a quick glimpse of symplectic topologyucahmko/glimpseofsymptop2.pdf · applications gromov’s...
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![Page 1: A Quick Glimpse of Symplectic Topologyucahmko/GlimpseOfSympTop2.pdf · Applications Gromov’s non-squeezing theorem Suppose there exists an embedding F : B2n(R) ,!B2(r) R2n 2 which](https://reader035.vdocuments.mx/reader035/viewer/2022070805/5f0396a77e708231d409cb45/html5/thumbnails/1.jpg)
A Quick Glimpse of Symplectic Topology
Momchil Konstantinov
London School of Geometry and Number TheoryUniversity College London
Momchil Konstantinov (LSGNT/UCL) Symplectic Topology 1 / 17
![Page 2: A Quick Glimpse of Symplectic Topologyucahmko/GlimpseOfSympTop2.pdf · Applications Gromov’s non-squeezing theorem Suppose there exists an embedding F : B2n(R) ,!B2(r) R2n 2 which](https://reader035.vdocuments.mx/reader035/viewer/2022070805/5f0396a77e708231d409cb45/html5/thumbnails/2.jpg)
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 / 17
![Page 3: A Quick Glimpse of Symplectic Topologyucahmko/GlimpseOfSympTop2.pdf · Applications Gromov’s non-squeezing theorem Suppose there exists an embedding F : B2n(R) ,!B2(r) R2n 2 which](https://reader035.vdocuments.mx/reader035/viewer/2022070805/5f0396a77e708231d409cb45/html5/thumbnails/3.jpg)
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 / 17
![Page 4: A Quick Glimpse of Symplectic Topologyucahmko/GlimpseOfSympTop2.pdf · Applications Gromov’s non-squeezing theorem Suppose there exists an embedding F : B2n(R) ,!B2(r) R2n 2 which](https://reader035.vdocuments.mx/reader035/viewer/2022070805/5f0396a77e708231d409cb45/html5/thumbnails/4.jpg)
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 / 17
![Page 5: A Quick Glimpse of Symplectic Topologyucahmko/GlimpseOfSympTop2.pdf · Applications Gromov’s non-squeezing theorem Suppose there exists an embedding F : B2n(R) ,!B2(r) R2n 2 which](https://reader035.vdocuments.mx/reader035/viewer/2022070805/5f0396a77e708231d409cb45/html5/thumbnails/5.jpg)
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 / 17
![Page 6: A Quick Glimpse of Symplectic Topologyucahmko/GlimpseOfSympTop2.pdf · Applications Gromov’s non-squeezing theorem Suppose there exists an embedding F : B2n(R) ,!B2(r) R2n 2 which](https://reader035.vdocuments.mx/reader035/viewer/2022070805/5f0396a77e708231d409cb45/html5/thumbnails/6.jpg)
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!Here is (the) one on R4:
ω((v1, v2, v3, v4), (w1,w2,w3,w4)) = v1w2 − v2w1 + v3w4 − v4w3
Definition 2:
A symplectic structure on a manifold M is a 2-form ω, which is everywherenon-degenerate and closed. A symplectic manifold is a manifold with afixed symplectic structure.
