Detalhes
GEOMETRIA SIMPLÉTICA
Nome da Disciplina: GEOMETRIA SIMPLÉTICA
Carga Horária: 60
Créditos: 4
Disciplina Regular: Sim
EMENTA
- From classical mechanics to symplectic geometry: Lagrangian and Hamiltonian mechanics, the Legendre transform.
- Linear symplectic algebra: Symplectic vector spaces and connection to Hermitian inner products, special subspaces (symplectic, Lagrangian, isotropic, coisotropic), linear symplectic reduction, the linear Darboux theorem, compatible linear complex structures, the Lie group of linear symplectomorphisms and its Lie algebra, polar decomposition.
- Symplectic manifolds and their morphisms: (Almost-)Symplectic forms and first examples (surfaces, Hermitian and Kähler manifolds, cotangent bundles, coadjoint orbits), induced volume form, (almost-)symplectomorphisms and their infinitesimal counterparts, symplectic and Hamiltonian vector fields, simple obstructions to the existence of (almost-)symplectic structures: dimensional and cohomological necessary conditions, submanifolds of constant rank, their induced geometry and their symplectic reduction.
- Local normal forms: Isotopies, diffeotopies and their relation to time-dependent vector fields and their flows, Moser's trick, Darboux's theorem, local normal forms near any embedded submanifold of a symplectic manifold, special cases: symplectic and Lagrangian submanifolds, applications.
- Symmetries: (Almost-)symplectic actions and their infinitesimal counterparts, quotients of (almost-)symplectic manifolds by discrete groups, the (co)adjoint action, (weakly) Hamiltonian actions, their infinitesimal counterparts and moment maps, Marsden-Weinstein-Meyer reduction.
- Possible extra topics: integrable systems and their action-angle variables, toric actions and the Atiyah-Guillemin-Sternberg convexity theorem, Duistermaat-Heckman formulas, Poisson manifolds and their local geometry.
- Linear symplectic algebra: Symplectic vector spaces and connection to Hermitian inner products, special subspaces (symplectic, Lagrangian, isotropic, coisotropic), linear symplectic reduction, the linear Darboux theorem, compatible linear complex structures, the Lie group of linear symplectomorphisms and its Lie algebra, polar decomposition.
- Symplectic manifolds and their morphisms: (Almost-)Symplectic forms and first examples (surfaces, Hermitian and Kähler manifolds, cotangent bundles, coadjoint orbits), induced volume form, (almost-)symplectomorphisms and their infinitesimal counterparts, symplectic and Hamiltonian vector fields, simple obstructions to the existence of (almost-)symplectic structures: dimensional and cohomological necessary conditions, submanifolds of constant rank, their induced geometry and their symplectic reduction.
- Local normal forms: Isotopies, diffeotopies and their relation to time-dependent vector fields and their flows, Moser's trick, Darboux's theorem, local normal forms near any embedded submanifold of a symplectic manifold, special cases: symplectic and Lagrangian submanifolds, applications.
- Symmetries: (Almost-)symplectic actions and their infinitesimal counterparts, quotients of (almost-)symplectic manifolds by discrete groups, the (co)adjoint action, (weakly) Hamiltonian actions, their infinitesimal counterparts and moment maps, Marsden-Weinstein-Meyer reduction.
- Possible extra topics: integrable systems and their action-angle variables, toric actions and the Atiyah-Guillemin-Sternberg convexity theorem, Duistermaat-Heckman formulas, Poisson manifolds and their local geometry.
BIBLIOGRAFIA
D. McDuff and D. Salamon, Introduction to Symplectic Topology, Second Edition, Oxford University Press, Oxford, UK, 2005.
A. Cannas da Silva, Lectures on Symplectic Geometry, Corrected 2nd printing, Lecture Notes in Mathematics 1764, Springer-Verlag, Berlin, Germany, 2008.
V. Guillemin and S. Sternberg, Symplectic Techniques in Physics, Cambridge University Press, Cambridge, UK, 1984.
A. Cannas da Silva, Lectures on Symplectic Geometry, Corrected 2nd printing, Lecture Notes in Mathematics 1764, Springer-Verlag, Berlin, Germany, 2008.
V. Guillemin and S. Sternberg, Symplectic Techniques in Physics, Cambridge University Press, Cambridge, UK, 1984.
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