symplectic geometry and hamiltonian group actions - …alpelayo/docs/gmclecture1.pdf · ·...

Outline Manifolds Actions Theory Poisson Manifolds

Symplectic Geometryand Hamiltonian Group Actions

Lecture 1, Miraflores de la Sierra

V School on Geometry, Mechanics and Control

Álvaro Pelayo

Washington University (USA)Institute for Advanced Study, Princeton (USA)

Partially supported by NSF CAREER Award, Spanish Ministry of Science GrantMTM 2010-21186-C02-01, NSF Postdoctoral Fellowship, Leibniz Fellowship, NSF

Grants DMS-0965738 and DMS-0635607 in Geometric Analysis, CSIC andMSRI-Berkeley

Álvaro Pelayo Symplectic Geometry and Hamiltonian Group Actions

A. Course Goals

GMCMinicourse

DynamicalSystems

SymplecticGeometry

GoalsIntroduce symplectic manifolds and their main propertiesGive overview of group actions and integrable systemsDescribe symplectic/spectral classification program ofintegrable systems

A. Course Goals

GMCMinicourse

DynamicalSystems

SymplecticGeometry

GoalsIntroduce symplectic manifolds and their main properties

Give overview of group actions and integrable systemsDescribe symplectic/spectral classification program ofintegrable systems

A. Course Goals

GMCMinicourse

DynamicalSystems

SymplecticGeometry

GoalsIntroduce symplectic manifolds and their main propertiesGive overview of group actions and integrable systems

Describe symplectic/spectral classification program ofintegrable systems

A. Course Goals

GMCMinicourse

DynamicalSystems

SymplecticGeometry

GoalsIntroduce symplectic manifolds and their main propertiesGive overview of group actions and integrable systemsDescribe symplectic/spectral classification program ofintegrable systems

B. Course Diagram

Symplectic Actions withSymplectic Orbit

HamiltonianActions

Symplectic Actions withLagrangian Orbit

Integrable Systems

Lec. 2

Lec. 1Lec. 2 Lec. 3, 4, 5

C. Course Lectures

TopicsLec. 1: Symplectic Geometry and Hamiltonian Actions

Lec. 2: Structure Theory for Symplectic Torus ActionsLec. 3: Integrable Systems, Semitoric SystemsLec. 4: Atiyah’s Connectivity, Morse Theory and Solution SetsLec. 5: Key Review, Spectral Theory, Classification

C. Course Lectures

TopicsLec. 1: Symplectic Geometry and Hamiltonian ActionsLec. 2: Structure Theory for Symplectic Torus Actions

Lec. 3: Integrable Systems, Semitoric SystemsLec. 4: Atiyah’s Connectivity, Morse Theory and Solution SetsLec. 5: Key Review, Spectral Theory, Classification

C. Course Lectures

TopicsLec. 1: Symplectic Geometry and Hamiltonian ActionsLec. 2: Structure Theory for Symplectic Torus ActionsLec. 3: Integrable Systems, Semitoric Systems

Lec. 4: Atiyah’s Connectivity, Morse Theory and Solution SetsLec. 5: Key Review, Spectral Theory, Classification

C. Course Lectures

TopicsLec. 1: Symplectic Geometry and Hamiltonian ActionsLec. 2: Structure Theory for Symplectic Torus ActionsLec. 3: Integrable Systems, Semitoric SystemsLec. 4: Atiyah’s Connectivity, Morse Theory and Solution Sets

Lec. 5: Key Review, Spectral Theory, Classification

C. Course Lectures

TopicsLec. 1: Symplectic Geometry and Hamiltonian ActionsLec. 2: Structure Theory for Symplectic Torus ActionsLec. 3: Integrable Systems, Semitoric SystemsLec. 4: Atiyah’s Connectivity, Morse Theory and Solution SetsLec. 5: Key Review, Spectral Theory, Classification

D. Classification Program

Figure: IAS Princeton and Bernoulli Center

Program LaunchingI will describe in the last lecture a Classification Program"Symplectic/Spectral Classification of Integrable Systems"

Two launching events:Minicourse by P. at IAS Princeton 2011-12Semester at Bernoulli Center, Switzerland July-December 2013

E. Course Main Object: Integrable Systems

What is an Integrable System?Many notions. Roughly, it is a “system of differential equations”:

Models physical systemGeometry describes solutions, exactly solvableAppears in mathematics, physics, chemistry, engineering ...

