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Geometric Combinatorics

May 23-27, 2004
June 6-9, 2005
Mathematical Sciences Research Institute
Berkeley, CA

Cosponsored by the Mathematical Sciences Research Institute (MSRI)

Geometric combinatorics refers to a growing body of mathematics concerned with counting properties of geometric objects described by a finite set of building blocks. Polytopes (which are bounded polyhedra) and complexes built up from them are primary examples. Other examples include arrangements of points, lines, planes, convex sets, and their intersection patterns. There are many connections to linear algebra, discrete mathematics, analysis, and topology, and there are exciting applications to game theory, computer science, and biology. The beautiful yet accessible ideas in geometric combinatorics are perfect for enriching courses in these areas.

The target audience is professors who desire to learn about this exciting field, enrich a variety of courses with new examples and applications, or teach a stand-alone course in geometric combinatorics.

Some of topics we will cover include the geometry and combinatorics of polytopes, triangulations, combinatorial fixed point theorems, set intersection theorems, combinatorial convexity, lattice point counting, and tropical geometry. We will have fun visualizing polytopes and other constructions, and exploring neat applications to other fields such as the social sciences (e.g., fair division problems and voting) and biology (e.g, the space of phylogenetic trees). Many interesting problems in geometric combinatorics are easy to explain, but remain unsolved. Some of the material will reflect recent research trends from the Fall 2003 program at MSRI in this field.

Familiarity with linear algebra and discrete mathematics will be assumed for some of the topics considered. Participants will receive some reading materials beforehand as well as some fun problems in the field to whet their appetite.

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