Integrable Systems and Group Actions, a Love Story

Home , Constant of motion

Integrable systems and group actions, a love story

Eva Miranda

UPC-Barcelona & Observatoire de Paris

Junior Global Poisson

Miranda (UPC) A love story 2020 1 / 42 Outline

1 Symmetries and brackets

2 Integrable systems and toric manifolds

3 Applications to Quantization

4 Integrable systems as cotangent lifts

5 Action-angle coordinates for singular structures

6 Singular cotangent models

7 A singular Delzant theorem

Miranda (UPC) A love story 2020 1 / 42 The usual suspects, September, 2020

Miranda (UPC) A love story 2020 2 / 42 Emmy Noether, 1882-1935

Miranda (UPC) A love story 2020 2 / 42 Noether’s principle

Noether principle and group actions Conserved quantities give rise to symmetries in physical systems and can be encoded as group actions on these manifolds.

Simple pendulum:

Miranda (UPC) A love story 2020 3 / 42 Motivation

The existence of symmetries in a diﬀerential equation allows reduction of degrees of freedom. An example is given by, Hamilton’s equations.

Hamilton’s equations correspond to the equations of the ﬂow of the Hamiltonian vector ﬁeld XH determined by the equation

ιXH ω = −dH P taking as form ω = i dpi ∧ dqi. There is a geometrical structure behind this formula symplectic form ω (closed non-degenerate 2-form).

Miranda (UPC) A love story 2020 4 / 42 Sim´eon Denis Poisson, 1781-1840

Miranda (UPC) A love story 2020 4 / 42 Poisson, 1809

Figure: Poisson bracket Miranda (UPC) A love story 2020 4 / 42 Example 1: Symplectic manifolds

Xf (f) = df(Xf ) = ω(Xf ,Xf ) = 0 (Noether’s principle).

Indeed for symplectic manifolds the Poisson bracket is:

{f, g} = ω(Xf ,Xg) and Noether’s principle reads {f, f} = 0.

µ = h

Miranda (UPC) A love story 2020 5 / 42 Example 2: Lie algebras

The commutator [A, B] = AB − BA is antisymmetric and satisﬁes [X, [Y,Z]] + [Y, [Z,X]] + [Z, [X,Y ]] = 0, (Jacobi). Consider so(3,R) := {A ∈ M(3, R),AT + A = 0. T r(A) = 0}. The brackets are determined on a basis  0 1 0  0 0 1 0 0 0 e1 := −1 0 0 , e2 :=  0 0 0 , e3 := 0 0 1 0 0 0 −1 0 0 0 −1 0

by [e1, e2] = −e3, [e1, e3] = e2, [e2, e3] = −e1. ∗ Deﬁne the (Poisson) bracket using the dual basis x1, x2, x3 in so(3,R)

{x1, x2} = −x3, {x1, x3} = x2, {x2, x3} = −x1

It satiﬁes Jacobi {xi, {xj, xk}} + {xj, {xk, xi}} + {xk, {xi, xj}} = 0

Miranda (UPC) A love story 2020 6 / 42 From Lie algebras to Poisson structures

Another way to write the Poisson bracket ∂ ∂ ∂ ∂ ∂ ∂ −x3 ∧ + x2 ∧ − x1 ∧ ∂x1 ∂x2 ∂x1 ∂x3 ∂x2 ∂x3

Using the properties of the Poisson bracket, 2 2 2 2 2 2 {x1 + x2 + x3, xi} = 0, i = 1, 2, 3 and the function f = x1 + x2 + x3 is a constant of motion.

Each sphere is endowed with an area form (symplectic structure).

Miranda (UPC) A love story 2020 7 / 42 General Poisson case

Theorem ( Local structure, Weinstein) Let (P n, Π) be a smooth Poisson manifold and let p be a point of P of rank 2k, then there is a smooth local coordinate system (x1, y1, . . . , xk, yk, z1, . . . , zn−2k) near p, in which the Poisson structure Π can be written as

k ∂ ∂ ∂ ∂ Π = X ∧ + X f (z) ∧ , ∂x ∂y ij ∂z ∂z i=1 i i ij i j

where fij vanish at the origin.

For linearizable transverse Poisson structures Example 1 and 2 are the building blocks of the theory.

Miranda (UPC) A love story 2020 8 / 42 Soﬁa Kovalevskaya, 1850-1891

Miranda (UPC) A love story 2020 8 / 42 Integrability and dynamical systems

Importance of integrability of the associated system and dynamical properties of its solutions such as stability.

