## Leonidas J. Guibas (Stanford University)

Leonidas Guibas obtained his Ph.D. from Stanford under the supervision of Donald Knuth. His main subsequent employers were Xerox PARC, Stanford, MIT, and DEC/SRC. He has been at Stanford since 1984 as Professor of Computer Science (and by courtesy, Electrical Engineering), where he heads the Geometric Computation group within the Graphics Laboratory. He is also part of the AI Laboratory, the Bio-X Program, and the Institute for Computational and Mathematical Engineering. Professor Guibas' interests span computational geometry, geometric modeling, computer graphics, computer vision, robotics, ad hoc communication and sensor networks, and discrete algorithms - all areas in which he has published and lectured extensively. Some well-known past accomplishments include the analysis of double hashing, red-black trees, the quad-edge data structure, Voronoi-Delaunay algorithms, the Earth Mover's distance, Kinetic Data Structures (KDS), and Metropolis light transport. At Stanford he has developed new courses in algorithms and data structures, geometric modeling, geometric algorithms, sensor networks, and biocomputation. Professor Guibas is an ACM Fellow and a winner of the ACM Allen Newell award.

### Talk: "Intrinsic Multiscale Geometry"

The geometric analysis of point cloud data as well as of triangle meshes is intimately tied to the notion of scale at which the data is examined. It is well known that different features of the data manifest at different scales and the discovery of scales at which prominent features appear in a stable form is a challenging research problem. This question has recently been addressed by both classical multiresolution analysis tools, as well as by persistent homology approaches. Many (though not all) of these tools, however, approach the data in an extrinsic form, requiring an explicit or implicit embedding of the shape or space from which the samples are taken and/or of any simplicial approximations obtained.

In this talk we focus on multiscale analysis in the intrinsic setting, an area that has been much less explored. Our aim is to study interesting properties of geometric data when only geodesic distances between the samples are given (and, if occasionally extrinsic information is needed as well, we aim to limit it to local embeddings of small neighborhoods of the shape). Specifically, we look at

- the reconstruction of shapes from samples using the witness complex, under mild sampling conditions - including the ability to have different reconstructions at different scales
- multiscale descriptors for shape neighborhoods obtained by examining heat diffusion on manifolds and restricting the heat kernel to the temporal domain - what we call heat kernel signatures (HKS)
- the analysis of scalar fields over shapes using persistent homology, exploiting recent advances in the stability of persistence diagrams.

The approaches presented come with correctness guarantees and can be realized by algorithms with reasonable complexities. An advantage of the intrinsic approach is that the corresponding analysis can be used to compare different isometric embeddings of the same shape. It also allows us to consider the same questions in more general data analysis settings where the ambient dimension may be high or where no natural embedding of the data in a Euclidean space may be available.

## Günter M. Ziegler (Technische Universität Berlin)

Günter M. Ziegler was born in München, Germany, in 1963. He got a Ph.D. at M.I.T. with Anders Björner in 1987. Since 1995 he is a Professor of Mathematics at TU Berlin, and a member of the DFG Research Center Matheon.

His interests connect discrete and computational geometry (especially polytopes), algebraic and topological methods in combinatorics, discrete mathematics and the theory of linear and integer programming. He is the author of "Lectures on Polytopes" (Springer-Verlag 1995) and of "Proofs from THE BOOK" (with Martin Aigner, Springer-Verlag 1998), which has by now appeared in 14 languages.

His honors include a "Leibniz Prize" (2001) of the German Science Foundation DFG, the "Chauvenet Prize" (2004) of the Mathematical Association of America, and the 2008 "Communicator Award" of DFG and Stifterverband. He is a member of the Berlin-Brandenburg Academy of Sciences. 2006-2008 he was the President of the German Mathematical Society DMV, and now directs the DMV Media Office.

### Talk: "Face-vectors and complexity of geometric structures"

The "set of face vectors" is a nice way to graph various types of geometric structures, such as planar triangulations, 3-dimensional polytopes, triangulated surfaces, 3-dimensional meshes, or 4-dimensional polytopes.

It also lets us easily ask many difficult questions.
In my lecture I will give a survey. It will focus on the case of high-complexity, which features some amazing constructions, but also many open problems.