Examples: spherical pendulum, Lagrange top, spin-oscillator ...

Models physical system

Geometry describes solutions, exactly solvableAppears in mathematics, physics, chemistry, engineering ...

Models physical systemGeometry describes solutions, exactly solvable

Appears in mathematics, physics, chemistry, engineering ...Examples: spherical pendulum, Lagrange top, spin-oscillator ...

F. Classification Program

Goals for next few years

Two goals:To classify symplectically finite dimensional integrable systems

To prove the Spectral Conjecture for integrable systems:

Á. Pelayo and S. Vu Ngo. c: Symplectic theory of completelyintegrable Hamiltonian systems, Bull. AMS 48 (2011) 409-455Á. Pelayo and S. Vu Ngo. c: Hamiltonian dynamics and spectraltheory for spin oscillators, Comm. Math. Phys., in press

First goal achieved for semitoric systems:Á. Pelayo and S. Vu Ngo. c: Semitoric integrable systems onsymplectic 4-manifolds. Invent. Math. 177 (2009) 571-597Á. Pelayo and S. Vu Ngo. c: Constructing integrable systems ofsemitoric type. Acta Math. 206 (2011) 93-125

Two goals:To classify symplectically finite dimensional integrable systemsTo prove the Spectral Conjecture for integrable systems:

G. Parallel Program

Type Theory Project

A joint program with Vladimir Voevodsky (IAS Princeton)

“Integrable Systems in the eyes of Type Theory”

A paper in progress dealing with 4d-case

Lec. 1: Symplectic Geometry and Hamiltonian Actions

1 Outline

2 Manifolds

3 Actions

4 Theory

5 Poisson Manifolds

Glimpse of Symplectic Manifolds in Dynamics/GeometryOrigins: Hamilton’s deep formulation of Lagrangian mechanics (around 1835). Reformulation ofideas of Galileo (1600), Lagrange and Newton about orbits of planetary systemsSymplectic Geometry becomes subject with Alan Weinstein’s seminal work in 1970sNow: Symplectic Dynamics on its way to become a "subject" too (Hofer, IAS 2011-12)J-holomorphic curves: Gromov (Introduced them in 1985), Hofer, Eliashberg, McDuff ...

Integrable systems, dynamical systems, microlocal analysis, PDEs (1950s–): Arnold, Duistermaat,Hofer, Eliasson, Kolmogorov, Moser, Kostant, Uhlenbeck, Fomenko, Marsden, Vu Ngo.c, Colin deVerdiere, Guillemin, Weinstein, de la Llave, Zelditch, Zworski, Zehnder ...

Figure: V. Arnold (USSR 1937 - Paris 2010)Structure theory of Lie group actions, representation theory and connections to algebraic geometry(1980s–): Atiyah, Audin, Berline, Kostant, Vergne, Bott, Gross, Etingof, Guillemin, Delzant,Benoist, Kirwan, Reshetikhin, Weinstein, Duistermaat, Souriau ...Fourier, phase-space analysis (1970s–):Duistermaat, Hörmander, Colin de Verdiere, Bismut ...Topology in low dimensions, symplectic topology, gauge theory, connections to physics (1980s–):McDuff, Hofer, Gompf, Eliashberg, Mrowka, Kronheimer, Uhlenbeck, Freed, Taubes, Witten,Perutz, Donaldson, Auroux, Muñoz, Seidel, Polterovich...Connections to mechanics, Poisson Geometry: Weinstein, Ratiu, Loja Fernandes, Martín deDiego, de León, Xu, Crainic, Zambon, Montgomery, Bursztyn, Padrón, Iglesias, Rodríguez Olmo,Balseiro, Alekseev, Miranda, Marrero, Cattaneo,... (and many, many more)

Glimpse of Symplectic Manifolds in Dynamics/GeometryOrigins: Hamilton’s deep formulation of Lagrangian mechanics (around 1835). Reformulation ofideas of Galileo (1600), Lagrange and Newton about orbits of planetary systemsSymplectic Geometry becomes subject with Alan Weinstein’s seminal work in 1970sNow: Symplectic Dynamics on its way to become a "subject" too (Hofer, IAS 2011-12)J-holomorphic curves: Gromov (Introduced them in 1985), Hofer, Eliashberg, McDuff ...Integrable systems, dynamical systems, microlocal analysis, PDEs (1950s–): Arnold, Duistermaat,Hofer, Eliasson, Kolmogorov, Moser, Kostant, Uhlenbeck, Fomenko, Marsden, Vu Ngo.c, Colin deVerdiere, Guillemin, Weinstein, de la Llave, Zelditch, Zworski, Zehnder ...