Integrable system 2n A set of functions f1, . . . , fn on (M , ω) such that

f1, . . . , fn Poisson commute, i.e., {fi, fj} = 0, ∀i, j.

df1 ∧ · · · ∧ dfn 6= 0.

2n n The mapping F : M −→ R given by F = (f1, . . . , fn) is called moment map.

Miranda (UPC) A love story 2020 9 / 42 Coupling two simple harmonic oscillators

∗ 2 Phase space: (T (R ), ω = dx1 ∧ dy1 + dx2 ∧ dy2). Total energy: 1 1 H = (y2 + y2) + (x2 + x2) 2 1 2 2 1 2 . H = h is a sphere S3.

We have rotational symmetry on this sphere the angular momentum is a constant of motion, L = x1y2 − x2y1, XL = (−x2, x1, −y2, y1) and

XL(H) = {L, H} = 0.

Miranda (UPC) A love story 2020 10 / 42 Cauchy-Riemann equations

2 Take a holomorphic function on F : C −→ C decompose it as 4 F = G + iH with G, H : R −→ R. Cauchy-Riemann equations for F in coordinates zj = xj + iyj, j = 1, 2

∂G ∂H ∂G ∂H = , = − ∂xi ∂yi ∂yi ∂xi

Reinterpret these equations as the equality

{G, ·}0 = {H, ·}1

with {·, ·}j the Poisson brackets associated to the real and imaginary part of the symplectic form ω = dz1 ∧ dz2 (ω = ω0 + iω1).

Check {G, H}0 = 0 and {H,G}1 = 0 (integrable system).

Miranda (UPC) A love story 2020 11 / 42 Classical integrable systems

The compact regular level sets of an integrable system F = (f1, . . . , fn) are tori (Liouville tori). Theorem (Liouville-Mineur-Arnold) Semilocally around a Liouville torus:

There exist coordinates (action-angle) (p1, . . . , pn, θ1, . . . , θn) with n n Pn values in B × T such that ω = i=1 dpi ∧ dθi. The level sets of the coordinates p1, . . . , pn correspond to the Liouville tori of F . The problem of global existence of action-angle variables is related to monodromy and the Chern class of the ﬁbration given by the moment map.

Miranda (UPC) A love story 2020 12 / 42 The Liouville-Mineur-Arnold theorem

Figure: Liouville tori

In action-angle coordinates (pi, θi) the ﬁbers of F are the tori {pi = ci} Pn and the symplectic structure is simple (Darboux) ω = i=1 dpi ∧ dθi.

Miranda (UPC) A love story 2020 13 / 42 Henri Mineur, 1899-1954

Miranda (UPC) A love story 2020 13 / 42 Action coordinates and Liouville 1-form

It was an astronomer and mathematician (Henri Mineur) who gave an explicit formula of action coordinates: Z pi = α γi where γi is a cycle of a Liouville torus and( ω = dα). Miranda (UPC) A love story 2020 14 / 42 Hans Duistermaat, 1942-2010

Miranda (UPC) A love story 2020 14 / 42 Semilocal and global toric actions

Deﬁnition (Hamiltonian action) Let G be a compact Lie group acting symplectically on (M, ω). The action is Hamiltonian if there exists an equivariant map µ : M → g∗ such that for each element X ∈ g,

X − dµ = ιX# ω, (1)

with µX =< µ, X >. The map µ is called the moment map.

Moment maps and reduction provide an eﬀective tool to study symmetries in geometrical models in mechanics.

Miranda (UPC) A love story 2020 15 / 42 Torus actions in the proof

1 Topology of the foliation. The fibration in a neighbourhood of a compact connected fiber is a trivial fibration by compact fibers 2 These compact fibers are tori: We recover a Tn-action tangent to the leaves of the foliation This implies a process of uniformization of periods. Φ: Rr × (Tr × Bs) → Tr × Bs (1) (r) (2) ((t1, . . . , tr), m) 7→ Φt1 ◦ · · · ◦ Φtr (m).

3 We prove that this action is symplectic (we use the fact that if Y is a complete vector field of period 1 and ω is a symplectic form for which 2 LY ω = 0, then LY ω = 0). 4 As ω is exact in a neighbourhood of the Liouville torus the action is Hamiltonian. 5 To construct action-angle coordinates we use Darboux-Carathéodory theorem and the constructed Hamiltonian action of Tn to drag normal forms from a neighbourhood of a point to a neighbourhood of a fiber.