Momchil Konstantinov (LSGNT/UCL) Symplectic Topology 4 / 17
![Page 7: A Quick Glimpse of Symplectic Topologyucahmko/GlimpseOfSympTop2.pdf · Applications Gromov’s non-squeezing theorem Suppose there exists an embedding F : B2n(R) ,!B2(r) R2n 2 which](https://reader035.vdocuments.mx/reader035/viewer/2022070805/5f0396a77e708231d409cb45/html5/thumbnails/7.jpg)
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 5 / 17
![Page 8: A Quick Glimpse of Symplectic Topologyucahmko/GlimpseOfSympTop2.pdf · Applications Gromov’s non-squeezing theorem Suppose there exists an embedding F : B2n(R) ,!B2(r) R2n 2 which](https://reader035.vdocuments.mx/reader035/viewer/2022070805/5f0396a77e708231d409cb45/html5/thumbnails/8.jpg)
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 5 / 17
![Page 9: A Quick Glimpse of Symplectic Topologyucahmko/GlimpseOfSympTop2.pdf · Applications Gromov’s non-squeezing theorem Suppose there exists an embedding F : B2n(R) ,!B2(r) R2n 2 which](https://reader035.vdocuments.mx/reader035/viewer/2022070805/5f0396a77e708231d409cb45/html5/thumbnails/9.jpg)
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 5 / 17
![Page 10: A Quick Glimpse of Symplectic Topologyucahmko/GlimpseOfSympTop2.pdf · Applications Gromov’s non-squeezing theorem Suppose there exists an embedding F : B2n(R) ,!B2(r) R2n 2 which](https://reader035.vdocuments.mx/reader035/viewer/2022070805/5f0396a77e708231d409cb45/html5/thumbnails/10.jpg)
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 5 / 17
![Page 11: A Quick Glimpse of Symplectic Topologyucahmko/GlimpseOfSympTop2.pdf · Applications Gromov’s non-squeezing theorem Suppose there exists an embedding F : B2n(R) ,!B2(r) R2n 2 which](https://reader035.vdocuments.mx/reader035/viewer/2022070805/5f0396a77e708231d409cb45/html5/thumbnails/11.jpg)
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 5 / 17
![Page 12: A Quick Glimpse of Symplectic Topologyucahmko/GlimpseOfSympTop2.pdf · Applications Gromov’s non-squeezing theorem Suppose there exists an embedding F : B2n(R) ,!B2(r) R2n 2 which](https://reader035.vdocuments.mx/reader035/viewer/2022070805/5f0396a77e708231d409cb45/html5/thumbnails/12.jpg)
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, analysis, homological algebra, sheaftheory...
Momchil Konstantinov (LSGNT/UCL) Symplectic Topology 6 / 17
![Page 13: A Quick Glimpse of Symplectic Topologyucahmko/GlimpseOfSympTop2.pdf · Applications Gromov’s non-squeezing theorem Suppose there exists an embedding F : B2n(R) ,!B2(r) R2n 2 which](https://reader035.vdocuments.mx/reader035/viewer/2022070805/5f0396a77e708231d409cb45/html5/thumbnails/13.jpg)
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, analysis, homological algebra, sheaftheory...
Momchil Konstantinov (LSGNT/UCL) Symplectic Topology 6 / 17
![Page 14: A Quick Glimpse of Symplectic Topologyucahmko/GlimpseOfSympTop2.pdf · Applications Gromov’s non-squeezing theorem Suppose there exists an embedding F : B2n(R) ,!B2(r) R2n 2 which](https://reader035.vdocuments.mx/reader035/viewer/2022070805/5f0396a77e708231d409cb45/html5/thumbnails/14.jpg)
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, analysis, homological algebra, sheaftheory...
Momchil Konstantinov (LSGNT/UCL) Symplectic Topology 6 / 17
![Page 15: A Quick Glimpse of Symplectic Topologyucahmko/GlimpseOfSympTop2.pdf · Applications Gromov’s non-squeezing theorem Suppose there exists an embedding F : B2n(R) ,!B2(r) R2n 2 which](https://reader035.vdocuments.mx/reader035/viewer/2022070805/5f0396a77e708231d409cb45/html5/thumbnails/15.jpg)
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, analysis, homological algebra, sheaftheory...
Momchil Konstantinov (LSGNT/UCL) Symplectic Topology 6 / 17
![Page 16: A Quick Glimpse of Symplectic Topologyucahmko/GlimpseOfSympTop2.pdf · Applications Gromov’s non-squeezing theorem Suppose there exists an embedding F : B2n(R) ,!B2(r) R2n 2 which](https://reader035.vdocuments.mx/reader035/viewer/2022070805/5f0396a77e708231d409cb45/html5/thumbnails/16.jpg)
Why “topology”?
Darboux Theorem
Let (M2n, ω) be a symplectic manifold and p ∈ M be a point. Then thereexists an open set U ⊆ M, containing p, an open set V ⊆ R2n and adiffeomorphism ϕ : V → U such that
ϕ∗ω = d x1 ∧ d y1 + · · ·+ d xn ∧ d yn.
