A Quick Glimpse of Symplectic Topologyucahmko/GlimpseOfSympTop.pdf · A Quick Glimpse of Symplectic...
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A Quick Glimpse of Symplectic Topology Momchil Konstantinov London School of Geometry and Number Theory University College London Momchil Konstantinov (LSGNT/UCL) Symplectic Topology 1 / 10
Transcript of A Quick Glimpse of Symplectic Topologyucahmko/GlimpseOfSympTop.pdf · A Quick Glimpse of Symplectic...
A Quick Glimpse of Symplectic Topology
Momchil Konstantinov
London School of Geometry and Number TheoryUniversity College London
Momchil Konstantinov (LSGNT/UCL) Symplectic Topology 1 / 10
Two immediate questions:
Topology has simplices...are you sure you don’t mean simplicialtopology?
Ok then, what does symplectic mean?
Momchil Konstantinov (LSGNT/UCL) Symplectic Topology 2 / 10
Two immediate questions:
Topology has simplices...are you sure you don’t mean simplicialtopology?
Ok then, what does symplectic mean?
Momchil Konstantinov (LSGNT/UCL) Symplectic Topology 2 / 10
Two immediate questions:
Topology has simplices...are you sure you don’t mean simplicialtopology?
Ok then, what does symplectic mean?
Momchil Konstantinov (LSGNT/UCL) Symplectic Topology 2 / 10
Why Symplectic?
On page 165 of his book “The Classical Groups” Hermann Weyl starts achapter on the Symplectic Group. In a footnote he writes:
Momchil Konstantinov (LSGNT/UCL) Symplectic Topology 3 / 10
Some Definitions
Definition 1:
A symplectic form on a real vector space V is a non-degenerateantisymmetric bilinear form
ω : V × V −→ R
Exercise: Such a thing exists if and only if the dimension of V is even! Infact, there is always a basis for which the matrix of the symplectic formbecomes: (
0 In−In 0
)Here is (the) one on R4:
ω((v1, v2, v3, v4), (w1,w2,w3,w4)) = v1w3 + v2w4 − v3w1 − v4w2
Momchil Konstantinov (LSGNT/UCL) Symplectic Topology 4 / 10
Definition 2:
A symplectic structure ω on a manifold M is a smoothly-varying choice ofsymplectic form on each tangent space. We also require ω to be “closed”.
Momchil Konstantinov (LSGNT/UCL) Symplectic Topology 5 / 10
Some Examples
R2n, ωstd = dx1 ∧ dy1 + · · ·+ dxn ∧ dyn
The cotangent bundle π : T ∗M → M of any manifold M. It isendowed with its canonical symplectic form ωcan = dλcan, whereλcan(m, ξ) = π∗ξ.
Any orientable surface with fixed volume form.
Complex projective varieties
many more...
Momchil Konstantinov (LSGNT/UCL) Symplectic Topology 6 / 10
Some Examples
R2n, ωstd = dx1 ∧ dy1 + · · ·+ dxn ∧ dyn
The cotangent bundle π : T ∗M → M of any manifold M. It isendowed with its canonical symplectic form ωcan = dλcan, whereλcan(m, ξ) = π∗ξ.
Any orientable surface with fixed volume form.
Complex projective varieties
many more...
Momchil Konstantinov (LSGNT/UCL) Symplectic Topology 6 / 10
Some Examples
R2n, ωstd = dx1 ∧ dy1 + · · ·+ dxn ∧ dyn
The cotangent bundle π : T ∗M → M of any manifold M. It isendowed with its canonical symplectic form ωcan = dλcan, whereλcan(m, ξ) = π∗ξ.
Any orientable surface with fixed volume form.
Complex projective varieties
many more...
Momchil Konstantinov (LSGNT/UCL) Symplectic Topology 6 / 10
Some Examples
R2n, ωstd = dx1 ∧ dy1 + · · ·+ dxn ∧ dyn
The cotangent bundle π : T ∗M → M of any manifold M. It isendowed with its canonical symplectic form ωcan = dλcan, whereλcan(m, ξ) = π∗ξ.
Any orientable surface with fixed volume form.
Complex projective varieties
many more...
Momchil Konstantinov (LSGNT/UCL) Symplectic Topology 6 / 10
Some Examples
R2n, ωstd = dx1 ∧ dy1 + · · ·+ dxn ∧ dyn
The cotangent bundle π : T ∗M → M of any manifold M. It isendowed with its canonical symplectic form ωcan = dλcan, whereλcan(m, ξ) = π∗ξ.
Any orientable surface with fixed volume form.
