First steps in symplectic and spectral theory of integrable systems

The paper intends to lay out the first steps towards constructing a unified framework to understand the symplectic and spectral theory of finite dimensional integrable Hamiltonian systems. While it is difficult to know what the best approach to such a large classification task would be, it is possible to single out some promising directions and preliminary problems. This paper discusses them and hints at a possible path, still loosely defined, to arrive at a classification. It mainly relies on recent progress concerning integrable systems with only non-hyperbolic and non-degenerate singularities. This work originated in an attempt to develop a theory aimed at answering some questions in quantum spectroscopy. Even though quantum integrable systems date back to the early days of quantum mechanics, such as the work of Bohr, Sommerfeld and Einstein, the theory did not blossom at the time. The development of semiclassical analysis with microlocal techniques in the last forty years now permits a constant interplay between spectral theory and symplectic geometry. A main goal of this paper is to emphasize the symplectic issues that are relevant to quantum mechanical integrable systems, and to propose a strategy to solve them.


Program outline
This paper suggests an approach to work towards a symplectic and spectral classification of finite dimensional integrable Hamiltonian systems.
It is a bottom to top path, consisting of many problems, in which each one tries to deal with only one major difficulty. Some of these problems are labelled as "Core Problems" to indicate that they are essential milestones towards the so-called "Classification Problems" that are the main incentives of this paper. Our proposed strategy to deal with each of these problems is inspired by our articles [48,49]. In order to facilitate the reading of the paper we briefly review the main results and proofs of these articles, as well as some essential ingredients from the literature used in them.
We are optimistic about the potential of the ideas we present in this paper, while we recognize many technical and conceptual challenges to implement them. Of course, we do not know if some of the challenges could be impossible to surpass. Two key tools in the modern approaches to integrable systems are: (i) the study of singular affine structures, and (ii) the symplectic linearization theorems for non-degenerate singularities 1 .
These tools dominate, implicitly or explicitly, the proof strategies that we outline. 1 In the C ∞ -smooth category, we expect major difficulties in the cases where the integrable systems have degenerate singularities (this is not so in the algebraic setting, as seen by the works of Garay and van Straten). The lack of symplectic linearization theorems for degenerate singularities makes their study challenging. At this time we neither have much information about such degenerate integrable systems, nor do we have methods to analyze them (see Section 2.7 for further comments). Algebraic geometry seems a natural setting for the study degenerate singularities. The case where a system has hyperbolic singularities is rich and interesting, as we know from examples by Zhilinskií and a few results by Vũ Ngo . c, Dullin, Zung, and Bolsinov, among others. In addition, a key tool on the spectral theory side is the microlocal analysis of Toeplitz operators developed in the past decade. 1 In this framework, the known integrable systems should be understood in terms of a collection of invariants. Six motivational examples of the program are: the coupled spin-oscillator, the spherical pendulum, the Lagrange top, the two-body problem, the Kowalevski top, and the three wave interaction; these examples are explained in more detail in Section 2.1.
Some of these examples have hyperbolic or degenerate singularities. Analyzing the case of integrable systems which have hyperbolic singularities, will probably involve major breakthroughs (see Section 2.3); it remains largely unexplored. Our proposal for a unifying approach aims at identifying the essential features of integrable systems through the computation of invariants that classify them. In this sense, a success in our approach would help to reconcile the vast amount of literature for specific systems, with the theoretical approaches developed in recent times (see Section 1.3).
The present paper does not intend to be a survey, but rather a fast description of a few preliminary ideas which fit into a larger plan. In this sense it is more a "work guide" than a research paper. It is evident from the very few articles we cite in the paper, that our choice of references is mostly a practical one. We refer to Section 5.1 for further discussion in this direction, and for references to survey papers where more extensive bibliographies may be found.
The paper could be read independently, but it is probably best read simultaneously or after our article [50] which gives a succinct overview of the current research in the subject, with references to previous works. Both papers complement each other and have different focuses.
1.1. Origins. The first steps in the approach to classifying integrable systems advocated in this paper originated in our attempts to develop technology to answer inverse type questions about quantum systems in molecular spectroscopy.
While the notion of a quantum integrable system dates back to the early times of quantum mechanics (e.g., in the works of Bohr, Sommerfeld, and Einstein), the basic results in the symplectic theory of classical integrable systems could not have been used in Schrödinger's quantum setting for the simple reason that the analysis of pseudodifferential operators in phase space was developed later. In addition to pseudodifferential operators, this paper advocates the use of microlocal analysis of Toeplitz operators to analyze quantization and inverse spectral problems on arbitrary phase spaces (not necessarily cotangent bundles).
In [48,49], we carried out the symplectic part of this "program" for non-hyperbolic systems with two degrees of freedom in four dimensional phase space for which one component of the system is periodic and proper -these systems are called semitoric systems. A physical example of semitoric system is the coupled spin-oscillator, mentioned above. {f i , f j } := ω(H f i , H f j ) = 0, for all 1 i, j n.
We also refer to the map F := (f 1 , . . . , f n ) : M → R n as the integrable system itself. Although the path to arrive at it is still loosely defined, the first goal of the program is clear. 1.2.2. Spectral Geometry. The driving force behind this paper, and behind the goal above, is the inverse question: if we know the spectrum of a quantum physical system which is integrable (see Figure 19), can we reconstruct the classical system from it? Answering this question involves studying symplectic and spectral invariants. The study of these invariants is the common theme of the present paper. It is connected, at a fundamental level, with the analysis of the singularities of integrable systems which encode information about the global behavior of the system (Figure 1). ¿From a mathematical point of view, this question leads to the second goal of this program: to develop an inverse spectral result along the following lines. A quantum integrable system is defined as a family of commuting operators T 1 , . . . , T n on Hilbert spaces H , indexed by → 0, whose principal symbols f 1 , . . . , f n form a classical integrable system on a symplectic manifold (M, ω). The semiclassical joint spectrum of the system is given by the collection of joint spectra of T 1 , . . . , T n (see Figure 19 for one such element.) Next us explain the terminology more concretely when M is a compact manifold (but the spectral goal refers to all manifolds, compact or not).
In the compact case the commuting operators which we consider are "Toeplitz operators", and the quantization of the symplectic manifold is the "geometric quantization". Note that the Spectral Goal assumes that we have a quantization of the manifold, and the existence of such a quantization is far from obvious. A now standard procedure by B. Kostant and J.M. Souriau in the 1960s is to introduce a prequantum bundle L → M , that is a Hermitian line bundle with curvature 1 i ω and a complex structure j compatible with ω. One then defines the quantum space as the space H k := H 0 (M, L k ) of holomorphic sections of tensor powers L k of L. Here the semiclassical parameter is = 1/k. The parameter k is a positive integer, the semi-classical limit corresponds to the large k limit. Associated to such a quantization there is an algebra T (M, L, j) of operators, called Toeplitz operators. A Toeplitz operator is any sequence (T k : H k → H k ) k∈N * of operators of the form where f (·, k), viewed as a multiplication operator, is a sequence in C ∞ (M ) with an asymptotic expansion f 0 + k −1 f 1 + . . . for the C ∞ topology, and the norm of R k is O(k −∞ ). The algebra of Toeplitz operators plays the same role as the algebra of semiclassical pseudodifferential operators for a cotangent phase space. A Toeplitz operator has a principal symbol, which is a smooth function on the phase space M . If T and S are Toeplitz operators, then (T k + k −1 S k ) k∈N * is a Toeplitz operator with the same principal symbol as T . If T k is Hermitian (i.e. self-adjoint) for k sufficiently large, then the principal symbol of T is real-valued. Two Toeplitz operators (T k ) k∈N * and (S k ) k∈N * commute if T k and S k commute for every k.
Spectral Goal of Program for Integrable Systems: Show that the semiclassical joint spectrum of a quantum integrable system recovers the classical integrable system, up to symplectic isomorphisms (a program proposal to arrive at this result is outlined in Section 3).
As we will see later, to complete this second goal, partial success with the first goal is necessary. Put differently, achieving the second goal for quantum integrable systems (of a certain type) passes through the solution of the first goal for classical integrable systems (of the type corresponding to the quantum system at hand). 1.3. Approach comparison. We put forward some ideas to unify the existing knowledge of concrete integrable systems. The final goals were presented in Subsection 1.2. Our approach differs from the traditional one, where concrete individual or families of integrable systems are studied. We do not intend to replace this classical method but to complement it by drawing attention to an additional research focus. For instance, the coupled spin-oscillator system fits in the classification theory developed in our papers [48,49]. However, these articles do not determine the invariants of the coupled spin-oscillator, which is a difficult computational problem addressed in our later paper [51]; all invariants of the coupled spin-oscillator are found, with the exception of one for which only a rigorous linear approximation is given (this invariant is a formal power series).
The existing literature contains sophisticated techniques for computing features of integrable systems explicitly. These methods, and their conclusions, are spread over many thousands of papers. Our theory for semitoric systems (developed in [48,49]) and its extension proposed here, neither improve these traditional approaches nor replace the invariants they compute. Although superficially, many of these characteristic traits seem unrelated to our symplectic invariants, they are, of course, linked to them. Whenever a property of an integrable system does not change under symplectic isomorphisms which preserve the system, it must be encoded in the finite list of invariants we construct. Our initial investigations have relied on existing literature and have been motivated by it. The ideas suggested in this paper go hand in hand with the traditional work and are not independent from it.
1.4. Strategies and proofs. The purpose of this paper is to present many open problems, not to outline avenues for their solution. The reason for this is two-fold. First, we have not solved these problems ourselves and it is very likely that surprises may lurk in the technical details. Second, we believe that at least some of them may have proofs that would follow the outline of those in [48,49]: construct local symplectic invariants (analogues of the invariants defined in these papers) and then symplectically glue these pieces. The proofs of our classification theorems in [48,49] -for the so called semitoric systems -are divided into separate steps, so it is easy to follow and understand the overall approach. We review in Subsection 2.2 the strategy to prove these classification theorems (Theorem 2.21 and Theorem 2.24), emphasizing which methods therein could be applicable in more general cases. In addition, we discuss the difficulties encountered in each part of the overall setup. The arguments proper to each step in [48,49] depend on the symplectic invariants given in these articles.
The building of these invariants is somewhat "rigid" and is designed for the type of systems classified in [48,49] (i.e., semitoric systems). Slightly changing the assumption on the type of integrable system can invalidate major parts of the original construction of the invariant (for semitoric systems). How the assumptions of these theorems (i.e., the type of integrable system under consideration) can be weakened corresponds to the titles of the subsections 2.3, 2.4, 2.5, 2.6, 2.7, and 2.8: hyperbolic systems, non-proper systems, higher dimensional systems, non-periodic systems, degenerate systems, and independent topological and geometric questions (see Section 2.1 for examples which fit into one of these categories, or in the overlap of several categories.) The goal of subsections 2.3, 2.4, 2.5, 2.6, 2.7, and 2.8 is to outline, and briefly explain, the major difficulties we foresee, based on our previous experience dealing with integrable systems. Then, in subsection 2.9 (with the title "Collaborative efforts"), we show how to gather all the information gleaned from the previous subsections 2.4, 2.5, 2.6, 2.7, and 2.8 to achieve a general classification. In our opinion, each of these subsections presents a challenge on its own because they address somewhat unrelated technical difficulties; so we suggest that they be considered individually. Precisely because of the likely "independence" of these assumptions, putting all of them together in Section 2.9 should not be a major technical problem. Instead, we expect it to be a book-keeping problem: if all goes well, the general answer in subsection 2.9 would be a "direct sum" of the answers provided in subsections 2.3, 2.4, 2.5, 2.6, 2.7, and 2.8.

