- Pankaj K. Agarwal, Duke University
An Odyssey beyond Flatland: Arrangements, Envelopes, and UnionsThe arrangement of a finite collection of geometric objects is the decomposition of the space into connected cells induced by them. This talk will survey combinatorial and algorithmic results on arrangements, and their substructures, over the last 25 years, and how Micha has influenced this field.
- Noga Alon, Tel-Aviv University
Hypergraph List Coloring and Euclidean Ramsey TheoryIt is well known that one can color the plane by 7 colors with no monochromatic configuration consisting of the two endpoints of a unit segment. In sharp contrast we show that for any finite set of points X in the plane, and for any finite integer k, one can assign a list of k distinct colors to each point of the plane, so that any coloring of the plane that colors each point by a color from its list contains a monochromatic isometric copy of X.
Joint work with A. Kostochka.
- Dan Halperin, Tel-Aviv University
From Piano Movers to Piano Makers: Constructing and Deconstructing Minkowski SumsThe Minkowski sum of two sets P and Q in Euclidean space is the result of adding every point in P to every point in Q. Minkowski sums constitute a fundamental tool in geometric computing, often in relation to motion planning (Piano Movers), as well as to many other problems. We survey results on the structure, complexity, algorithms, and implementation of Minkowski sums in two and three dimensions. We then consider the reverse, deconstruction, problem: Can a given shape be expressed as the Minkowski sum of certain types of objects. This question too arises in various domains and in particular in connection with wood-cutting machines (Piano Makers). We review a few recent results on the deconstruction question.
- Sariel Har-Peled, UIUC
Finding Haystacks (and Other Structures) in GeometryOne of the key ideas in geometric computing is the usage of small subsets that represents well the (considerably larger) original input. We will survey some notions in geometry that were defined to this end, including eps-nets, eps-approximations, relative approximations, and coresets.
- Klara Kedem, Ben-Gurion University
Perverse and Non-Perverse Geometry: from Hausdorff Distance to GPUThe title was inspired by an early 1990's quote distinguishing between non-perverse (pure) and perverse (applied) computational geometry. Applications can dominate a field, especially computational geometry which has so many of them, such as in computer games, bioinformatics, computer graphics, image processing and on and on. But as in the case of classical Euclidean geometry the pure intellectual contributions, for which Micha is known, are immeasurably more influential and lasting.
I will talk about my (non-perverse) work on the minimum Hausdorff distance with Micha and Dan Huttenlocher, and on my combined perverse and non perverse work with Dror Aiger (my recent PhD student) on partial point matching and its implementation in the GPU framework. I will briefly mention two applications from my work in bioinformatics and image processing which are based on distance measures between shapes.
I will take this opportunity to tell stories that only an advisee can tell about her PhD advisor.
- Raimund Seidel, Saarland University
Issues in Geometric RoundingLoosely speaking, ``geometric rounding'' refers to approximating a geometric object by another one that admits a simple representation using ``simple'' coordinate values. An interesting example concerns approximating a plane straight-edge graph by another one whose vertices have real coordinates that can be expressed as fixed point numbers using few bits. Already this seemingly easy example raises a number of interesting algorithmic and complexity questions. I will address some of them.