Miranda (UPC) A love story 2020 16 / 42 Torus actions in the proof

Miranda (UPC) A love story 2020 16 / 42 Toric symplectic manifolds

Theorem (Delzant) Toric manifolds are classiﬁed by Delzant’s polytopes. The bijective correspondence between these two sets is given by the image of the {toric manifolds} −→ {Delzant polytopes} moment map: 2n n (M , ω, T ,F ) −→ F (M)

µ = h µ 2 CP

it it (t1, t2) · [z0 : z1 : z2] = [z0 : e 1 z1 : e 2 z2]

Miranda (UPC) A love story 2020 17 / 42 Classical vs. Quantum: Another love story.

1 Classical systems 1 Quantum System

2 Observables C∞(M) 2 Operators in H (Hilbert)

3 3 2πi Bracket {f, g} Commutator [A, B]h = h (AB − BA)

Miranda (UPC) A love story 2020 18 / 42 Geometric Quantization in a nutshell

(M 2n, ω) symplectic manifold with integral [ω].

(L, ∇) a complex line bundle with a connection ∇ such that curv(∇) = −iω (prequantum line bundle). A real polarization P is a Lagrangian foliation. Integrable systems provide natural examples of real polarizations.

Flat sections equation: ∇X s = 0, ∀X tangent to P.

Deﬁnition A Bohr-Sommerfeld leaf is a leaf of a polarization admitting global ﬂat sections.

1 ∂ Example: Take M = S × R with ω = dt ∧ dθ, P =< ∂θ >, L the trivial bundle with connection 1-form Θ = tdθ ∇X σ = X(σ) − i < Θ, X > σ Flat itθ sections: σ(t, θ) = a(t).e Bohr-Sommerfeld leaves are given by the condition t = 2πk, k ∈ Z. Liouville-Mineur-Arnold ! this example is the canonical one.

Miranda (UPC) A love story 2020 19 / 42 Quantization via action-angle coordinates

Theorem (Guillemin-Sternberg) If the polarization is a regular ﬁbration with compact leaves over a simply connected base B, then the Bohr-Sommerfeld set is given by, n BS = {p ∈ M, (f1(p), . . . , fn(p)) ∈ Z }where f1, . . . , fn are global action coordinates on B. For toric manifolds the base B is the image of the moment map. Quantization “Quantize” these systems counting Bohr-Sommerfeld leaves. For integrable systems Bohr-Sommerfeld leaves are just “integral” Liouville tori.

Miranda (UPC) A love story 2020 20 / 42 Liouville-Mineur-Arnold via cotangent lifts

Cotangent lifts Given a Lie group action ρ : G × M −→ M, its cotangent lift to T ∗M is Hamiltonian with moment map

∗ ∗ # µ : T M → g , hµ(α),Xi = hλ|α,XT ∗M |αi with λ the Liouville one-form.

Example For M = G = Tn and the action is translations of the torus on itself the moment map is µ : Tn × Rn =∼ T ∗Tn → t∗ =∼ Rn :(θ, p) 7→ p.

In particular: Cotangent model in the symplectic case Semilocally around a Liouville torus, an integrable system is equivalent to the moment map of the cotangent lift of the action by translations of Tn on itself.

Miranda (UPC) A love story 2020 21 / 42 Action-angle coordinates with singularities

The theorem of Marle-Guillemin-Sternberg for ﬁxed points of toric actions can be generalized to non-degenerate singularities of integrable systems. Theorem (Eliasson, M., Zung) There exists symplectic Morse normal forms for non-degenerate singularities.

k k 2(n−k) Pk Pn−k Local model: N = D × T × D , ω = i=1 dpi ∧ dθi + i=1 dxi ∧ dyi. with components of the moment map:

1 Regular fi = pi for i = 1, ..., k;

2 2 2 Elliptic fi = xi + yi for i = k + 1, ..., ke;

3 Hyperbolic fi = xiyi for i = ke + 1, ..., ke + kh;

4 focus-focus fi = xiyi+1 − xi+1yi, fi+1 = xiyi + xi+1yi+1 for i = ke + kh + 2j − 1, j = 1, ..., kf .

The system is semitoric if there are no hyperbolic components.