Figure: V. Arnold (USSR 1937 - Paris 2010)Structure theory of Lie group actions, representation theory and connections to algebraic geometry(1980s–): Atiyah, Audin, Berline, Kostant, Vergne, Bott, Gross, Etingof, Guillemin, Delzant,Benoist, Kirwan, Reshetikhin, Weinstein, Duistermaat, Souriau ...

Fourier, phase-space analysis (1970s–):Duistermaat, Hörmander, Colin de Verdiere, Bismut ...Topology in low dimensions, symplectic topology, gauge theory, connections to physics (1980s–):McDuff, Hofer, Gompf, Eliashberg, Mrowka, Kronheimer, Uhlenbeck, Freed, Taubes, Witten,Perutz, Donaldson, Auroux, Muñoz, Seidel, Polterovich...Connections to mechanics, Poisson Geometry: Weinstein, Ratiu, Loja Fernandes, Martín deDiego, de León, Xu, Crainic, Zambon, Montgomery, Bursztyn, Padrón, Iglesias, Rodríguez Olmo,Balseiro, Alekseev, Miranda, Marrero, Cattaneo,... (and many, many more)

Glimpse of Symplectic Manifolds in Dynamics/GeometryOrigins: Hamilton’s deep formulation of Lagrangian mechanics (around 1835). Reformulation ofideas of Galileo (1600), Lagrange and Newton about orbits of planetary systemsSymplectic Geometry becomes subject with Alan Weinstein’s seminal work in 1970sNow: Symplectic Dynamics on its way to become a "subject" too (Hofer, IAS 2011-12)J-holomorphic curves: Gromov (Introduced them in 1985), Hofer, Eliashberg, McDuff ...Integrable systems, dynamical systems, microlocal analysis, PDEs (1950s–): Arnold, Duistermaat,Hofer, Eliasson, Kolmogorov, Moser, Kostant, Uhlenbeck, Fomenko, Marsden, Vu Ngo.c, Colin deVerdiere, Guillemin, Weinstein, de la Llave, Zelditch, Zworski, Zehnder ...

Figure: V. Arnold (USSR 1937 - Paris 2010)Structure theory of Lie group actions, representation theory and connections to algebraic geometry(1980s–): Atiyah, Audin, Berline, Kostant, Vergne, Bott, Gross, Etingof, Guillemin, Delzant,Benoist, Kirwan, Reshetikhin, Weinstein, Duistermaat, Souriau ...Fourier, phase-space analysis (1970s–):Duistermaat, Hörmander, Colin de Verdiere, Bismut ...Topology in low dimensions, symplectic topology, gauge theory, connections to physics (1980s–):McDuff, Hofer, Gompf, Eliashberg, Mrowka, Kronheimer, Uhlenbeck, Freed, Taubes, Witten,Perutz, Donaldson, Auroux, Muñoz, Seidel, Polterovich...Connections to mechanics, Poisson Geometry: Weinstein, Ratiu, Loja Fernandes, Martín deDiego, de León, Xu, Crainic, Zambon, Montgomery, Bursztyn, Padrón, Iglesias, Rodríguez Olmo,Balseiro, Alekseev, Miranda, Marrero, Cattaneo,... (and many, many more)

I.1. Definition and Examples of Symplectic Manifolds

DefinitionSymplectic form on vector space V: non-degenerate, skew,bilinear map V×V→ R

Symplectic manifold is a pair (M,ω):M smooth manifoldω := ωpp∈M symplectic form on M, i.e. ω is smoothcollection of symplectic forms ωp, one on each space Tp M

A little more: differential equation dω = 0 (closedness)

Examples of Symplectic Manifolds

(Σg, ω), surface of genus g with area form

(R2n, ∑ dxi∧ dyi) where (x1, y1, . . . , xn, yn) ∈ R2n

DefinitionSymplectic form on vector space V: non-degenerate, skew,bilinear map V×V→ RSymplectic manifold is a pair (M,ω):

M smooth manifoldω := ωpp∈M symplectic form on M, i.e. ω is smoothcollection of symplectic forms ωp, one on each space Tp M

I.2.a. Why Symplectic Geometry?

