POLYTOPES AND HAMILTONIAN GEOMETRY: STACKS, TORIC DEGENERATIONS, AND PARTIAL TROPICALIZATION
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I present three papers written on the theme of the interaction between polyhedra and Hamil- tonian mechanics. In the first, I extend Delzantās classification of toric symplectic mani- folds to a classification of toric symplectic stacks. These are singular objects whose moment polytopes may be irrational. In the second, with Jeremy Lane we construct a completely integrable system on the dual of the Lie algebra of a compact group. This generalizes the celebrated Gelfand-Zeitlin system. The third concerns a construction called partial tropi- calization, which was motivated by considering the limits of families of Poisson structures on certain Poisson-Lie groups. Together with Anton Alekseev, Jeremy Lane, and Yanpeng Li, we develop basic results about partial tropicalizations and use them to build symplectic embeddings into multiplicity-free spaces.
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239 pages
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2020-05
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Symplectic Geometry
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Sjamaar, Reyer
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Riley, Tara
Berest, Yuri
Berest, Yuri
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Mathematics
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Ph. D., Mathematics
Degree Level
Doctor of Philosophy
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Government Document
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Attribution-NoDerivatives 4.0 International
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dissertation or thesis