1.5.
Directions. At the time of the writing of this paper, most of the problems we state are open. We have begun work on a few of them, and where this is the case, we explicitly say so.
The widely open directions outlined in this paper concern the symplectic theory of systems with degenerate singularities, systems with hyperbolic singularities, and the corresponding spectral theory for these systems; this is explained in subsections 2.3, 2.7, and parts of 3. There are also many open questions of geometric and topological nature that fit within the program; some of these can be found in subsection 2.8. The nonperiodic case, treated in subsection 2.6, is also mostly open and our initial investigations indicate the presence of challenges already in dimension four. Subsection 2.9 is open, but it can be undertaken only after all the problems of the previous subsections have been solved.
1.6. Timeline. We are writing this paper with the intention of raising awareness of what we believe is a known fact among some specialists on integrable systems: the time is ripe for great advances. The reason is that several new effective tools and ideas are now understood, which was not the case, say, ten years ago. As already mentioned, there are three key ingredients that are understood today much better: singular affine structures, symplectic singularity theory, and the microlocal analysis for Toeplitz operators. The technology and ideas developed by many since the 1970 and 1980s, but particularly in the past ten years, could be greatly developed with a view towards a better understanding of the symplectic and spectral aspects of integrable systems.
We are optimistic about the chances of success of the ideas presented here, judging from our own investigations in the past few years. Nevertheless, we admit that there are still many technical and conceptual challenges that need resolution and do not know if some of our scenarios could lead to a dead-end. The program is at the beginning stages, we explain its success highlights so far, and outline expected challenges. It is difficult to predict how long it will be till major progress has been made.
We hope that this paper may be of help to readers who wish to do research on some part of the program.
1.7. How to read this paper. The paper could be read independently, but it is probably better to consult simultaneously our review [50], or even to familiarize oneself first with it. In [50] we gave an overview of the state of the art in the subject. Both articles together could serve as a speedy way to be immersed in the subject and have easy access to some prominent problems. Also, the paper need not be read linearly.
Section 2 deals with the symplectic theory of integrable systems and proposes a self-contained program on its own. Section 3 addresses the spectral theory of integrable systems, but it relies heavily on Section 2.
The plan suggested in the present paper is dynamic and we expect that the approaches to some of the problems, and the strategies we describe, will change as the ideas presented here evolve, and new ones arise. In this sense, this paper is far from giving a definite approach to the problems that it proposes.

Symplectic theory of integrable systems
The unifying topic of this section is symplectic geometry, which is a mathematical language that clearly and concisely formulates many problems in theoretical physics. It also provides a framework in which classical and quantum physics are treated simultaneously. Symplectic geometry is closely connected with many areas of mathematics. Within symplectic geometry, we focus on integrable systems which are a fundamental class of "explicitly" solvable dynamical systems of interest in classical dynamics, semiclassical analysis, partial differential equations, low-dimensional topology, algebraic geometry, representation theory, and theoretical physics.
Many integrable systems are found in simple physical models of classical and quantum physics. Two examples are the spherical pendulum (see Figure 4) and the quantum coupled spin oscillator (see Figure 19). Integrable systems also appear in other applied fields, such as locomotion generation in robotics, elasticity theory, plasma physics, and planetary mission design. In spite of being, in some sense, "solvable", they exhibit a rich behavior. For instance, the symplectic dynamics around the so called focus-focus singularities is highly complex (it has infinitely many symplectic invariants, for example). Figure 3. According to the Arnold-Liouville-Mineur theorem, a tubular neighborhood U of a regular fiber embeds symplectically into T * (T n ) ≃ T n × R n , and the integrable system on this neighborhood is given by the n canonical projections to R. Hence the dynamics around a regular fiber is simple.
2.1. Results, tools, and methodology. Our initial motivation for studying integrable systems was to develop a theory capable of answering questions raised by physicists and chemists, working on quantum molecular spectroscopy. The main question is simple to state: can a classical integrable physical system be reconstructed from the spectrum of its (semi-classical) quantization? Mathematically, this is a fascinating problem. The answer seems to be "yes" for a number of cases. For instance, it is the case for toric systems. Not only symplectic geometry is the tool to answer this question, but it also serves as a method to help understand, even predict, features of the spectral theory of systems in quantum chemistry and quantum physics.
Studying this inverse question in particular, and spectral geometry of quantum integrable systems in general, requires introducing and combining a number of mathematical tools; this is what this project is all about.
A. Previous Results. The behavior of many complicated integrable systems is well understood. However, few general results are known. Among these, the Arnold-Liouville action angle theorem stands out as a classical result to be found in any text on mechanics. In addition, there have been some remarkable successes describing local and semilocal behavior of integrable systems such as Eliasson's linearization theorem for non-degenerate singularities or the work of Vũ Ngo . c and Zung on singularities. Shortly before Eliasson proved his theorems in the mid 1980s, Duistermaat formulated his global action-angle obstruction theorem which remains, to this day, one of the landmark results in the theory of integrable systems. ¿From a topological point of view, the work of the Fomenko school on the classification of of singular Lagrangian fibrations is ground breaking. These results completely classify the topology of non-degenerate two degrees of freedom systems, give a method for handling finite dimensional integrable systems in any dimension, and sometimes even include global descriptions. The important results on compact Hamiltonian group actions by Kostant, Atiyah, Guillemin, Sternberg, Kirwan, and Delzant, provide another model and motivation for the program proposed in this paper.

B. New Tools.
A number of mathematical developments to which many people have contributed, principally since the 1970/1980s, but particularly in the past fifteen years, have provided effective tools that can be used to study global aspects of integrable systems. These include methods from functional analysis and partial differential equations (particularly, geometric techniques involving Fourier integral operators, Toeplitz operators, linearization theorems), symplectic geometry (developments on Lie theory, monodromy, symplectic actions, singular reduction, relations between quantization and reduction, global action-angle coordinates), and algebraic geometry (singular Lagrangian fibrations, affine structures, and the development of the theory of toric varieties).
C. Program methodology. This program relies on the symplectic local and semilocal theory of integrable systems as well as the detailed description of several integrable systems available in the mathematics and physics literature. We propose to investigate global patterns using the aforementioned tools. With their aid, one can introduce a framework to describe complicated phenomena in four dimensions, i.e., Hamiltonian integrable systems with two degrees of freedom, for which one component of the system is periodic; these are called semitoric integrable systems. This program outlines a plan for developing these results, with the ultimate goal of achieving a full understanding of the symplectic and spectral theory of finite dimensional integrable systems.