Miranda (UPC) A love story 2020 22 / 42 Singularities in physical examples

Harmonic oscillator Simple pendulum Spherical pendulum

k m

Fg Fg Elliptic singularity Hyperbolic singularity Focus-focus singularity

2 2 f = xy f1 = x1y2 − x2y1 f = x + y 2 2 (orf = x − y ) f2 = x1y1 + x2y2

Miranda (UPC) A love story 2020 23 / 42 Pau Mir, 2020

Miranda (UPC) A love story 2020 23 / 42 Some applications

Cotangent models ! rigidity (stability) of integrable systems. Nondegenerate singularities can be seen as cotangent lifts (complexiﬁed) applications to quantization. S1-invariant degenerate singularities are rigid (in the sense of Palais: close actions are equivalent).

Miranda (UPC) A love story 2020 24 / 42 The solar system

‘‘We revolve around the Sun like any other planet.’’ 1514, Nicolaus Copernicus.

Miranda (UPC) A love story 2020 25 / 42 The solar system

Kepler: Planets spin in elliptical orbits.

Miranda (UPC) A love story 2020 26 / 42 The n-body problem

The n-body problem describes the movement of n bodies under mutual attraction.

Integrable for n = 2: The Hamiltonian function is

2 2 kp1k kp2k H(x1, x2, p1, p2) := Ekin − U = + − U. 2m1 2m2 where U := Gm m 1 is the gravitational potential. Integrals: 1 2 kx2−x1k 1 Total linear momentum: The problem reduces to determining the relative position r = x2 − x1. 2 Total angular momentum: makes the problem planar.

Miranda (UPC) A love story 2020 27 / 42 The restricted 3-body problem

Simpliﬁed version of the general 3-body problem. One of the bodies has negligible mass. The other two bodies move independently of it following Kepler’s laws for the 2-body problem.

Figure: Circular 3-body problem

Miranda (UPC) A love story 2020 28 / 42 Planar restricted 3-body problem

The time-dependent self-potential of the small body is 1−µ µ U(q, t) = + , with q1 = q1(t) the position of the planet with |q−q1| |q−q2| mass 1 − µ at time t and q2 = q2(t) the position of the one with mass µ. The Hamiltonian of the system is H(q, p, t) = p2/2 − U(q, t), (q, p) ∈ R2 × R2, where p =q ˙ is the momentum of the planet.

Consider the canonical change (X,Y,PX ,PY ) 7→ (r, α, Pr =: y, Pα =: G). 2 + Introduce McGehee coordinates (x, α, y, G), where r = x2 , x ∈ R , can be then extended to inﬁnity (x = 0). The symplectic structure becomes a singular object 4 ω = − x3 dx ∧ dy + dα ∧ dG. for x > 0 The integrable 2-body problem for µ = 0 is integrable with respect to the singular ω.

Miranda (UPC) A love story 2020 29 / 42 Model of these systems

1 ω = dx ∧ dy + X dx ∧ dy xm 1 1 i i 1 i≥2

Close to x1 = 0, the systems behave like,

and not like,

Miranda (UPC) A love story 2020 30 / 42 b-Poisson structures

Deﬁnition Let (M 2n, Π) be an (oriented) Poisson manifold such that the map

p ∈ M 7→ (Π(p))n ∈ Λ2n(TM) is transverse to the zero section, then Z = {p ∈ M|(Π(p))n = 0} is a hypersurface called the critical hypersurface and we say that Π is a b-Poisson structure on (M,Z).

Theorem

For all p ∈ Z, there exists a Darboux coordinate system x1, y1, . . . , xn, yn centered at p such that Z is deﬁned by x1 = 0 and

n ∂ ∂ X ∂ ∂ Π = x ∧ + ∧ 1 ∂x ∂y ∂x ∂y 1 1 i=2 i i

Miranda (UPC) A love story 2020 31 / 42 Examples

A Radko surface.

The product of (R, πR) a Radko compact surface with a compact symplectic manifold (S, ω) is a b-Poisson manifold. corank 1 Poisson manifold (N, π) and X Poisson vector ﬁeld ⇒ 1 ∂ (S × N, f(θ) ∂θ ∧ X + π) is a b-Poisson manifold if, 1 f vanishes linearly. 2 X is transverse to the symplectic leaves of N. We then have as many copies of N as zeroes of f.

Miranda (UPC) A love story 2020 32 / 42 Poisson Geometry of the critical hypersurface

This last example is semilocally the canonical picture of a b-Poisson structure .