Momchil Konstantinov (LSGNT/UCL) Symplectic Topology 7 / 17
![Page 17: A Quick Glimpse of Symplectic Topologyucahmko/GlimpseOfSympTop2.pdf · Applications Gromov’s non-squeezing theorem Suppose there exists an embedding F : B2n(R) ,!B2(r) R2n 2 which](https://reader035.vdocuments.mx/reader035/viewer/2022070805/5f0396a77e708231d409cb45/html5/thumbnails/17.jpg)
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.
Definition 4
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 / 17
![Page 18: A Quick Glimpse of Symplectic Topologyucahmko/GlimpseOfSympTop2.pdf · Applications Gromov’s non-squeezing theorem Suppose there exists an embedding F : B2n(R) ,!B2(r) R2n 2 which](https://reader035.vdocuments.mx/reader035/viewer/2022070805/5f0396a77e708231d409cb45/html5/thumbnails/18.jpg)
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
· · ·
Momchil Konstantinov (LSGNT/UCL) Symplectic Topology 9 / 17
![Page 19: A Quick Glimpse of Symplectic Topologyucahmko/GlimpseOfSympTop2.pdf · Applications Gromov’s non-squeezing theorem Suppose there exists an embedding F : B2n(R) ,!B2(r) R2n 2 which](https://reader035.vdocuments.mx/reader035/viewer/2022070805/5f0396a77e708231d409cb45/html5/thumbnails/19.jpg)
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
· · ·
Momchil Konstantinov (LSGNT/UCL) Symplectic Topology 9 / 17
![Page 20: A Quick Glimpse of Symplectic Topologyucahmko/GlimpseOfSympTop2.pdf · Applications Gromov’s non-squeezing theorem Suppose there exists an embedding F : B2n(R) ,!B2(r) R2n 2 which](https://reader035.vdocuments.mx/reader035/viewer/2022070805/5f0396a77e708231d409cb45/html5/thumbnails/20.jpg)
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
· · ·
Momchil Konstantinov (LSGNT/UCL) Symplectic Topology 9 / 17
![Page 21: A Quick Glimpse of Symplectic Topologyucahmko/GlimpseOfSympTop2.pdf · Applications Gromov’s non-squeezing theorem Suppose there exists an embedding F : B2n(R) ,!B2(r) R2n 2 which](https://reader035.vdocuments.mx/reader035/viewer/2022070805/5f0396a77e708231d409cb45/html5/thumbnails/21.jpg)
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
· · ·
Momchil Konstantinov (LSGNT/UCL) Symplectic Topology 9 / 17
![Page 22: A Quick Glimpse of Symplectic Topologyucahmko/GlimpseOfSympTop2.pdf · Applications Gromov’s non-squeezing theorem Suppose there exists an embedding F : B2n(R) ,!B2(r) R2n 2 which](https://reader035.vdocuments.mx/reader035/viewer/2022070805/5f0396a77e708231d409cb45/html5/thumbnails/22.jpg)
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 9 / 17
![Page 23: A Quick Glimpse of Symplectic Topologyucahmko/GlimpseOfSympTop2.pdf · Applications Gromov’s non-squeezing theorem Suppose there exists an embedding F : B2n(R) ,!B2(r) R2n 2 which](https://reader035.vdocuments.mx/reader035/viewer/2022070805/5f0396a77e708231d409cb45/html5/thumbnails/23.jpg)
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 9 / 17
![Page 24: A Quick Glimpse of Symplectic Topologyucahmko/GlimpseOfSympTop2.pdf · Applications Gromov’s non-squeezing theorem Suppose there exists an embedding F : B2n(R) ,!B2(r) R2n 2 which](https://reader035.vdocuments.mx/reader035/viewer/2022070805/5f0396a77e708231d409cb45/html5/thumbnails/24.jpg)
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 9 / 17
![Page 25: A Quick Glimpse of Symplectic Topologyucahmko/GlimpseOfSympTop2.pdf · Applications Gromov’s non-squeezing theorem Suppose there exists an embedding F : B2n(R) ,!B2(r) R2n 2 which](https://reader035.vdocuments.mx/reader035/viewer/2022070805/5f0396a77e708231d409cb45/html5/thumbnails/25.jpg)
Rigidity: Gromov’s Pseudoholomorphic Curves
Definition 5
An almost complex structure on a manifold M is a sectionJ ∈ Γ(End(TM)) such that J2 = −1. A pseudoholomorphic curve in(M, J) is a map f : (Σ, j)→ M from a surface Σ to M which satisfies theequation
df ◦ j = J ◦ df
Momchil Konstantinov (LSGNT/UCL) Symplectic Topology 10 / 17
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These things are good, but when J is not integrable they can be quitewild. But if there is a symplectic structure ω compatible with J (i.e. suchthat gJ = ω(·, J·) is a Riemannian metric) things work out great!
