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Applications of Convex and Algebraic Geometry to Graphs & Polytopes
Algebra & Discrete Mathematics| Speaker: | Mohamed Omar, UC Davis |
| Location: | 2212 MSB |
| Start time: | Thu, May 26 2011, 4:10PM |
Description
We explore the application of nonlinear algebraic tools to problems on
graphs and polytopes. We begin by exploring the use of systems of
polynomial equations to makes steps toward an algebraic obstruction theory
for some classical combinatorial properties: k-colorability in graphs,
unique Hamiltonicity, and graphs having a trivial automorphism group. We
then study the convex geometry of permutation polytopes - convex hulls of
permutation matrices. We find volumes by computing unimodular
triangulations and Ehrhart polynomials, particularly illuminating the
beautiful symmetry afforded from their intrinsic group structure.
Finally, we explore the foundational algebraic underpinnings of the theta
body hierarchy of Gouveia, Parrilo and Thomas, extending their results and
providing an algebraic characterization for nonnegativity on real
varieties.