D. Motivational examples.
The following are famous examples of finite dimensional integrable systems which fit in the scope of our program. The reader may revisit them after reading each of the upcoming sections of the paper. They have been thoroughly studied from many points of view. Examples (1), (2), (3), (4), and (5) are integrable systems with two degrees of freedom on fourdimensional phase space. Example (6) is an example of an integrable system with three degrees of freedom on a six-dimensional phase space. All of the examples below are on a non-compact phase space.
All sections of our program, with the exception of subsection 2.7 (on degenerate systems), use the linearization theorem of Eliasson for singularities. Hence one has to check the non-degeneracy of the singularities (see Definition 2.11). We have verified this for some of the examples below. One could probably deduce this non-degeneracy (or lack thereof) from the existing literature, but we are not aware of a reference where this has been explicitly done.
(1) Coupled spin-oscillator. This integrable system fits into the developed and finalized theory in subsection 2.2 (semitoric systems). We checked this in [51,Section 2]. In this article there is also a detailed study of the coupled spin-oscillator and its quantum counterpart. This example is of primary importance in physics, where it is called the Jaynes-Cummings model. There are natural extensions of this model to arbitrary dimensions. (2) Spherical pendulum. This integrable system fits into subsection 2.4 (non-proper systems).
We checked this in [46,Section 5]. (There is a recent paper by Dullin that computes invariant (ii) in Theorem 2.21 for the spherical pendulum). (3) Two-body problem. This integrable system fits into a combination of subsection 2.3 (hyperbolic systems) and subsection 2.4 (non-proper systems). One of its components generates a periodic flow, but it is not a proper map. The singular Lagrangian fibration defined by the system is not proper either (because the bifurcation set and the critical set do not coincide).
This integrable system has hyperbolic singularities. We have not checked whether all the singularities of this system are non-degenerate. If there are degenerate singularities, then the study of this example also would overlap with subsection 2.7 (degenerate systems), in addition to subsection 2.3 and subsection 2.4. (4) Lagrange top. The heavy top equations in body representation are known to be Hamiltonian on se(3) * . These equations describe a classical Hamiltonian system with 2 degrees of freedom on the coadjoint orbits of the Euclidean group SE(3). The generic coadjoint orbit is a magnetic cotangent bundle of the 2-sphere S 2 . This two degrees of freedom system has a conserved integral but it does not have, generically, additional integrals. However, in the Lagrange top, one can find one additional integral, namely, the momentum map associated with rotations about the axis of gravity, which makes the system integrable. It is classically known that the Lagrange Heavy Top is integrable. The Lagrange momentum map is, however, a non-proper map. But the singular Lagrangian fibration given by the system itself is a proper map. The Lagrange Top has hyperbolic singularities.
Hence, this example fits in subsection 2.3 (hyperbolic systems) and subsection 2.4 (non--proper systems). It may also overlap with subsection 2.7 (degenerate systems), we have not checked this non-degeneracy condition. (5) Kowalevski top. This integrable system was discovered by Sophie Kowalevski and is published in her seminal paper [39]. This integrable system fits into a combination of subsection 2.6 (non-periodic systems) and subsection 2.7 (degenerate systems).
Although it is an integrable system, to our knowledge, it is not known whether one of the components is periodic, i.e., whether it comes from a Hamiltonian S 1 -action. It is generally believed that this is not the case but, as far as we know, there is no proof of this fact. (6) Three wave interaction. This example fits in subsection 2.5 (higher-dimensional systems), and subsection 2.7 (degenerate systems). It may also overlap with subsection 2.3 (hyperbolic systems). The phase-space is 6-dimensional; we have checked that it has many degenerate singularities. The three wave interaction was brought up to our attention by D. Holm.
Examples (3), (4), (5), and (6) may not fit into any of the sections 2.3, 2.4, 2.5, 2.6, 2.7. However, they do fit in the program described in this paper, after the content of these sections has been put together, as indicated in a subsection 2.9.

Semitoric systems.
This section explains the technology, key tools, and recent techniques developed to study the symplectic theory of integrable systems of semitoric type. These methods have lead to a complete classification in terms of five symplectic invariants [48,49]. The overall strategy leading to these results should be applicable in a much more general context. In this section we recall this strategy, pointing out what technology particular to semitoric systems was used and what methods are likely to extend to a more general context. Let G be a Lie group with Lie algebra g whose dual is denoted by g * ; exp : g → G is the exponential map. for all X ∈ g, where X M is the infinitesimal generator vector field on M induced by X, i.e., for every m ∈ M ; here µ, X : M → R is a smooth function obtained by using the duality pairing ·, · between g * and g.
The map µ is called the momentum map of the Hamiltonian G-action. Since i X M ω := ω(X M , ·) is a closed 1-form, every symplectic action is Hamiltonian if H 1 (M, R) = 0. There are many examples of non-Hamiltonian G-actions: take for instance the standard translational action of the circle S 1 on the two-torus T 2 := S 1 × S 1 .
Our intuition on integrable systems has been guided by some remarkable results proven in the 80s by Atiyah, Guillemin-Sternberg and Delzant, in the context of Hamiltonian torus actions. Note that if g is m-dimensional, by choosing a basis of g we may see the momentum map as a map into R m (see Figure 5).

Theorem 2.3 (Atiyah [3]
and Guillemin-Sternberg [31]). If an m-dimensional torus G with Lie algebra g acts on a compact connected symplectic manifold (M, ω) in a Hamiltonian fashion, the image µ(M ) of M under the momentum map µ : M → g * ≃ R m is a convex polytope. Figure 6 shows the images of the momentum map described in Theorem 2.3 for the standard toric actions on complex projective spaces.
Example 2.4. Consider the projective space CP n equipped with a λ multiple of the Fubini-Study form and the standard rotational action of T n (for CP 1 = S 2 , we already drew the momentum map in Figure 5). This action is Hamiltonian, and the momentum map is given by . It follows that the image of µ equals the convex hull in R n of 0 and the scaled canonical vectors λe 1 , . . . , λe n , see Figure 6. Let g Z be the kernel of the exponential map exp : g → G. We denote the isomorphism g/g Z → G also by exp.
Definition 2.5. Let G be an n-dimensional torus, and let ∆ ⊂ g * be an n-dimensional convex polytope. Let F and V be the set of codimension one faces and vertices of ∆, respectively.
i) For each ℓ ∈ F there exists X ℓ ∈ g Z and λ ℓ ∈ R such that the hyperplane which contains ℓ is equal to the set of all ξ ∈ g * such that X ℓ , ξ + λ ℓ = 0. ii) For every v ∈ V, the vectors X ℓ with ℓ ∈ F v form a Z-basis of the integral lattice g Z in g (hence there are such n vectors for each fixed v.) Delzant showed the following stunning result. In the statement of the theorem, the term isomorphism between two symplectic manifolds M and M ′ respectively equipped with G-actions for which the momentum maps are µ and µ ′ , refers to a symplectomorphism 2 ϕ : M → M ′ such that If it is the case, then ϕ intertwines the torus actions. Theorem 2.6 (Delzant [19]). If the dimension n of the torus G is half the dimension of M , then the polytope in the Atiyah-Guillemin-Sternberg Theorem is a Delzant polytope, this polytope determines the isomorphism type of M , and M is a toric variety. Even more, starting from any Delzant polytope in g * , one can construct a symplectic manifold with a Hamiltonian G-action for which its associated polytope is the one we started with.
Remark 2.7. Delzant's theorem says that the polytopes in Figure 6 determine all the information about CP n , the symplectic form and the torus action on it.
It is natural to wonder whether these striking results persist in the case where the torus is replaced by a non-compact group acting in a Hamiltonian fashion on M . The seemingly simplest case happens when the group is R n ; the study of these R n -actions is precisely the goal of the theory of integrable systems. The image of the momentum map of an integrable system is usually not a convex polytope, see Figure 7. In most cases it is not even convex, and it would for instance have an annulus shape. The behavior of an integrable system -which essentially always has complicated singularitiesis much more flexible than that of a torus action. Actions of n-dimensional tori on 2n-dimensional manifolds may be viewed as examples of integrable systems in the sense that the components of the momentum map of the action form an integrable system (we may also say the momentum map itself is an integrable system).

2.2.2.
Integrable systems. Let (M, ω) be a symplectic manifold of dimension 2n. Definition 2.8. An integrable system consists of n functions f 1 , . . . , f n on M whose differentials df 1 , . . . , df n are almost everywhere linearly independent and which Poisson-commute, i.e., {f i , f j } = 0 for all integers 1 i, j n.
We call F := (f 1 , . . . , f n ) : M → R n the momentum map of the integrable system. Often we refer to the map F simply as the integrable system. Example 2.9. (Coupled spin-oscillator) A simple non-toric non-compact example of integrable system is the so called "coupled spin-oscillator" model, which is of fundamental importance in physics, and therein known as the Jaynes-Cummings model. In this case the symplectic manifold is M = S 2 × R 2 , where S 2 is viewed as the unit sphere in R 3 with coordinates (x, y, z), and the second factor R 2 is equipped with coordinates (u, v). The integrable system is given by the smooth maps J := (u 2 + v 2 )/2 + z and H := 1 2 (ux + vy).
Definition 2.10. A singularity 3 of an integrable system is a point m ∈ M for which the tangent mapping T m F : T m M → R n has rank less than n, i.e. df 1 , . . . , df n are linearly dependent one-forms at m. The fiber that contains m is a singular fiber.
The most interesting features of the integrable system are encoded in the singular fibers of the momentum map, some of which are depicted in Figure 8.
at m is zero. We say that m is non-degenerate, or transversally non-degenerate at m if this new integrable system on Σ has a rank zero non-degenerate singularity (in the sense defined above). It follows from the work Williamson in the 1930s that a Cartan subalgebra as in Definition 2.11 has a basis with three type of blocks: (1) one-dimensional ones: elliptic block q = x 2 + ξ 2 , -hyperbolic block q = xξ; (2) two-dimensional one: focus-focus block : q 1 = xη − yξ, q 2 = xξ + yη.
The following notion is due to N.T. Zung. Definition 2.13. Let m be a singularity of an integrable system F : M → R n . If k e , k h , k f respectively denote the number of elliptic, hyperbolic, and focus-focus components of m, we call (k e , k h , k f ) the Williamson type of m.
The following theorem is one of the key tools in the modern theory of integrable systems. It was priorly proven in the analytic case by Vey.
Theorem 2.14 (Eliasson [24]). The non-degenerate singularities of an integrable system F : M → R n are linearizable, i.e., if m ∈ M is a non-degenerate singularity of the completely integrable system where we have the following possibilities for the components q 1 , . . . , q n , each of which is defined on a small neighborhood of (0, . . . , 0) in R n : take any value 2 ≤ j ≤ n − 1 (note that this component appears as "pairs"). (iv) Non-singular component: q j = ξ j , where j may take any value 1 ≤ j ≤ n. Moreover if m does not have any hyperbolic component, then the system of commuting equations {f i , q j } = 0, for all indices i, j, may be replaced by the single equation where ϕ = (x 1 , . . . , x n , ξ 1 , . . . , ξ n ) −1 and g is a diffeomorphism from a small neighborhood of the origin in R n into another such neighborhood, such that g(0, . . . , 0) = (0, . . . , 0).
If the dimension of M is 4 and F has no hyperbolic singularities, we have the following possibilities for the map (q 1 , q 2 ), depending on the rank of the singularity: (1) if m is a singularity of F of rank zero, then q j is one of if m is a singularity of F of rank one, then (iii) q 1 = (x 2 1 + ξ 2 1 )/2 and q 2 = ξ 2 . Definition 2.15. Suppose that (M, ω) is a 4-dimensional symplectic manifold and that F : M → R 2 is an integrable system. A non-degenerate singularity is respectively called elliptic-elliptic, focus--focus, or transversally-elliptic if both components q 1 , q 2 above are of elliptic type, q 1 , q 2 together correspond to a focus-focus component, or one component is of elliptic type and the other component is ξ 1 or ξ 2 , respectively. Similar definitions hold for transversally-hyperbolic, hyperbolic-elliptic and hyperbolic-hyperbolic non-degenerate singularity.

