19115

SoSe 13: Forschungsmodul "Diskrete Geometrie" Seminar Realisierung von Polytopen

Raman Sanyal

Hinweise für Studierende

Zielgruppe:

Master/PhD students and motivated Bachelor students with interest in geometry

Zusätzl. Angaben / Voraussetzungen

Prior acquaintance of polytopes is required but no specialized knowledge. Basic knowledge of discrete mathematics and geometry and a solid background in linear algebra is sufficient.

Kommentar

Inhalt:

Polytopes arise naturally in many contexts in mathematics and, abstracting from the geometry, can be studied by combinatorial means. In particular combinatorially 'identical' polytopes can occur in geometrically 'different' realizations. The central theme of this seminar will be to make this statement precise and to explore its implications. To that end, we will view the realization space of a polytope, that is, the space of (coordinatizations of) polytopes with a fixed combinatorial structure from various perspectives.

* Realizations spaces can be arbitrarily complicated: The 'universality theorem' states that realizations spaces can take on arbitrary (semi-algebraic) shapes and thus can exhibit all kinds of pathologies. This is an instance where the combinatorics constrains the geometry of polytopes. To make this precise, we need to develop tools to turn semi-algebraic sets into combinatorics and combinatorics into polytopes.
* This is a high-dimensional (dimension 4 and up) phenomenon if we insist on convexity. Similar behavior, however, can be observerd for polyhedral surfaces in three-dimensional space as well as for planar point-line incidence configurations.
* In three dimensions realizations spaces of convex polytopes offer no surprises. There are various explicit techniques to obtain realizations from combinatorial data. These techniques make use of circle packings and variational principles from discrete differential geometry or equilibrium (rubber band) embeddings of graphs.
* It is more difficult to control the `complexity' of these realizations of 3-dimensional polytopes. Complexity here can refer to the fact that coordinates might get large or non-rational. We will discuss notions of complexity and intrinsic complexity of discrete geometric objects.

This seminar has strong ties to the Project A3 of the newly established Collaborative Research Center `Discretization in Geometry and Dynamics' (Collaborative Research Center `Discretization in Geometry and Dynamics'). The selected topics range from well-understood material (little/medium background required) to current research directions. Schließen

Literaturhinweise

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