mainly because...
∫Σf ∗ω =
∫ω(∂s f , ∂t f ) ds dt
=
∫ √ω(∂s f , J∂s f )ω(∂t f , J∂t f ) ds dt
=
∫ √det({gJ(∂i f , ∂j f )}i ,j) ds dt
=
∫Σ
dVolf ∗gJ
that is...
Symplectic area is actual area!
Momchil Konstantinov (LSGNT/UCL) Symplectic Topology 11 / 17
![Page 27: A Quick Glimpse of Symplectic Topologyucahmko/GlimpseOfSympTop2.pdf · Applications Gromov’s non-squeezing theorem Suppose there exists an embedding F : B2n(R) ,!B2(r) R2n 2 which](https://reader035.vdocuments.mx/reader035/viewer/2022070805/5f0396a77e708231d409cb45/html5/thumbnails/27.jpg)
Applications
Gromov’s non-squeezing theorem
Suppose there exists an embedding F : B2n(R) ↪→ B2(r)× R2n−2 whichrespects the standard symplectic structures. Then R ≤ r .
Lagrangians in R2n
There are no compact simply-connected Lagrangians in R2n.
Momchil Konstantinov (LSGNT/UCL) Symplectic Topology 12 / 17
![Page 28: A Quick Glimpse of Symplectic Topologyucahmko/GlimpseOfSympTop2.pdf · Applications Gromov’s non-squeezing theorem Suppose there exists an embedding F : B2n(R) ,!B2(r) R2n 2 which](https://reader035.vdocuments.mx/reader035/viewer/2022070805/5f0396a77e708231d409cb45/html5/thumbnails/28.jpg)
Applications
Gromov’s non-squeezing theorem
Suppose there exists an embedding F : B2n(R) ↪→ B2(r)× R2n−2 whichrespects the standard symplectic structures. Then R ≤ r .
Lagrangians in R2n
There are no compact simply-connected Lagrangians in R2n.
Momchil Konstantinov (LSGNT/UCL) Symplectic Topology 12 / 17
![Page 29: A Quick Glimpse of Symplectic Topologyucahmko/GlimpseOfSympTop2.pdf · Applications Gromov’s non-squeezing theorem Suppose there exists an embedding F : B2n(R) ,!B2(r) R2n 2 which](https://reader035.vdocuments.mx/reader035/viewer/2022070805/5f0396a77e708231d409cb45/html5/thumbnails/29.jpg)
Input from many walks of mathematics
For the past 30 years people have been working hard and have come upwith things like: Quantum Cohomology, Floer homology, Symplectichomology, Fukaya categories, Symplectic Field Theory.... All these thingsare machines which take information about the complicated ways in whichpseudoholomorphic curves bend and twist and repackage it into algebra.Then one can do some homological algebra and deduce more and moresurprising rigidity facts.
More recently people have applied microlocal sheaf theory to try andanswer the same questions without pseudoholomorphic curves.
Momchil Konstantinov (LSGNT/UCL) Symplectic Topology 13 / 17
![Page 30: A Quick Glimpse of Symplectic Topologyucahmko/GlimpseOfSympTop2.pdf · Applications Gromov’s non-squeezing theorem Suppose there exists an embedding F : B2n(R) ,!B2(r) R2n 2 which](https://reader035.vdocuments.mx/reader035/viewer/2022070805/5f0396a77e708231d409cb45/html5/thumbnails/30.jpg)
Open Problems
Symp(M , ω)
What is the topology of the group of diffeomorphisms of (M, ω) whichpreserve ω?