2.2.3.
Semitoric systems: definition and basics. The classification theory we outline next was introduced by the authors in [48,51]. The theory of semitoric integrable systems is at the early development stages, which have already yielded results beyond our initial expectations. Our estimate is that it may take many years of work to push the theory (both at the symplectic and spectral level) to cover integrable systems which are not necessarily semitoric 4 .
We start with the basic notions. For the remaining of Section 2.2 we work only with connected four-dimensional manifolds.
are isomorphic if there exists a symplectomorphism φ : M 1 → M 2 , and a smooth map f : Example 2.19. Perhaps the simplest non-toric, semitoric integrable system on a non-compact manifold is the coupled spin-oscillator model described Example 2.9.
A polygon ∆ ⊂ R 2 is said to be rational, if the normal vectors to its edges span a sublattice of Z 2 .
Theorem 2.20 (Vũ Ngo . c [61]). To a semitoric integrable system F = (J, H) : M → R 2 one can associate a convex rational polygon ∆ ⊂ R 2 , unique up to translations, and modulo the action of the matrix group with elements 1 0 k 1 .
In addition, ∆ = µ(F (M )), where µ is a homeomorphism and µ is integral-affine 5 on a dense open subset.
It is natural to wonder whether the polygon constructed in Theorem 2.20 characterizes the system, as in the Delzant's classification result (Theorem 2.6). We elaborate on this more in subsection 2.5. 4 Here we are including integrable systems on higher dimensional manifolds. Though semitoric systems may naturally be defined in all dimensions, they were originally defined for 4-dimensional manifolds. 5 For the exact technical definition of the notion of integral-affine, see the review [50].

2.2.4.
Uniqueness theorem for semitoric systems. The polygon in Theorem 2.20 does not characterize the system. However, a larger collection of invariants, which includes the polygon, does determine the system. This is the content of the following result.
Theorem 2.21 (Pelayo-Vũ Ngo . c [48]). A semitoric system is characterized, up to isomorphisms, by the following invariants: (1) number of singularities: the number of focus-focus singularities m f ; (2) singular foliation type: a formal Taylor series S(X, Y ) at each focus-focus singularity; (3) polygon invariant: a class of polygons equipped with m f oriented vertical lines (see Figure  10); (4) volume: m f points c 1 , . . . , c m f contained in the interior of the image of the system or, equivalently, a finite set of positive numbers giving the positions of these interior points; (5) twisting-index : a collections of m f integer (one integer between two consecutive nodes ordered according to their J-component).
(1) The number of singularities is an integer m f counting the number of isolated singularities of the integrable system (which correspond precisely to the images of singularities of focusfocus type). We write m f to emphasize that the singularities that m f counts are focus-focus singularities.
(2) The singular foliation type is a collection of m f infinite Taylor series on two variables which classifies symplectically a saturated neighborhood of the singular fiber. 16 (3) The polygon invariant is the equivalence class of a weighted rational convex 6 polygon , which is constructed from the image of the system by performing a very precise "cutting" which appears in the proof of Theorem 2.20 (see Figure 10). Here ∆ is a convex polygonal domain in R 2 , the ℓ j are vertical lines intersecting ∆ and the ǫ j are ±1 signs giving each line ℓ j an orientation. Proof Strategy for Theorem 2.21. The proof strategy for the uniqueness part is simple. We start with two integrable systems Step 1 (Construction of invariants). Construct local symplectic invariants which are analogues of the invariants (1)-(5) above.
Step 2 (Construction of local symplectomorphisms). Argue that one can reduce to a case where the images F 1 (M 1 ) and F 2 (M 2 ) are equal. Prove that this common image can be covered by open sets Ω α , above each of which F 1 and F 2 are symplectically interwined, i.e., above each Ω α one finds a part of the system F 1 and a part of the system F 2 that are symplectically the same because they have the same symplectic invariants. At this stage one has a collection of local symplectomorphisms which cover all the pieces of the manifolds M 1 and M 2 .
Step 3 (Symplectic gluing). Use symplectic gluing of these local pieces to prove a uniqueness and an existence type theorem. The last step is to glue together these local symplectomorphisms, thus constructing a global isomorphism φ : 6 generalizing the Delzant polygon and which may be viewed as a bifurcation diagram We expect that the same proof strategy applies to each of the upcoming subsections, so we do not repeat it therein. Instead, each of the following subsections is focused on explaining what the difficulties are, mainly in Step 1 (but we also comment on potential difficulties in Step 2 and Step 3).
Remark 2.23. The "analytic-combinatorial" data in Theorem 2.21 completely describes the moduli space of semitoric systems, as shown in the following reconstruction theorem. These invariants are depicted in Figure 11 for the case of the coupled spin-oscillator. Figure 11. The coupled spin-oscillator example is a semitoric system with one focus-focus point whose image is (1, 0). The invariants are depicted on the right hand-side. In this case m f = 1 and the class of generalized polygons for this system consists of two polygons. S 1 denotes the Taylor series invariant (the linear terms of which were computed in [51]), k 1 = 0 is the twisting index invariant (which is trivial because there is only one focus-focus value in this case), and h = 1 is the volume invariant, which determines the position of the focus-focus value node.  [49]). Given the following ingredients: (1) number of singularities: an integer number 0 ≤ m f < ∞; (2) singular foliation type: an m f -tuple of Taylor series the notion of Delzant semitoric polygon). We denote the representative Theorem 4.7 ]). Then there exists a symplectic 4-manifold (M, ω) and a semitoric integrable system F = (J, H) on M whose symplectic invariants of Theorem 2.21 given in that order coincide with (1)-(5) above.
Proof Strategy for Theorem 2.24. The proof strategy for the existence theorem is simple.
Step 1 (Start with abstract collection of items "corresponding" to invariants). Let be a representative of the polygon invariant with all ǫ j 's equal to +1. The strategy is to use a symplectic gluing procedure introduced in [49] in order to obtain a semitoric system by constructing a suitable singular torus fibration above ∆ ⊂ R 2 . For j = 1, . . . , m f , let c j ∈ R 2 be the point with coordinates where π 2 : R 2 → R is the projection on the second factor. Because of the assumption on h j , all points c j lie in the interior of the polygon ∆. We call these points nodes. We denote by ℓ + j the vertical half-line through c j pointing upwards. We call these half-lines cuts. Now construct a "convenient" (this is technical, we leave it to the papers) covering of the polygon ∆.
Step 2 (Local construction piece by piece). Construct a "semitoric system" over the part of the polygon away from the sets in the covering that contain the cuts ℓ + j ; then we attach to this "semitoric system" the focus-focus fibrations, i.e., the models for the systems in a small neighborhood of the nodes. Continue to glue the local models in a small neighborhood of the cuts. The "semitoric system" is given by a proper toric map only in the preimage of the polygon away from the cuts.
Step 3 (Recovering smoothness). Modify the system to recover the smoothness of the system (this is very delicate) and observe that the invariants of the system are precisely the items (1)-(5) we started with.
2.3. Hyperbolic systems. This branch of the program has been explored by Zhilinskií from the point of view of physics. However it is not rigorously formalized as a mathematical theory. Zhilinskií's explorations are intriguing, and should give a lot of insight when working towards a mathematical theory. There are also a few related results by Vũ Ngo . c, Dullin Zung, and Bolsinov, among others. The one-dimensional case (i.e. when the phase space is a symplectic 2-dimensional manifold) was solved by Toulet, Molino, and Dufour.
Next we elaborate on the difficulties to construct the symplectic invariants for this type of integrable systems. Once this is done, the goal is to achieve a classification (existence and uniqueness) result in terms of these invariants.
We propose a general strategy to prove such classification immediately after the statements of Theorem 2.21 and Theorem 2.24 in the previous sections, so we do not repeat it here. Instead, we explain what the expected difficulties are, as far as we can see, to construct the invariants in the  In what follows, we refer to the previous subsection for the basic terminology and results (and to [50, and the references therein for more detailed explanations).
2.3.1. Assumptions. We say that a non-degenerate singularity of an integrable system is of hyperbolic type if the Willamson type of m (see Definition 2.13) has k h = 0. In other words, a non-degenerate singularity has hyperbolic type if it has some hyperbolic component. Suppose that F : M → R 2 is an integrable system which fails to be semitoric only because it has some singularities containing components of hyperbolic type. Concretely, this means that (M, ω) is a connected At this time we are not aware of examples of systems satisfying all assumptions (a)-(e) but we do not see any a priori reason why they would not exist. In fact, it is through the study of invariants that we propose in this section that one should be able to construct large classes of them, if in fact they exist (this was the case in our article [49]), or disprove their existence. Nevertheless there are physically relevant examples like the two-body problem or the Lagrange Top that satisfy several of these assumptions, see Section 2.1.