Symplectic Geometry in Mathematics and PhysicsMathematically: symplectic form is natural object

Physically:Origin in mechanics: phase space = symplectic manifold

Figure: Symplectic Geometry is natural setting for Spherical Pendulum

Framework for many classical/quantum problems

Symplectic Geometry in Mathematics and PhysicsMathematically: symplectic form is natural objectPhysically:

Origin in mechanics: phase space = symplectic manifold

I.2.b. Why Symplectic Geometry? continuation

Where else is Symplectic Geometry?Applied: physics, chemistry, engineering eg:

Molecular spectroscopyPlasma physicsElasticity theoryString theoryRobotics ...

Pure: connected at a core level with major subjects eg:Representation theoryDynamicsComplex algebraic geometryLie theoryFourier theory, microlocal analysis and PDE ...

I.2.b. Why Symplectic Geometry? continuation

Where else is Symplectic Geometry?Applied: physics, chemistry, engineering eg:

Molecular spectroscopyPlasma physicsElasticity theoryString theoryRobotics ...

Pure: connected at a core level with major subjects eg:Representation theoryDynamicsComplex algebraic geometryLie theoryFourier theory, microlocal analysis and PDE ...

I.2.c. Why Symplectic Geometry? continuation

Conceptual framework for many problems in physicsComplex computations become quick

Without symplectic geometry language:

Hamilton’s PDEs:

dyidt (t) = −∂H

∂xi(γ(t))

dxidt (t) = ∂H

∂yi(γ(t))

With symplectic geometry language:

Hamilton’s PDEs: ω(Y , ·) = dH

Conceptual framework for many problems in physicsComplex computations become quickWithout symplectic geometry language:

Hamilton’s PDEs:

dyidt (t) = −∂H

∂xi(γ(t))

dxidt (t) = ∂H

∂yi(γ(t))

Conceptual framework for many problems in physicsComplex computations become quickWithout symplectic geometry language:

Hamilton’s PDEs:

dyidt (t) = −∂H

∂xi(γ(t))

dxidt (t) = ∂H

∂yi(γ(t))

I.2.d. Why Symplectic Geometry? continuation

Figure: A. Weinstein (1943 - )

Famous Weinstein Creed, 1970s“EVERYTHING IS A LAGRANGIAN SUBMANIFOLD”

Symplectic manifolds and their submanifolds (isotropic,coisotropic, symplectic, Lagrangian etc) are natural objectsWhat does this mean? Wait till end of lecture!Creed ⇒ EVERYTHING IS SYMPLECTIC GEOMETRY!

Symplectic manifolds and their submanifolds (isotropic,coisotropic, symplectic, Lagrangian etc) are natural objects

What does this mean? Wait till end of lecture!Creed ⇒ EVERYTHING IS SYMPLECTIC GEOMETRY!

Symplectic manifolds and their submanifolds (isotropic,coisotropic, symplectic, Lagrangian etc) are natural objectsWhat does this mean?

Wait till end of lecture!Creed ⇒ EVERYTHING IS SYMPLECTIC GEOMETRY!

Symplectic manifolds and their submanifolds (isotropic,coisotropic, symplectic, Lagrangian etc) are natural objectsWhat does this mean? Wait till end of lecture!

Creed ⇒ EVERYTHING IS SYMPLECTIC GEOMETRY!

I.2. Geometric Interpretation of "Symplectic"

(M, ω) symplectic manifold, i.e. ω closed and non-degenerateSymplectic area of a surface S :=

Closednessdω = 0⇒ ∫

S ω unchanged when deforming S keeping ∂S fixed

Figure: S and R have same symplectic area since ∂S = ∂R = cStriking difference with Riemannian geometry !

Non-degeneracyω non-degenerate gives isomorphism:

TM = vector fields on M −→ T∗M = 1-forms onMX 7→ ω(X , ·)

Figure: S and R have same symplectic area since ∂S = ∂R = c

Striking difference with Riemannian geometry !

I.3. Questions about Symplectic Manifolds

Can you put a symplectic form on the 3-sphere S3?

S3 := (x, y, z, t) ∈ R4 | x2 + y2 + z2 + t2 = 1

Can you put a symplectic form on the Klein bottle?