1 The critical hypersurface Z has an induced regular Poisson structure of corank 1.

2 There exists a Poisson vector ﬁeld v transverse to the symplectic foliation induced on Z (modular vector ﬁeld).

3 (Guillemin-M. Pires) Z is a mapping torus with glueing diﬀeomorphism the ﬂow of v.

Miranda (UPC) A love story 2020 33 / 42 Singular forms

A vector field v is a b-vector field if vp ∈ TpZ for all p ∈ Z. The b-tangent bundle bTM is defined by ( ) b-vector fields Γ(U, bTM) = on (U, U ∩ Z)

The b-cotangent bundle bT ∗M is (bTM)∗. Sections of Λp(bT ∗M) are b-forms, bΩp(M).The standard diﬀerential extends to

d : bΩp(M) → bΩp+1(M)

A b-symplectic form is a closed, nondegenerate, b-form of degree 2. This dual point of view, allows to prove a b-Darboux theorem and semilocal forms via an adaptation of Moser’s path method because we can play the same tricks as in the symplectic case.

Miranda (UPC) A love story 2020 34 / 42 Kiesenhofer-M.-Scott, 2016

Miranda (UPC) A love story 2020 34 / 42 b-integrable systems

Deﬁnition a 2n b-integrable system A set of b-functions f1, . . . , fn on (M , ω) such that

f1, . . . , fn Poisson commute. n b ∗ df1 ∧ · · · ∧ dfn 6= 0 as a section of Λ ( T (M)) on a dense subset of M and on a dense subset of Z ac log |x| + g

Example 1 The symplectic form h dh ∧ dθ deﬁned on the interior of the upper hemisphere 2 2 H+ of S extends to a b-symplectic form ω on the double of H+ which is S . The triple (S2, ω, log|h|) is a b-integrable system.

Example

If (f1, . . . , fn) is an integrable system on M, then (log |h|, f1, . . . , fn) on 2 H+ × M extends to a b-integrable on S × M.

Miranda (UPC) A love story 2020 35 / 42 Action-angle coordinates for b-integrable systems

The compact regular level sets of a b-integrable system are (Liouville) tori. Theorem (Kiesenhofer-M.-Scott) Around a Liouville torus there exist coordinates n n (p1, . . . , pn, θ1, . . . , θn): U → B × T such that

c n ω| = dp ∧ dθ + X dp ∧ dθ , (3) U p 1 1 i i 1 i=2

and the level sets of the coordinates p1, . . . , pn correspond to the Liouville tori of the system.

Reformulation a n Integrable systems semilocally ! twisted cotangent lift of a T action ∗ n by translations on itself to (T T ). a Pn We replace the Liouville form by log |p1|dθ1 + i=2 pidθi.

Miranda (UPC) A love story 2020 36 / 42 The cotangent model for b-integrable systems

Pn Consider the “log” Liouville one-form λ = c log |p1|dθ1 + i=2 pidθi. Deﬁnition c n ω := dλ = dp ∧ dθ + X dp ∧ dθ p 1 1 i i 1 i=2 is called the twisted b-symplectic form on T ∗Tn with modular weight c.

The moment map of the twisted b-cotangent lift is

µ := (c log |p1|, p2, . . . , pn).

Theorem (Cotangent model in the b-symplectic case) Semilocally around a Liouville torus, a b-integrable system is equivalent to the moment map of the twisted b-cotangent lift of the action by translations of Tn on itself.

Miranda (UPC) A love story 2020 37 / 42 Proof

1 Topology of the foliation. In a neighbourhood of a compact connected ﬁber the b-integrable system F is diﬀeomorphic to the b-integrable system on n n W := T × B given by the projections p1, . . . , pn−1 and log |pn|.

2 Uniformization of periods: We want to deﬁne integrals whose (b-)Hamiltonian vector ﬁelds induce a Tn action. Start with Rn-action:

Φ: Rn × (Tn × Bn) → Tn × Bn (1) (n) ((t1, . . . , tn), m) 7→ Φt1 ◦ · · · ◦ Φtn (m).

n Uniformize to get a T action with fundamental vector ﬁelds Yi.

3 The vector fields Yi are Poisson vector fields (check LYi LYi ω = 0). b ∞ 4 The vector fields Yi are Hamiltonian with primitives σ1, . . . , σn ∈ C (W ). In this step the properties of b-cohomology are essential.Use this action to drag a local normal form (Darboux-Carathéodory) in a whole neighbourhood.