2.3.2.
Expected difficulties. The expected difficulties are: Figure 13. Hyperbolic critical fiber with Z 2 symmetry.
(i) Polygon invariant. There is no obvious candidate for the polygon invariant.
The original construction of the polygon invariant from the image of F (M ) is by cutting along the vertical lines that pass through the focus-focus values of F . At an ideological level, the construction works because focus-focus singularities are isolated and the behavior of the integrable system around a focus-focus singularity (while very complicated and having infinitely many symplectic invariants) can be controlled in a small semiglobal neighborhood U of the focus-focus singularity. In particular, we know that focus-focus singular fibers of semitoric systems are connected. However, singular fibers over hyperbolic points may not be connected, and the number of connected components changes when passing through it. This fiber connectivity is essential in the construction of the polygon invariant for semitoric systems 7 .
The image of U corresponds then to a vertical band in the image F (M ), which contains the vertical line passing through the focus-focus singularity. This is, however, not the case with hyperbolic singularities, where the singularity may, for instance, come as a curve of singularities (e.g., suppose that the singularity has a hyperbolic component and a regular component) and it is not clear what the image of this curve is inside of M , and its relation to the integral affine structure induced by the the system.
Understanding what the image of the hyperbolic singularity and its surrounding singularities are is a must if one wants to construct any kind of polygon invariant. Once this is clarified, it is not clear whether an invariant as simple as the polygon invariant may be constructed. It is more likely that a new more complicated invariant is going to replace the polygon invariant, probably a foliation type invariant: a collection of 2-dimensional leaves with some structure. Foliations, and foliation type invariants, are important in many contexts, see for instance Bolsinov-Fomenko The original construction of the twisting-index invariant encodes the "difference" between consecutive (according to the ordered J-values) normalizations of the integrable system F , which is given by a 2 by 2 integer matrix with all entries constant, with the exception of one entry which, in general, is a non-zero integer. This non-zero integer is precisely the twisting-index. This matrix is very much linked to the nature of the focus-focus singularities.
Because we are in dimension four, and hence focus-focus singularities are isolated, it is quite possible that we have to keep track of two objects: a twisting-invariant for the focus--focus singularities (which is the same as the original one) and an invariant which keeps track of the normalization around the hyperbolic singularities. These objects should take into account the foliation type invariant replacing the polygon invariant in (i). Figure 14. Symplectic gluing over polygon preimage. The point where the axes cross is a focus-focus value. D i is a simply connected set containing this point. The outer part of A δ i is a subset of R 2 containing only regular values and on which we have a toric momentum map. The u-shaped region separates this region from a region of R 2 which contains a focus-focus value over which the system is not toric. These regions must be glued using a careful control of the smoothness and symplectic geometry of the system around the u-shaped region. Symplectic gluing arguments of this nature lie at the heart of the classification proofs (Theorem 2.21 and Theorem 2.24).
(iii) Symplectic gluing . Once the invariants are defined, the symplectic gluing constructions outlined in the proof strategies of Theorems 2.21 and 2.24 need to be adapted. Indeed, the proofs need to incorporate the fact that in Step 1 we no longer have an invariant as simple as a polygon and hence the "convenient covering" of the polygon mentioned therein ( [48,49]) has to be changed. This convenient covering was such that it isolated the focus--focus singularities of the integrable system, so that the preimage of the system over such element of the covering contained at most a focus-focus singularity, which made the gluing construction feasible. Now we are presented with the fact that the image of the hyperbolic singularities in F (M ) can be a curve and the preimage contains many non-isolated singularities. How does one choose the elements U α covering ∆, so that the preimages F −1 (U α ) can be glued symplectically and smoothly? In the case of semitoric systems outlined in the theorems, we were essentially gluing along regular points, but it is not clear if we can do this now: it seems quite likely that one needs to glue pieces taking care of the gluing along curves of singularities which, of course, must match. One could probably use a covering of the curve of hyperbolic critical values, but we are presented with two problems: (1) Curves of the systems: we do not know whether one can assume that for two given integrable systems, the aforementioned curves corresponding to each system, coincide. This was essential in the proof for semitoric systems. Moreover, does this information need to be included as an invariant? Solving this problem will involve studying the singularities of the integral affine structure of the system. (2) Semiglobal Normal Forms: we do not know how to glue because we do not have semiglobal normal forms around the singularities, these need to be proven first. This poses a real challenge.
2.4. Non-proper systems. The authors are working together with T.S. Ratiu on this part; the fiber connectivity is proved in [46] and the construction of the polygon invariant will be given in an upcoming article. (i) Fiber connectivity and the polygon invariant. In contrast with the standard semitoric setting, the fibers of F are not necessarily connected (see our paper [46] for the known results in this direction). In order to guarantee that the fibers of F are connected, in general, one can assume a non-vertical tangency condition on the bifurcation diagram of the system, up to smooth deformations (see [46,Theorem 1 and Theorem 2]). This sufficient condition seems to us to be very close to necessary, but we do not know at this time if it can be weakened (there are many integrable systems with disconnected fibers that nevertheless do not satisfy this vertical tangency condition). Now, assuming that the fibers of F are connected, it should be possible to construct a generalization of the polygon invariant, which will likely not be as simple as a family of polygons, but it will still be a rigid object that can be drawn in R 2 . We have investigated this case, and although we have no proofs yet, we think that analogue of the polygon invariant will be given in this case by a collection of regions of R 2 (under some discrete group action), where each region is bounded by the graphs, epigraphs, or hypographs of two piecewise linear functions.
Note that, before constructing the polygon invariant, it is necessary to have a complete understanding of the image F (M ) of the system. It is precisely because we can now describe this image concretely ([46, Theorem 3 and Theorem 4]) as bounded by the hypograph and epigraph of two lower/upper semicontinuous functions, that we can guess that the polygon invariant is going to be bounded by the graphs, epigraphs, or hypographs of two piecewise linear functions (it should not be too difficult to do the transition from F (M ) to the polygon invariant, the proof should be very close to the original construction for semitoric systems).
If, on the other hand, the map F is proper but does not have connected fibers, then we do not know whether it is possible to construct a reasonable analogue of the polygon invariant. The original construction of the polygon invariant uses in an essential way the connectivity of the fibers of F . Even in the construction of the polytope corresponding to a toric integrable system F (i.e., F is the momentum map of a Hamiltonian 2-torus action), the connectivity of the fibers of F is intimately connected to the structure of F (M ) itself as a convex polytope. This is the context of the famous convexity theorem by Atiyah and Guillemin-Sternberg (see Theorem 2.3), which says that F (M ) is, moreover, equal to the convex hull of the images of the fixed points of the action. (ii) Other invariants. Until the analogue of the polygon invariant in (i) is constructed we do not see a way to anticipate what the other invariant which depends on the polygon (twisting--index invariant) will be. The cardinality, singular foliation type, and volume invariant should remain the same as for semitoric systems. In the case that F has connected fibers we expect that the construction of these other invariants should not be too different from the case of semitoric systems. (iii) Symplectic gluing . The difficulties here will depend on the construction of the polygon.
As in subsection 2.3, the constructions outlined in the proof strategies of Theorem 2.21 and Theorem 2.24 need to be adapted to deal with the fact that in Step 1 we no longer have an invariant as simple as a polygon, but rather a more complicated, but still "rigid" invariant which is close to a family of polygons inside of R 2 . The method of finding a suitable covering of a representative polygon of this family, as outlined in Step 1, should work. Of course, the covering here will be more complicated, but the fact that there are no hyperbolic singularities should make this step technically challenging but within reach.

Assumptions.
When M is a 6-dimensional symplectic manifold, we think that the natural definition of a semitoric system should be the following. (e) F has only non-degenerate singularities and non-hyperbolic singularities. 24 The following is the classification problem of the section.
Classification Problem 2.28. The problem has three parts.
(I) Give explicit constructions of "analogues" of the symplectic invariants in Theorem 2. 21 (which refer to semitoric systems only), for integrable systems satisfying assumptions (a)-(e). (II) Define an abstract list of all such possible invariants which occur in (I).
2.5.2. Expected difficulties. The difficulties we expect are as follows.
(i) Fiber connectivity and the polygon invariant. A major difficulty is to define a 3--dimensional polytope invariant, because the singularities of F are now more complicated. One can have singularities whose Williamson type (Definition 2.13) is (k e , k h , k f ) = (1, 0, 1) or (0, 0, 1), for example. The singularities with the Williamson type (0, 0, 1) have a focus-focus component, but are no longer isolated as was the case for semitoric systems. This is already a major difference with the semitoric case in dimension four, and we believe it is a very substantial problem.