S3 := (x, y, z, t) ∈ R4 | x2 + y2 + z2 + t2 = 1

I.4. Properties of Symplectic Manifolds

Three Main Properties

Even dimensional. Hence S3 is not symplectic

Orientable: volume form Ω := ω ∧ . . .(n times) . . .∧ω = ωn,where 2n = dimM. So Klein Bottle is not symplectic

Topologically “non-trivial” : if M is compact

H2kdR(M) 6= 0 since [ωk] ∈ H2k

dR(M), 0≤ k ≤ n.

So spheres S4, S6, S8, . . . ,S24, ... are not symplectic

dR(M), 0≤ k ≤ n.

I.5. The Symplectic Category

I.6. Local Classification of Symplectic Manifolds

Figure: Jean-Gaston Darboux (Nimes 1842 - Paris 1917)

Theorem (Darboux 1882)

Near each point in (M, ω), ∃ Darboux coordinates (x1,y1, . . . ,xn,yn)

∑i=1

dxi∧ dyi

So symplectic manifolds have no local invariants, expect dimension

∑i=1

dxi∧ dyi

∑i=1

dxi∧ dyi

II.1. Hamiltonian Dynamics

Originated with Galileo, Lagrange and NewtonGreatly generalized and reformulated by Hamilton ' 1835

Figure: William R. Hamilton (Dublin 1805-1865)

II.2. Dynamics of Vector Fields

Next: (M, ω) connected symplectic manifold

DefinitionY symplectic vector field on M if its flow preserves ω

Y Hamiltonian vector field on M if the system

ω(Y , ·) = dH (Hamilton’s Equations)

has a solution H : M→ R. If so, notation:

Y := HH

H called Hamiltonian or Energy Function

Y := HH

II.3. Exercise — Example of Vector Field

Vector field on sphereConsider

X :=√

1−h2 ∂

on 2-sphere S2 with coordinates (θ , h) and form dθ ∧ dh

Is X symplectic? YesIs X Hamiltonian? Yes. Really?Define H : S2→ R by H(θ , h) := h. Then:

ω(X , ·) = dH

X :=√

1−h2 ∂

on 2-sphere S2 with coordinates (θ , h) and form dθ ∧ dhIs X symplectic? Yes

Is X Hamiltonian? Yes. Really?Define H : S2→ R by H(θ , h) := h. Then:

ω(X , ·) = dH

X :=√

1−h2 ∂

on 2-sphere S2 with coordinates (θ , h) and form dθ ∧ dhIs X symplectic? YesIs X Hamiltonian?

Yes. Really?Define H : S2→ R by H(θ , h) := h. Then:

ω(X , ·) = dH

X :=√

1−h2 ∂

on 2-sphere S2 with coordinates (θ , h) and form dθ ∧ dhIs X symplectic? YesIs X Hamiltonian? Yes.

Really?Define H : S2→ R by H(θ , h) := h. Then:

ω(X , ·) = dH

X :=√

1−h2 ∂

on 2-sphere S2 with coordinates (θ , h) and form dθ ∧ dhIs X symplectic? YesIs X Hamiltonian? Yes. Really?

Define H : S2→ R by H(θ , h) := h. Then:

ω(X , ·) = dH

X :=√

1−h2 ∂

on 2-sphere S2 with coordinates (θ , h) and form dθ ∧ dhIs X symplectic? YesIs X Hamiltonian? Yes. Really?Define H : S2→ R by H(θ , h) := h. Then:

ω(X , ·) = dH

II.4. Exercise – Another Example of Vector Field

Vector field on torus

Consider ∂

∂θon 2-torus T2 := (R/Z)2 with coordinates (θ , α)

Is it symplectic? YesIs it Hamiltonian? No

If ω(∂

∂θ, ·) = dH =⇒ H(θ ,α) = α locally for some H : T2→ R

H is not a function: multivalued!!!(Can you make this argument rigorous?)

