Edelsbrunner. Her current research interests are in the areas of computational geometry, algorithms, and include sphere packing/covering, delaunay/voronoi diagrams, persistence and pockets. She has publish articles in the areas of fingerprint matching algorithms, structural representations for human motion in video images, the computation of topological invariants up to cohomology/homology in graph pyramids and in the new area of relaxed sphere packing and covering. She is a member of the Cuban Association of Pattern Recognition and the International Association in Pattern Recognition since 2005.

 

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