Arnold’s Nearby Lagrangian Conjecture
Is it true that every compact exact Lagrangian in the cotangent bundle ofa compact manifold can be mapped to the zero section by a Hamiltonianisotopy?
Momchil Konstantinov (LSGNT/UCL) Symplectic Topology 14 / 17
![Page 31: A Quick Glimpse of Symplectic Topologyucahmko/GlimpseOfSympTop2.pdf · Applications Gromov’s non-squeezing theorem Suppose there exists an embedding F : B2n(R) ,!B2(r) R2n 2 which](https://reader035.vdocuments.mx/reader035/viewer/2022070805/5f0396a77e708231d409cb45/html5/thumbnails/31.jpg)
Open Problems
Symp(M , ω)
What is the topology of the group of diffeomorphisms of (M, ω) whichpreserve ω?
Arnold’s Nearby Lagrangian Conjecture
Is it true that every compact exact Lagrangian in the cotangent bundle ofa compact manifold can be mapped to the zero section by a Hamiltonianisotopy?
Momchil Konstantinov (LSGNT/UCL) Symplectic Topology 14 / 17
![Page 32: A Quick Glimpse of Symplectic Topologyucahmko/GlimpseOfSympTop2.pdf · Applications Gromov’s non-squeezing theorem Suppose there exists an embedding F : B2n(R) ,!B2(r) R2n 2 which](https://reader035.vdocuments.mx/reader035/viewer/2022070805/5f0396a77e708231d409cb45/html5/thumbnails/32.jpg)
a weaker question...
Is it true that if M and N are compact manifolds and there exists adiffeomorphism F : (T ∗M, ωcan)→ (T ∗N, ωcan) which respects thesymplectic structures, then M is diffeomorphic to N?
a rephrasing...
Is all of differential topology just a part of symplectic topology?
Momchil Konstantinov (LSGNT/UCL) Symplectic Topology 15 / 17
![Page 33: A Quick Glimpse of Symplectic Topologyucahmko/GlimpseOfSympTop2.pdf · Applications Gromov’s non-squeezing theorem Suppose there exists an embedding F : B2n(R) ,!B2(r) R2n 2 which](https://reader035.vdocuments.mx/reader035/viewer/2022070805/5f0396a77e708231d409cb45/html5/thumbnails/33.jpg)
a weaker question...
Is it true that if M and N are compact manifolds and there exists adiffeomorphism F : (T ∗M, ωcan)→ (T ∗N, ωcan) which respects thesymplectic structures, then M is diffeomorphic to N?
a rephrasing...
Is all of differential topology just a part of symplectic topology?
Momchil Konstantinov (LSGNT/UCL) Symplectic Topology 15 / 17
![Page 34: A Quick Glimpse of Symplectic Topologyucahmko/GlimpseOfSympTop2.pdf · Applications Gromov’s non-squeezing theorem Suppose there exists an embedding F : B2n(R) ,!B2(r) R2n 2 which](https://reader035.vdocuments.mx/reader035/viewer/2022070805/5f0396a77e708231d409cb45/html5/thumbnails/34.jpg)
Disclaimer!
This last thing was a massive piece of propaganda and exaggeration! Thisis a good time for you to stop listening to me!
Momchil Konstantinov (LSGNT/UCL) Symplectic Topology 16 / 17
![Page 35: A Quick Glimpse of Symplectic Topologyucahmko/GlimpseOfSympTop2.pdf · Applications Gromov’s non-squeezing theorem Suppose there exists an embedding F : B2n(R) ,!B2(r) R2n 2 which](https://reader035.vdocuments.mx/reader035/viewer/2022070805/5f0396a77e708231d409cb45/html5/thumbnails/35.jpg)
Thank you!
Momchil Konstantinov (LSGNT/UCL) Symplectic Topology 17 / 17