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