The fundamental issues here are:
what is the stratification of the singular affine structure of the image F (M ), and can one describe concretely the singular stratum S corresponding to the focus-focus singularities? The structure of S ought to be essential if one wants to cut F (M ) and, from it, construct a 3-dimensional polytope. At the moment, this is an outstanding and, we believe, a difficult problem. In [62], Wacheux proved that S does not contain any embedded circle : it has to connect the stratum of transversally elliptic singularities. This opens the way to the construction of the polytope invariant.
Note that in the case when the system F : M → R 3 is the momentum map for a Hamiltonian action of a 3-torus, the polygon invariant is precisely the image F (M ), which was shown to be equal to the convex hull of the images of the fixed points of the action by the Atiyah-Guillemin-Sternberg theorem (see Theorem 2.3 and Figure 6). In this case, the image of F (M ) is not "cut". (ii) Other invariants. Once the polytope invariant is constructed, we expect difficulties, both conceptual and technical, in defining the other invariants with the possible exception of the singular foliation type invariant (which should be close to a twisted direct sum of the singular foliation type invariant in the semitoric case, with the additional component(s) corresponding to the other singularities).
To arrive at a classification, both in the six-dimensional, as well as the higher dimensional cases, it is necessary to understand the semiglobal classification of singularities. For instance, what are the semiglobal symplectic invariants near a singularity that has a focus--focus and an elliptic component? This should not be too difficult, but as far as we know, it has not been done. See [50, Section 5.2.3] for a summary and references of a semiglobal analysis of focus-focus singularities in the 4-dimensional case, which was previously carried out by the second author.
The twisting index invariant in this case should compare normalizations of the system near singularities which have focus-focus components. An expected difficulty will come from the fact that in dimension 6 a focus-focus component is always coupled with either an elliptic or a regular component. Thus these singularities come in families, which makes it difficult to isolate them or order them in some meaningful way (in the case of semitoric systems, the focus-focus singularities were ordered according to the value of the J-component). Altogether, how to handle these issues is going to depend on how the polytope invariant is constructed. (iii) Dimensions greater than 6. In part a) above, we only considered integrable systems on 6-dimensional manifolds. After this case is dealt with, the analogous problem in any dimension would be to consider an integrable system given by smooth functions . . , f n−1 generates a vector field with a 2π-periodic flow. We shall call these systems semitoric systems with n degrees of freedom (what we have been calling "semitoric systems" are therefore the same as "semitoric systems with 2 degrees of freedom"). (iv) Symplectic gluing . A discussion analogous to that in subsections 2.3 and 2.4 applies. The construction of the covering of the 3-dimensional polytope by sets U α will depend heavily on how the polytope is constructed and what the structure of the set of singularities inside of the polytope is.

Expected difficulties.
(i) Fiber connectivity and the polygon invariant. Similarly to subsection 2.3, one should be able to define, from the image F (M ) of the integrable system F , a geometric invariant which resembles the polytope invariant. The fibers of F may not be connected, which is essential in the construction of the polygon invariant for semitoric systems, as we have mentioned already. However, we know that these fibers are connected, for instance when the fibers of J or the fibers of H are connected ([46, Theorem 3.7]). The fibers of F can also be guaranteed to be connected by assuming that the bifurcation diagram of F has no vertical tangencies, up to smooth deformations.
(ii) Polygon invariant, lack of momentum map, and other invariants. The fiber connectivity for F is only a requirement in the construction of the polygon invariant; it allows us to describe F (M ) precisely, in terms of hypographs/epigraphs of lower/upper semicontinuous functions. However, in the construction of the polygon invariant it is essential that J is the momentum map for a Hamiltonian S 1 -action on M . Indeed, this gives a "vertical invariance" to the image of F (M ), which allows one to cut F (M ) along vertical lines going through the focus-focus values of F , and hence to construct a polygon from F (M ). If we drop this periodicity assumption on J, the construction of the polygon needs to be rethought; we believe this to be a challenging task (at least a priori). The work of Leung and Symington [42] is directly relevant to these problems, in case the fibers are connected. (iii) Symplectic gluing . A discussion analogous to that of subsections 2.3 and 2.4 applies.