Consider ∂

Is it symplectic? Yes

Is it Hamiltonian? No

If ω(∂

Consider ∂

If ω(∂

∂θ, ·) = dH

=⇒ H(θ ,α) = α locally for some H : T2→ R

Consider ∂

If ω(∂

Consider ∂

If ω(∂

II.5. Lie Groups

DefinitionA (G, ?) Lie group is a pair where

G is smooth manifold

? is internal operation ? : G×G→ G

? is smooth

The Torus: a compact, connected, Abelian Lie group(R/Z, +

z ∈ C | |z|= 1, ·)

((R/Z)k, +

II.5. Lie Groups

G is smooth manifold

? is internal operation ? : G×G→ G

? is smooth

z ∈ C | |z|= 1, ·)

((R/Z)k, +

II.5. Lie Groups

G is smooth manifold? is internal operation ? : G×G→ G

? is smooth

z ∈ C | |z|= 1, ·)

((R/Z)k, +

II.5. Lie Groups

? is smooth

z ∈ C | |z|= 1, ·)

((R/Z)k, +

II.5. Lie Groups

? is smooth

z ∈ C | |z|= 1, ·)

((R/Z)k, +

)Álvaro Pelayo Symplectic Geometry and Hamiltonian Group Actions

II.6. Lie Group Actions

Definition(G, ?) Lie group, M smooth manifold

A G-action is a smooth map G×M→M,

(g, x) 7→ g · x,

such thate · x = x

g · (h · x) = (g?h) · x

Torus acting on Complex Space

An S1-action S1×Cn→ Cn on Cn:

(θ , (z1, z2, . . . , zn)) 7→ (θ z1, z2, . . . , zn)

Definition(G, ?) Lie group, M smooth manifoldA G-action is a smooth map G×M→M,

(g, x) 7→ g · x,

such that

e · x = xg · (h · x) = (g?h) · x

(θ , (z1, z2, . . . , zn)) 7→ (θ z1, z2, . . . , zn)

(g, x) 7→ g · x,

such thate · x = xg · (h · x) = (g?h) · x

(θ , (z1, z2, . . . , zn)) 7→ (θ z1, z2, . . . , zn)

(g, x) 7→ g · x,

such thate · x = xg · (h · x) = (g?h) · x

(θ , (z1, z2, . . . , zn)) 7→ (θ z1, z2, . . . , zn)

II.7. Another Lie Group Action Example

Figure: Rotational action of G := S1 on the 2-torus M := T2

II.8. Dynamics Generated by Torus Actions

Generating Vector Fields from an Action

T ' Tk := (S1)k torus. Assume T acts smoothly on MLet X ∈ t= Lie(T)

Vector field G (X) on M generated by T-action from X is

G (X)p := tangent vector to t 7→curve in T︷︸︸︷exp(tX) ·p︸︷︷︸

curve in M through p

at t = 0

DefinitionA smooth T-action on (M, ω) is

symplectic if all vector fields it generates are symplecticHamiltonian if all vector fields it generates are Hamiltonian

at t = 0

symplectic if all vector fields it generates are symplectic

Hamiltonian if all vector fields it generates are Hamiltonian

at t = 0

III.1. The Momentum MapLie-Kostant-Souriau: Map from Hamiltonian Action 1882, 1965-66

Suppose dimT = m and dimM = 2ne1, . . . ,em integral basis of tE1, . . . ,Em corresponding v.f. with Hamiltonians H1, . . . ,Hm

µ := (H1, . . . ,Hm) : M→ Rm momentum map(µ unique up to GL(m, Z) and translations in Rm)

(0,0,− 1)

(0,0,0)

(0,0,1)

(α,h) θ · (α,h)= (θ + α,h)

µ(θ,h) = h

(0,0,− 1)

(0,0,0)

(0,0,1)

(α,h) θ · (α,h)= (θ + α,h)

µ(θ,h) = h

(0,0,− 1)

(0,0,0)

(0,0,1)

(α,h) θ · (α,h)= (θ + α,h)

µ(θ,h) = h

III.2. Structure Theorems for Hamiltonian Torus Actions

Theorem (Atiyah-Guillemin-Sternberg, Invent Math/Bull LMS 1982)

If a torus T of dimension m acts Hamiltonianly on compact (M,ω),

µ(M) = convex hull

µ(fixed point set)⊂ Rm

Other: Kostant (Ann ENS 1973), Kirwan (Invent Math 1984)

Theorem (Delzant, Bull SMF 1988)

If a torus T of dimension n acts Hamiltonianly on compact2n-manifold (M,ω), then µ(M) classifies (M, ω) and the T-action

