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Symplectic Topology

Symplectic Topology Quiz

15 Questions Published
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Questions

Question 1 Multiple Choice (Single Answer)

What is the symplectic form on a symplectic manifold?

  1. A closed 2-form.
  2. A non-degenerate 2-form.
  3. A closed and non-degenerate 2-form.
  4. A symplectic form is a closed and non-degenerate 2-form.
Question 2 Multiple Choice (Single Answer)

What is the Darboux theorem?

  1. Every symplectic manifold is locally symplectomorphic to the standard symplectic space.
  2. Every symplectic manifold is globally symplectomorphic to the standard symplectic space.
  3. Every symplectic manifold is locally symplectomorphic to a Darboux chart.
  4. Every symplectic manifold is globally symplectomorphic to a Darboux chart.
Question 3 Multiple Choice (Single Answer)

What is a symplectic vector field?

  1. A vector field that preserves the symplectic form.
  2. A vector field that is tangent to the symplectic foliation.
  3. A vector field that is Hamiltonian.
  4. All of the above.
Question 4 Multiple Choice (Single Answer)

What is a symplectic Hamiltonian system?

  1. A system of differential equations that describes the motion of a particle in a symplectic manifold.
  2. A system of differential equations that describes the motion of a fluid in a symplectic manifold.
  3. A system of differential equations that describes the motion of a wave in a symplectic manifold.
  4. All of the above.
Question 5 Multiple Choice (Single Answer)

What is the Arnold conjecture?

  1. Every symplectic manifold is symplectomorphic to a cotangent bundle.
  2. Every symplectic manifold is symplectomorphic to a product of spheres.
  3. Every symplectic manifold is symplectomorphic to a complex manifold.
  4. None of the above.
Question 6 Multiple Choice (Single Answer)

What is the Weinstein conjecture?

  1. Every symplectic manifold is symplectomorphic to a Weinstein manifold.
  2. Every symplectic manifold is symplectomorphic to a product of Weinstein manifolds.
  3. Every Weinstein manifold is symplectomorphic to a cotangent bundle.
  4. None of the above.
Question 7 Multiple Choice (Single Answer)

What is a symplectic embedding?

  1. An embedding that preserves the symplectic form.
  2. An embedding that is symplectomorphic to the standard symplectic embedding.
  3. An embedding that is Hamiltonian.
  4. All of the above.
Question 8 Multiple Choice (Single Answer)

What is a symplectic isotopy?

  1. A smooth family of symplectic embeddings.
  2. A smooth family of symplectomorphisms.
  3. A smooth family of Hamiltonian embeddings.
  4. All of the above.
Question 9 Multiple Choice (Single Answer)

What is the Floer homology of a symplectic manifold?

  1. A homology theory that is defined on symplectic manifolds.
  2. A homology theory that is defined on cotangent bundles.
  3. A homology theory that is defined on Weinstein manifolds.
  4. All of the above.
Question 10 Multiple Choice (Single Answer)

What is the Gromov-Witten theory?

  1. A theory that counts the number of pseudoholomorphic curves in a symplectic manifold.
  2. A theory that counts the number of symplectic embeddings of a symplectic manifold into another symplectic manifold.
  3. A theory that counts the number of symplectic isotopies of a symplectic manifold.
  4. None of the above.
Question 11 Multiple Choice (Single Answer)

What is the symplectic Lefschetz theorem?

  1. A theorem that relates the symplectic homology of a symplectic manifold to its cohomology.
  2. A theorem that relates the Floer homology of a symplectic manifold to its cohomology.
  3. A theorem that relates the Gromov-Witten theory of a symplectic manifold to its cohomology.
  4. None of the above.
Question 12 Multiple Choice (Single Answer)

What is the Weinstein conjecture?

  1. Every symplectic manifold is symplectomorphic to a Weinstein manifold.
  2. Every Weinstein manifold is symplectomorphic to a cotangent bundle.
  3. Every symplectic manifold is symplectomorphic to a product of Weinstein manifolds.
  4. None of the above.
Question 13 Multiple Choice (Single Answer)

What is the Arnold conjecture?

  1. Every symplectic manifold is symplectomorphic to a cotangent bundle.
  2. Every symplectic manifold is symplectomorphic to a product of spheres.
  3. Every symplectic manifold is symplectomorphic to a complex manifold.
  4. None of the above.
Question 14 Multiple Choice (Single Answer)

What is the Gromov-Witten invariant?

  1. A number that counts the number of pseudoholomorphic curves in a symplectic manifold.
  2. A number that counts the number of symplectic embeddings of a symplectic manifold into another symplectic manifold.
  3. A number that counts the number of symplectic isotopies of a symplectic manifold.
  4. None of the above.
Question 15 Multiple Choice (Single Answer)

What is the Floer homology group?

  1. A homology group that is defined on symplectic manifolds.
  2. A homology group that is defined on cotangent bundles.
  3. A homology group that is defined on Weinstein manifolds.
  4. All of the above.
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