Miranda (UPC) A love story 2020 38 / 42 Proof

2 Uniformization of periods: We want to deﬁne integrals whose (b-)Hamiltonian vector ﬁelds induce a Tn action. Start with Rn-action:

Φ: Rn × (Tn × Bn) → Tn × Bn (1) (n) ((t1, . . . , tn), m) 7→ Φt1 ◦ · · · ◦ Φtn (m).

n Uniformize to get a T action with fundamental vector ﬁelds Yi.

Miranda (UPC) A love story 2020 38 / 42 Proof

2 Uniformization of periods: We want to deﬁne integrals whose (b-)Hamiltonian vector ﬁelds induce a Tn action. Start with Rn-action:

Φ: Rn × (Tn × Bn) → Tn × Bn (1) (n) ((t1, . . . , tn), m) 7→ Φt1 ◦ · · · ◦ Φtn (m).

n Uniformize to get a T action with fundamental vector ﬁelds Yi.

Miranda (UPC) A love story 2020 38 / 42 Proof

2 Uniformization of periods: We want to deﬁne integrals whose (b-)Hamiltonian vector ﬁelds induce a Tn action. Start with Rn-action:

Φ: Rn × (Tn × Bn) → Tn × Bn (1) (n) ((t1, . . . , tn), m) 7→ Φt1 ◦ · · · ◦ Φtn (m).

n Uniformize to get a T action with fundamental vector ﬁelds Yi.

Miranda (UPC) A love story 2020 38 / 42 A b-Delzant theorem

We can reconstruct the b-Delzant polytope from the Delzant polytope on a mapping torus via symplectic cutting close to the critical hypersurface.

µ = log |h| µ

Guillemin-M.-Pires-Scott There is a one-to-one correspondence between b-toric manifolds and b-Delzant polytopes and toric b-manifolds are either:

bT2 × X (X a toric symplectic manifold of dimension (2n − 2)). obtained from bS2 × X via symplectic cutting. Miranda (UPC) A love story 2020 39 / 42 (Singular) symplectic manifolds

bm -Symplectic

Symplectic

Folded symplectic

Miranda (UPC) A love story 2020 40 / 42 D´ej`a-vu...

Symplectic b-Symplectic Folded symplectic manifolds manifolds manifolds • Darboux theorem • Darboux theorem • Darboux theorem • Delzant and • Delzant and (Martinet) convexity theorems convexity theorems • Delzant-type • Action-Angle • Action-Angle theorems (Cannas da coordinates theorem Silva-Guillemin-Pires) • Action-agle theorem (M-Cardona)

Miranda (UPC) A love story 2020 40 / 42 Desingularizing bm-symplectic structures

Theorem (Guillemin-M.-Weitsman) Given a bm-symplectic structure ω on a compact manifold (M 2n,Z):

If m = 2k, there exists a family of symplectic forms ω which coincide with the bm-symplectic form ω outside an -neighbourhood of Z and for which −1 2k−1 the family of bivector ﬁelds (ω) converges in the C -topology to the Poisson structure ω−1 as → 0 .

If m = 2k + 1, there exists a family of folded symplectic forms ω which coincide with the bm-symplectic form ω outside an -neighbourhood of Z.

In particular: Any b2k-symplectic manifold admits a symplectic structure. Any b2k+1-symplectic manifold admits a folded symplectic structure. The converse is not true: S4 admits a folded symplectic structure but no b-symplectic structure.

Miranda (UPC) A love story 2020 41 / 42 Deblogging everything...

What to deblog? • Integrable Systems • Toric actions • Forms..

Miranda (UPC) A love story 2020 42 / 42 Cardona, Guillemin-Weitsman, Matveeva, Planas

Miranda (UPC) A love story 2020 42 / 42 To be continued...

Geometric quantization of bm-symplectic manifolds and [Q, R] = 0 with Guillemin and Weitsman. Action-angle coordinates and KAM for bm-symplectic manifolds with Planas. Slice theorem and reduction for bm-symplectic manifolds with Matveeva. Action-angle coordinates for singular symplectic manifolds (including folded symplectic manifolds): From local to global (with Cardona).

Miranda (UPC) A love story 2020 42 / 42 Thanks for your attention!

Figure: The usual suspects in November 26, 2019, vs September 14, 2020 Miranda (UPC) A love story 2020 42 / 42