Complex projective varieties
many more...
Momchil Konstantinov (LSGNT/UCL) Symplectic Topology 6 / 10
Why?
Why people do it...
Cotangent bundles are models for phase space of classical mechanicalsystems.
Modern theoretical physics - string theory and mirror symmetry
It is fun and mysterious!
Why I like it...
Flexibility vs. rigidity
Employs differential geometry, functional analysis, homologicalalgebra, sheaf theory...
Momchil Konstantinov (LSGNT/UCL) Symplectic Topology 7 / 10
Why?
Why people do it...
Cotangent bundles are models for phase space of classical mechanicalsystems.
Modern theoretical physics - string theory and mirror symmetry
It is fun and mysterious!
Why I like it...
Flexibility vs. rigidity
Employs differential geometry, functional analysis, homologicalalgebra, sheaf theory...
Momchil Konstantinov (LSGNT/UCL) Symplectic Topology 7 / 10
Why?
Why people do it...
Cotangent bundles are models for phase space of classical mechanicalsystems.
Modern theoretical physics - string theory and mirror symmetry
It is fun and mysterious!
Why I like it...
Flexibility vs. rigidity
Employs differential geometry, functional analysis, homologicalalgebra, sheaf theory...
Momchil Konstantinov (LSGNT/UCL) Symplectic Topology 7 / 10
Why?
Why people do it...
Cotangent bundles are models for phase space of classical mechanicalsystems.
Modern theoretical physics - string theory and mirror symmetry
It is fun and mysterious!
Why I like it...
Flexibility vs. rigidity
Employs differential geometry, functional analysis, homologicalalgebra, sheaf theory...
Momchil Konstantinov (LSGNT/UCL) Symplectic Topology 7 / 10
Lagrangian submanifolds
Definition 3
• A Lagrangian subspace of (V 2n, ω) is an n−dimensional subspace L ≤ Vsuch that for every v ,w ∈ L one has ω(v ,w) = 0.• A Lagrangian submanifold of (M2n, ω) is an n−dimensional submanifoldL ⊆ M such that for every x ∈ L the tangent space TxL is a Lagrangiansubspace of TxM.
Momchil Konstantinov (LSGNT/UCL) Symplectic Topology 8 / 10
Momchil Konstantinov (LSGNT/UCL) Symplectic Topology 9 / 10
Two immediate questions
1 What are the closed Lagrangian submanifolds of R2?
2 What are the compact Lagrangian submanifolds of R4?a) Orientable:
· · ·
b) Non-orientable
RP2
Momchil Konstantinov (LSGNT/UCL) Symplectic Topology 10 / 10
Two immediate questions
1 What are the closed Lagrangian submanifolds of R2?
2 What are the compact Lagrangian submanifolds of R4?a) Orientable:
· · ·
b) Non-orientable
RP2
Momchil Konstantinov (LSGNT/UCL) Symplectic Topology 10 / 10
Two immediate questions
1 What are the closed Lagrangian submanifolds of R2?
2 What are the compact Lagrangian submanifolds of R4?
a) Orientable:
· · ·
b) Non-orientable
RP2
Momchil Konstantinov (LSGNT/UCL) Symplectic Topology 10 / 10
Two immediate questions
1 What are the closed Lagrangian submanifolds of R2?
2 What are the compact Lagrangian submanifolds of R4?a) Orientable:
· · ·
b) Non-orientable
RP2
Momchil Konstantinov (LSGNT/UCL) Symplectic Topology 10 / 10
Two immediate questions
1 What are the closed Lagrangian submanifolds of R2?
2 What are the compact Lagrangian submanifolds of R4?a) Orientable:
· · ·
b) Non-orientable
RP2 RP2#RP2
RP2#RP2#RP2
Momchil Konstantinov (LSGNT/UCL) Symplectic Topology 10 / 10
Two immediate questions
1 What are the closed Lagrangian submanifolds of R2?
2 What are the compact Lagrangian submanifolds of R4?a) Orientable:
· · ·
b) Non-orientable
RP2
RP2#RP2
RP2# · · ·#RP2︸ ︷︷ ︸4k+2
Momchil Konstantinov (LSGNT/UCL) Symplectic Topology 10 / 10
Two immediate questions
1 What are the closed Lagrangian submanifolds of R2?
2 What are the compact Lagrangian submanifolds of R4?a) Orientable:
· · ·
b) Non-orientable
RP2
RP2#RP2
RP2# · · ·#RP2︸ ︷︷ ︸4k+2
Momchil Konstantinov (LSGNT/UCL) Symplectic Topology 10 / 10