2.7.
Degenerate systems. This part of the program, as was the case with subsection 2.3, is wide open. We do not know how realistic it is to achieve a classification for systems that have degenerate singularities. Degenerate singularities appear in many systems, so the study of degenerate systems should be considered an important case. Some important systems, such as the three wave interaction (see subsection 2.1), are known to have degenerate singularities.
We believe that the work of Garay [26,27], Sevenheck and van Straten [54,55] on Lagrangian rigidity, and Zung's work on analytic Birkhoff normal forms [64], make very substantial advances towards the understanding of degenerate singularities.
The one-dimensional C ∞ smooth case was done by de Verdière. It involves Gauss-Manin and techniques by Malgrange related to Milnor fibers, and usual singularity theory. (d) J is a proper map; Note that unlike in previous subsections, there is no item (e) above. If there are degenerate singularities there is no a priori reason to single out hyperbolic singularities. We have the following outstanding preliminary problem, which is self-contained.
Core Problem 2. 30. Determine what kinds of singularities can be allowed for a system satisfying (a)-(d), so that one can assure that the system has connected fibers.
The following is the classification problem of the section.  [48,49,51,46]), holds only for non-degenerate singularities.
However, degenerate singularities occur often in systems arising from physics, so it is important to develop a theory which covers systems with degenerate singularities. A first step is to investigate whether any part of Eliasson's theorem can be rescued in the degenerate case (and the hope is small of getting a result which is not much weaker than Eliasson's original theorem).
There has been, however, some work for degenerate singularities in [50,Section 4.2.2], and it is conceivable that the classification theorems we proved in [48,49] could be extended to cover the degenerate case by reformulating these results. This is definitely a difficult, but very important, part of the classification program announced in this paper. This part stands somewhat independently of the rest of the program and is the one for which we have fewer ideas or expectations of where it can lead.
We do not know how having degenerate singularities will affect the construction of the symplectic invariants. First one needs to classify some subclass of degenerate singularities (cf. Arnold-Gusein-Zade-Varchenko's books [2]) We expect this part of the program to be of great interest for the physics community.
2.8. Self-contained topological and geometric questions. There are many self-contained questions in this program that are of interest on their own right. The following gives a small sample. In addition, answering some of them is a prerequisite, as far as we can tell, for the construction of the polygon invariant in the previous sections. This is the case for Problem 2.33 and Problem 2.34 below; therefore it can be a good strategy to start by solving these problems in the context of the corresponding section (i.e., hyperbolic systems, non-proper systems, non-periodic systems etc.). Because the Fomenko school has done work on topological aspects of integrable systems, we would like to point out how the following questions, though topological, differ from those they answered. The Fomenko school gave an extensive treatment of the topological properties of integrable systems viewed as fibrations over R n (we refer to [46] for references); this work has exerted a major influence on several parts of modern mathematics, including our own work. The goal and results of the symplectic bifurcation theory introduced in [46] are fundamentally different from those of the Fomenko school. For instance, from the Fomenko school view-point, if the fibration given by an integrable system has disconnected fibers, one may try to slightly modify the system, to consider a topologically equivalent system with connected fibers ("topologicaly equivalent" in the sense that sets of connected components of the fibers of both systems coincide).
In general, such a change does not respect the symplectic structure, so the modified system can be topologically equivalent to the original one but its symplectic geometry and dynamical behavior can be different. Since we are interested in quantization and spectral theory of the integrable system, such an operation defeats our purpose: the symplectic structure cannot be altered.
In fact, the fibrations of two integrable systems may be globally the same at a topological level, while their symplectic invariants are inequivalent [48]. From the point of view of symplectic and spectral theory the two systems have little to do with each other: there is no global isomorphism which preserves the integrable system with the symplectic structure, which in turn changes the way quantization and spectral theory are carried out.
The development of the symplectic bifurcation theory in [46] requires the introduction of methods to construct Morse-Bott functions which, from the point of view of symplectic geometry, behave well near the singularities of integrable systems. These methods use Eliasson's theorems on linearization of non-degenerate singularities as well as the symplectic topology of integrable systems, topics that have been developed by many.
Theorem 2.32 (Pelayo-Ratiu-Vũ Ngo . c). Suppose that (M, ω) is a compact connected symplectic four-manifold. Let F : M → R 2 be a non-degenerate integrable system without hyperbolic singularities. Denote by Σ F the bifurcation set of F . Assume that there exists a diffeomorphism g : F (M ) → R 2 onto its image such that g(Σ F ) does not have vertical tangencies (see Figure 15). Then F has connected fibers. Figure 16. Description of the image of an integrable system. The image is first transformed to remove vertical tangencies, and then it can be described as a region bounded by two graphs.
Core Problem 2.33. Prove Theorem 2.32 under the assumptions of each of the previous sections of the paper. For instance, in higher dimensions: suppose that (M, ω) is a compact connected symplectic 2n-manifold. Let F : M → R n be a non-degenerate integrable system without hyperbolic singularities. Denote by Σ F the bifurcation set of F . Assume that there exists a diffeomorphism g : F (M ) → R n onto its image such that g(Σ F ) does not have "vertical tangencies". Does F always have connected fibers? The first problem is to determine precisely what we mean by "vertical tangency" for a domain in R n when n > 2.
Core Problem 2.34. There is also an analogue of Theorem 2.32 for non-compact manifolds in [46], so we can also pose the problem above for non-compact manifolds.
Problem 2.35. Find formulas, or recipes, to compute the symplectic invariants constructed in the previous section. For instance, is there a general strategy to compute some terms of the Taylor series invariant (i.e., the singularity type invariant)? For instance, if the system is toric, then the image of the momentum map, which is the simplest case of a polygon invariant, can be computed 29 in terms of the images of the fixed points of the action. Can one write a recipe to compute it in each of the cases described in the previous sections of this paper?
Problem 2.36. Are there Duistermaat-Heckman formulas suitable for the polygon invariant?
The following result describes the image of an integrable system.
Theorem 2.37 (Pelayo-Ratiu-Vũ Ngo . c). Suppose that (M, ω) is a compact connected symplectic four-manifold. Let F : M → R 2 be a non-degenerate integrable system without hyperbolic singularities. Denote by Σ F the bifurcation set of F . Assume that there exists a diffeomorphism g : F (M ) → R 2 onto its image such that g(Σ F ) does not have vertical tangencies (see Figure 15). Then: (1) the image F (M ) is contractible and the bifurcation set can be described as  Core Problem 2.38. Prove Theorem 2.37 under the assumptions of each of the previous sections of the paper. For instance, in higher dimensions: suppose that (M, ω) is a compact connected symplectic 2n-manifold. Let F : M → R n be a non-degenerate integrable system without hyperbolic singularities. Denote by Σ F the bifurcation set of F . Assume that there exists a diffeomorphism g : F (M ) → R n onto its image such that g(Σ F ) does not have "vertical tangencies". Does F always have connected fibers?
Core Problem 2.39. There is also an analogue of Theorem 2.37 for non-compact manifolds in [46], so we can also pose the problem above for non-compact manifolds. Figure 18. A disk, a disk with a conic point, a disk with two conic points, or a compact convex polygon.
As a matter of fact, some cases in which vertical tangencies are present, can also be addressed. In the following result a neighborhood of a conic point is, by definition, locally diffeomorphic to some proper cone C α,β . (b) there exists a diffeomorphism g such that g(F (M )) is either a disk, a disk with a conic point, a disk with two conic points, or a compact convex polygon (see Figure 18).
Then the fibers of F are connected. We next pose three, in our opinion, important moduli problems which are independent of the classification program.
Problem 2.42. Describe the topology of the moduli space of symplectic toric systems (using Theorem 2.6). The moduli of semitoric systems of a given dimension may be viewed inside of the set all of integrable systems in that dimension -can one tell how "big" it is?
As far as we know there are no definite results regarding Problem 2.42. There is collection of notes about this problem by A.R. Pires and the first author with several preliminary observations. The following should be a much more challenging problem. The following problem is vaguely defined, but probably worth exploring.
Problem 2.44. Can one define a suitable notion of "limit", such that an integrable system with connected fibers in dimension four could be obtained as a "limit" of a sequence of semitoric systems (at least for some type of non-semitoric integrable systems)?
The following problem has the same flavor as Problem 2.44, and is motivated by an informal, famous question of Anatole Katok [36] in low dimensions (meaning 2 for maps, 3 for flows): "Is every conservative dynamical system that has zero topological entropy a limit of integrable systems?" Problem 2.45. Consider the set of integrable systems: Similarly one can define I toric and I semitoric , by requiring, respectively, the integrable systems in I to be toric or semitoric.
(a) What is the closure of I ⊂ C ∞ (M, R 2 ), in some adequate topology to be defined? (b) One can ask the same question for I toric and I semitoric . The same problem can be formulated for the systems in Sections 2.3, 2.4, 2.5, 2.6, and 2.7.
We conclude by raising some broad problems about semitoric systems. 2.9. Collaborative work. There are three levels we suggest to go from the material described in the previous sections to a symplectic classification of completely integrable Hamiltonian systems.
Level 1 (Putting the classifications together ). This level should involve no major technical or conceptual difficulties. We see it as a serious book-keeping and organizational challenge: the effort will be into coordinating the results obtained by the authors that have respectively worked in subsections 2.4, 2.5, 2.6, 2.7, 2.8, and putting their classification results together. As we mentioned in the introduction, we believe that, for the most part, each of these sections deals with difficulties that are somewhat unrelated. Precisely because of the likely "independence", the general answer in 2.9 should be a "direct sum" of the answers provided in the subsections 2.3, 2.4, and 2.6.
At this level, putting the classifications obtained in subsections 2.3, 2.4, 2.7 together, we expect to arrive at a result of the following type: Goal: Symplectic Classification Theorem for Semitoric Systems. An integrable system where each f 1 , . . . , f n−1 generates a vector field with a 2π-periodic flow is characterized -in the sense of uniqueness and existence as in Theorems 2.21 and 2.24 -up to symplectic isomorphisms preserving the system, by a list of symplectic invariants which are explicitly defined (in subsections 2.3, 2.4, and 2.6).
Level 2 (Complexity two semitoric systems). We call complexity of an integrable system the number of non-periodic components (so a semitoric system has complexity one). This level holds the major conceptual and technical jump of the program and will rely on techniques analogous to those developed in subsection 2.5. We believe that going from the four to the six-dimensional case already contains the essential difficulties and those need to be clarified first. The type of theorem that we expect to have is the following. Level 3 (Increasing the complexity of a semitoric system to arrive to a general integrable systems). Here there is another really substantial step: to proceed inductively using the theorem in level 2 above, to obtain a general classification theorem of the following form.
Goal: Symplectic Classification Theorem for Integrable Systems. An integrable system (or at least a large class of integrable systems) is characterized -in the sense of uniqueness and existence as in Theorems 2.21 and 2.24 -up to symplectic isomorphisms preserving the system, by a list of symplectic invariants which are explicit (possibly defined recursively in terms of invariants of lower complexity systems).
We believe that it is possible to achieve a classification in terms of explicit invariants, but these are likely to be defined inductively in terms of invariants of lower complexity systems. Even if one deals with the quite rigid case of integrable systems coming from torus actions, the goal seems still very challenging, and at this point we cannot provide much information about it. However, once the the six-dimensional case in Section 2.5 is understood, one should be able to compare the invariants therein with the invariants of semitoric systems in Theorems 2.21, 2.24, and conjecture an approach to go from six-dimensional manifolds to eight-dimensional etc. In this sense, the following problem is a (possible) preliminary step towards the goal above.
Problem 2.48. Describe to the greatest possible extent the symplectic invariants built in Section 2.5 for semitoric systems on 6-dimensional manifolds, in terms of symplectic invariants of semitoric systems of 4-dimensional symplectic manifolds (in Theorems 2.21, 2.24).