We call (M, ω) a symplectic toric manifold

µ(M) = convex hull

III.3. Example: Complex Projective Spaces for SymplecticGeometers are Polytopes

(0,0) (2,0,0)

(0,0,2)

(0,2,0)

Figure: Polytopes of (CP2,3 ·ωFS) and (CP3,2 ·ωFS)

Example

(CPn, λ ·ωFS) with rotational Tn-action

Hamiltonian, momentum map µ(z) = ( λ |z1|2∑

ni=0 |zi|2 , . . . ,

λ |zn|2∑

ni=0 |zi|2 )

Polytope ∆ = convex hull0, λ e1, . . . , λ en

(0,0) (2,0,0)

(0,0,2)

(0,2,0)

Example

ni=0 |zi|2 , . . . ,

λ |zn|2∑

ni=0 |zi|2 )

(0,0) (2,0,0)

(0,0,2)

(0,2,0)

Example

ni=0 |zi|2 , . . . ,

λ |zn|2∑

ni=0 |zi|2 )

IV.1. Poisson Manifolds

DefinitionM a smooth manifold

A Poisson bracket on C∞(M) is a bilinear map

·, · : C∞(M)× C∞(M)→ C∞(M)

satisfying:(skew-symmetry) f ,g=−g, f(Jacobi identity) f ,g,h+g,h, f+h,f ,g= 0

(Leibniz’s Rule) f g,h= fg,h+gf ,h

DefinitionThe pair (M, ·, ·) is a Poisson manifold

DefinitionM a smooth manifoldA Poisson bracket on C∞(M) is a bilinear map

·, · : C∞(M)× C∞(M)→ C∞(M)

satisfying:

(skew-symmetry) f ,g=−g, f(Jacobi identity) f ,g,h+g,h, f+h,f ,g= 0

·, · : C∞(M)× C∞(M)→ C∞(M)

satisfying:(skew-symmetry) f ,g=−g, f

(Jacobi identity) f ,g,h+g,h, f+h,f ,g= 0

·, · : C∞(M)× C∞(M)→ C∞(M)

IV.2. Symplectic vs Poisson Manifolds

Symplectic ⇒ Poisson

(M, ω) symplectic manifold

Set f , g := ω(Hf , Hg)

Then (M, ·, ·) is Poisson manifold

Poisson but not Symplectic: (h∗, ·, ·)(h, [·, ·]) finite dimensional real Lie algebrah' (h∗)∗ =⇒ h⊂ C∞(h∗)

Define ·, · on h by A, B := [A, B]

One can show ∃! extension of ·, · from h to C∞(h∗)

(h∗, ·, ·) is Poisson manifold

Poisson but not Symplectic: (h∗, ·, ·)(h, [·, ·]) finite dimensional real Lie algebra

h' (h∗)∗ =⇒ h⊂ C∞(h∗)

IV.3. Explicit Poisson Brackets

Four-dimensional Euclidean Space

R4 with coordinates (x, y, z, t) and dx∧ dy+ dz∧ dt

Let f : R4→ R and g : R4→ R smoothCorresponding Hamiltonian vector fields, eg:

Hf :=(

∂ f∂ z

, − ∂ f∂x

,∂ f∂ t

, −∂ f∂y

Define brackets

f , g= ∂ f∂ z· ∂g

∂x− ∂ f

∂x· ∂g

∂ z+

∂ f∂ z· ∂g

∂y− ∂ f

∂y· ∂g

(R4, ·, ·) is Poisson bracket

Let f : R4→ R and g : R4→ R smooth

Corresponding Hamiltonian vector fields, eg:

Hf :=(

∂ f∂ z

, − ∂ f∂x

,∂ f∂ t

, −∂ f∂y

Define brackets

f , g= ∂ f∂ z· ∂g

∂x− ∂ f

∂x· ∂g

∂ z+

∂ f∂ z· ∂g

∂y− ∂ f

∂y· ∂g

Hf :=(

∂ f∂ z

, − ∂ f∂x

,∂ f∂ t

, −∂ f∂y

Define brackets

f , g= ∂ f∂ z· ∂g

∂x− ∂ f

∂x· ∂g

∂ z+

∂ f∂ z· ∂g

∂y− ∂ f

∂y· ∂g

Hf :=(

∂ f∂ z

, − ∂ f∂x

,∂ f∂ t

, −∂ f∂y

Define brackets

f , g= ∂ f∂ z· ∂g

∂x− ∂ f

∂x· ∂g

∂ z+

∂ f∂ z· ∂g

∂y− ∂ f

∂y· ∂g

Hf :=(

∂ f∂ z

, − ∂ f∂x

,∂ f∂ t

, −∂ f∂y

Define brackets

f , g= ∂ f∂ z· ∂g

∂x− ∂ f

∂x· ∂g

∂ z+

∂ f∂ z· ∂g

∂y− ∂ f

∂y· ∂g

IV.4. Symplectic and Poisson Structure on CotangentBundles

Cotangent BundlesM smooth manifold

(U, x1, . . . , xn) coordinate chart for M

Associated cotangent chart (T∗U, x1, . . . , xn, ξ1, . . . , ξn)

Define 2-form ω on T∗U

ω T∗M :=n

∑i=1

dxi∧ dξi.

Coordinate independent!: canonical cotangent formProof. Note ω =− dα where α := ∑ξi dxi

Easy to check α intrinsically defined (⇒ω intrinsically defined)Canonical Poisson structure: f , g := ω T∗M(Hf , Hg)

More on Poisson manifolds: Rui Loja Fernandes’ Course

Cotangent BundlesM smooth manifold(U, x1, . . . , xn) coordinate chart for M

ω T∗M :=n

∑i=1

dxi∧ dξi.

ω T∗M :=n

∑i=1

dxi∧ dξi.

ω T∗M :=n

∑i=1

dxi∧ dξi.

Coordinate independent!: canonical cotangent form

Proof. Note ω =− dα where α := ∑ξi dxi

ω T∗M :=n

∑i=1

dxi∧ dξi.

Easy to check α intrinsically defined (⇒ω intrinsically defined)

Canonical Poisson structure: f , g := ω T∗M(Hf , Hg)

ω T∗M :=n

∑i=1

dxi∧ dξi.

ω T∗M :=n

∑i=1

dxi∧ dξi.

More on Poisson manifolds: Rui Loja Fernandes’ CourseÁlvaro Pelayo Symplectic Geometry and Hamiltonian Group Actions

IV.5. Weinstein’s Creed Revisited

Any smooth M is a Lagrangian submanifold of (T∗M, ω T∗M):

the zero section!

IV.5. Weinstein’s Creed Revisited

Any smooth M is a Lagrangian submanifold of (T∗M, ω T∗M):

the zero section!

In Memoriam: Prof. Johannes J. Duistermaat (1942-2010)

Figure: Johannes J. Duistermaat and Álvaro Pelayo at the Duistermaat’sfamily residence in Holland. Photo by Saskia Duistermaat in June 2005.

Summary and References for Lecture 1

Lecture 1 SummarySymplectic and Poisson manifolds;Dynamics of vector field, torus actions;Structure theorems for Hamiltonian torus actions

Lecture 1 ReferencesM. Atiyah: Convexity and commuting Hamiltonians.Bull. Lond. Math. Soc. 14 (1982) 1-15

A. C. da Silva: Lectures on Symplectic Geometry. Springer 2000

T. Delzant: Hamiltoniens périodiques et image convexe de l’application moment. Bull. Soc.Math. France 116 (1988) 315–339.

V. Guillemin and S. Sternberg: Convexity properties of the moment mappingInvent. Math. 67 (1982) 491–513

Symplectic theory of completely integrable Hamiltonian systems (with S. Vu Ngo.c)Bull. Amer. Math. Soc. 48 (2011) 409-455

Summary and References for Lecture 1

Lecture 1 SummarySymplectic and Poisson manifolds;Dynamics of vector field, torus actions;Structure theorems for Hamiltonian torus actions

Lecture 1 ReferencesM. Atiyah: Convexity and commuting Hamiltonians.Bull. Lond. Math. Soc. 14 (1982) 1-15

A. C. da Silva: Lectures on Symplectic Geometry. Springer 2000

T. Delzant: Hamiltoniens périodiques et image convexe de l’application moment. Bull. Soc.Math. France 116 (1988) 315–339.

V. Guillemin and S. Sternberg: Convexity properties of the moment mappingInvent. Math. 67 (1982) 491–513

Symplectic theory of completely integrable Hamiltonian systems (with S. Vu Ngo.c)Bull. Amer. Math. Soc. 48 (2011) 409-455

The end. THANK YOU!

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