Spectral theory of integrable systems
Our original motivation to develop a symplectic theory of semitoric systems in the papers above was that, at least from our point of view, it is a prerequisite for studying the quantization and spectral theory of integrable systems.
3.1. Inverse spectral theory. Let M be a symplectic manifold of dimension 2n and H a family of Hilbert spaces, > 0, associated to M in such a way that linear operators on H admit principal symbols in C ∞ (M ).
For example, such a family of Hilbert spaces can be obtained by geometric quantization with complex polarization ( [14]), or by hand like in quantum mechanics ( [51]), or if M is a cotangent bundle minus the zero section some completion of the Schwartz space. The simplest example is L 2 (X) when M = T * X, a case in which there is no dependance on . acting on H which commute (i.e., [f i, ,f j, ] = 0 for all i, j = 1, . . . , n, for all ) and whose principal symbols f 1 , . . . , f n form an integrable system on M . Instead of a single spectrum one considers the joint spectrum off 1 , . . . ,f n (assumed discrete), i.e. the sequence of points as goes to 0, given by: Question 3.2. Does the joint spectrum determine the integrable system f 1 , . . . , f n up to symplectic isomorphism?
Example 3.3. (Quantum coupled spin-oscillator) Recall the classical coupled spin-oscillator model in Example 2.9. Now we are interested in its quantum version. For any > 0 such that 2 = (n + 1), for some non-negative integer n ∈ N, let H denote the standard n + 1-dimensional Hilbert space quantizing the sphere S 2 (see [51] for details). Consider on R 2 coordinates (u, v) and on S 2 coordinates (x, y, z). The quantization of x, y, z is given by restricting, respectively, the following operators to H: where The unbounded operators which we callĴ andĤ respectively, on the Hilbert space are self-adjoint and commute. The spectrum ofĴ is discrete and consists of eigenvalues in ( 1−n 2 + N). This is natural way to quantize the coupled-spin oscillator. The joint spectrum for a fixed value of is depicted in Figure 19. The question of whether one can recover a system from its spectrum is fundamental in the theory of integrable systems. The symplectic classification in terms of concrete invariants described in the previous section serves as a tool to quantize semitoric systems.  Figure 20 with a so called "metaplectic correction". Introducing a metaplectic correction refers to twisting the prequantum bundle (or its powers) by a half-form bundle. The metaplectic correction allows to obtain an easier control of the subprincipal terms in the semiclassical limit, see our article [14].
In Delzant's theory, the image of the momentum map, for a toric integrable action, completely determines the system. In the quantum theory, the image of the momentum map is replaced by the joint spectrum. Can one determine the system from the joint spectrum? In this vast program, one can ask the following question, less ambitious but still quite spectacular: does the semiclassical (i.e., for a sequence of small values of ) joint spectrum determine the underlying classical system? This was one of the primary goals of the program described in this paper: Spectral Goal of Program for Integrable Systems: Show that from the semiclassical joint spectrum of a quantum integrable system one can recover the integrable system of principal symbols, up to symplectic isomorphisms (a program proposal to arrive at this result is outlined in subsection 3.2).
We believe that it is possible to make definite progress towards achieving this goal. In our article [50] we formulated this goal as a conjecture in the case of semitoric systems ([50, Conjecture 9.1]).
As it is evident from the strategy we describe below to achieve this goal, the achievement of the symplectic program described Section 2 will to a large extent determine the success of this part of the program. At this stage, we can formulate a general conjecture. Note that the notion of quantum system is not universally agreed upon and the conjecture may also hold knowing the semiclassical joint spectrum up to some order. Therefore, the following statement should be understood with a certain flexibility depending on the context. For instance, in the case of toric systems, a very complete answer can be given, which has as corollary a quantization theorem.
Conjecture 3.4 (Inverse Spectral Conjecture for Integrable Systems). From the semiclassical joint spectrum of a quantum integrable system one can completely recover the integrable system given by the principal symbols, up to symplectic isomorphisms.
We have proved this conjecture in the case of toric systems (Theorem 3.5) in our recent paper [14]. In the case that M is a 2-dimensional cotangent bundle andf 1 is a pseudodifferential operator, the conjecture was proved by Vũ Ngo . c [60]. The case when M is a 2-dimensional symplectic manifold andf 1 is a Toeplitz operator is a field of current interest. In this direction, a local quantum normal form near elliptic singularities has been obtained by Le Floch [41]. Note that in 2-dimensions, a singularity cannot have focus-focus components.
3.2. An approach to Conjecture 3.4 for semitoric systems. We begin with the following preliminary remarks.
(i) Toeplitz operators (on any manifold) versus pseudodifferential operators (on cotangent bundles). A first difficulty is to define precisely the space of quantum operators that are considered. We believe that the natural setting to formulate this question is semiclassical Toeplitz operators. Nevertheless, the approach below is based on ideas coming from the theory of pseudodifferential operators. Pseudodifferential quantization implies that the classical phase space be a cotangent bundle M = T * X, which is a serious restriction from the point of view of symplectic geometry. However, it is well known that a microlocal level the algebra of pseudodifferential operators is microlocally equivalent to the algebra of Toeplitz operators, and one can reasonably expect that the techniques that were developed for pseudodifferential operators have their analogues for Toeplitz operators. (ii) Semitoric systems versus general integrable systems. The strategy to prove the inverse spectral conjecture for general integrable systems should be quite close to the semitoric case that we describe below. The only reason why one can do this for semitoric systems is because we have a complete symplectic theory in which a semitoric system is determined by some computable symplectic invariants. With our approach, having these invariants is a must, and for the moment we only have them for semitoric systems -in the previous section we described a program to build symplectic invariants for more general systems. (iii) Local and semilocal inverse spectral formulas. The challenge of proving a global result such as the inverse spectral conjecture passes by first proving local and semilocal inverse spectral results for singularities (focus-focus, elliptic, and hyperbolic), involving Bohr-Sommerfeld rules. This can be avoided in the case of toric systems, where a global symplectic description of the system exists in terms of a single invariant: the image of the system, which is a polytope.
For each of the subsections 2.4, 2.5, 2.6, and 2.7 tentatively leading to a symplectic classification of integrable systems for the particular type of systems covered therein, one can try to push the strategy of proof outlined below to prove the inverse spectral conjecture for that type of systems.
Unlike for Theorems 2.21 and 2.24, we have not yet carried out the proof sketched below for semitoric systems, so the arguments are not as precise (and of course not guaranteed to work). However, we believe that this outline can develop into an efficient proof strategy.
Proof Strategy for Conjecture 3.4. We describe the strategy for the case of semitoric systems only. Even though the setting to answer this question, in general, is the theory Toeplitz operators, let's illustrate how to do this whenĴ,Ĥ are pseudodifferential operators and the phase space (M, ω) is a cotangent bundle. The general strategy we present should be adaptable to the general case of Toeplitz operators but this should be technically more complicated and we will comment on this throughout the proof.
Part A (Compute symplectic invariants from spectrum). To recover the number of singularities invariants, the volume invariant, and the polygon invariant we equivalently recover: the image F (M ) together with the J-coordinates of the focus-focus values, and the affine structure of F (M ).
Step 1 (Recovering F (M ) and J-values of focus-focus points). We are able to do this because we have the local density of states, i.e., the number of points of the spectrum in a small ball of radius δ multiplied by 2 and divided by the volume of the ball (we take the ball of radius δ , where δ ∈ (0, 1); we are choosing δ , but we could choose anything that goes to 0 smoothly and slowly.) The key point is that for pseudodifferential operators we have Bohr-Sommerfeld rules and, as a consequence of these rules, we have that, as → 0, at a regular point the density function tends to a smooth function of the center of a ball. At an elliptic point, the density function tends to a piecewise smooth but discontinuous function. At a focus-focus point, the density function tends to infinity. To make this rigorous we have to work as in Vũ Ngo . c's recent paper [60], and look at the local density of states to find F (M ), the set of critical values, and the set of non-values. This method works, in general, for all integrable systems, not necessarily semitoric. For semitoric systems it may be possible to even get an easier answer by applying the method line by line for each line J = constant to cover the entire plane and then analyze the eigenvalue spacing. The singular Bohr-Sommerfeld rules would give the required result, in caseĴ andĤ were pseudodifferential operators. Of course they are not, since the phase space is not usually a cotangent bundle. HoweverĴ,Ĥ are semiclassical Toeplitz operators and it is known that the algebra of Toeplitz operators is "microlocally equivalent" to the algebra of pseudodifferential operators. This works whenĴ,Ĥ are (self-adjoint) Toeplitz operators. What we have to do is to prove the Bohr--Sommerfeld rules for two degrees of freedom for general Toeplitz operators. The one-dimensional case has been done recently by Charles (for Toeplitz operators; for pseudodifferential operators it is well-known). Because this is completely microlocal, there is no expected difficulty in proving this (at least in theory. But precise formulas are not easy to get; for instance, there is no Maslov bundle in the Toeplitz case).
Step 2 (Recovering the affine structure of F (M )). The polygon invariant and the singular foliation type invariant (a Taylor series on two variables) can both be defined in terms of the behavior of the affine structure of F (M ) at its boundary. The affine structure is itself defined by action integrals. Therefore, it should be possible to recover these invariants as soon as one can recover the action integrals.
Because Bohr-Sommerfeld rules give the spectrum in terms of action integrals, they can be used, conversely, to compute actions from the spectrum (see Figure 23). cut Figure 23. The joint spectrum of the quantum coupled spin-oscillator has an integral affine structure on the regular values, which can be extended to the boundaries. Except along a vertical cut through the focus-focus critical value, one can develop this affine structure such that the joint eigenvalues become elements of Z 2 . The top picture is the joint spectrum of the quantum spherical pendulum. At the bottom, we have developed the joint eigenvalues into a regular lattice. The number of eigenvalues in each vertical line is the same in both pictures. The convex hull of the resulting set converges, as → 0, to the semitoric polygon invariant.
More precisely, the Bohr-Sommerfeld rules should give the affine structure in the regular part (=set of regular values). The singular part of the affine structure is then obtained by taking limits of the regular part. Let (γ 1 (c), γ 2 (c)) be a basis of H 1 (F −1 (c), Z) depending smoothly on c. If c 0 ∈ F (M ) is a regular point, then there exists a ball B around c 0 such that and α is any 1-form on some neighborhood of the fiber Λ c containing c such that dα = ω. The action integrals in (1) have a logarithmic divergence at the focus-focus singularities.
We claim that if one knows then one should know f ( Z 2 ) mod O( 2 ). Of course, one needs to recover the whole function, not just the discrete set, so this implies taking limits. A crucial remark is that joint eigenvalues as → 0 can approach any arbitrary value in F (M ); from this we should be able to recover f mod O( 2 ).
In particular, the spectrum should completely determine f 0 , and thus its inverse, thereby giving the action integrals.
Step 3 (Recovering the singular foliation type invariant). Let's try to recover the Taylor series S ∞ at focus-focus singularities. When we know the action-integrals, in order to recover the Taylor series, we use the formula: We get A(c) = (A 1 (c), A 2 (c)) = A 1 (c) + i A 2 (c) from the spectrum, which is the original data we are given. The only difficulty is thatc has to be expressed in Eliasson's normal coordinates, so we need the complete change of coordinates.
We know that A(c) has a logarithmic behavior and it is unique. From the theory we know that there existsc = g(c), with g a local diffeomorphism of (R 2 , c j ) such that We know c, we don't knowc is C ∞ onc. Now, how many such g are there? We claim that there exists a unique function g (unique in the sense of its Taylor series being unique) such that A(c) ±c ln(c) is smooth.
The strategy to recover the last invariant, the twisting-index invariant, should become more clear once the proof strategy above is carried out. At the moment we have not made progress on this.
Part B (Use the symplectic theory in [48,49] and Part A to recover integrable system). Once we have computed the symplectic invariants, we can symplectically recover the integrable system by Theorem 2.21 and Theorem 2.24.
This concludes the comments on the general strategy to prove the conjecture.
3.3. Conjecture 3.4 in the case of toric systems. In the case of toric systems the only invariant is the polytope. We have recently proved [14] Conjecture 3.4 for toric integrable systems. Recall from Section 1.2 the geometric quantization procedure introduced by Kostant and Souriau. Let (M, ω) be a compact symplectic manifold. Given a prequantum bundle L → M , that is a Hermitian line bundle with curvature 1 i ω and a complex structure j compatible with ω, one defines the quantum space as the space H k := H 0 (M, L k ) of holomorphic sections of tensor powers L k of L, where = 1/k (when there is such a bundle, we say that M is prequantizable). Associated to such a quantization there is an algebra T (M, L, j) of Toeplitz operators (T k : H k → H k ) k∈N * . Two Toeplitz operators (T k ) k∈N * and (S k ) k∈N * commute if T k and S k commute for every k. Figure 24. Sequence of images of the spectra of a quantum toric integrable system as the spectral parameter goes to 0. In the Hausdorff limit, corresponding to = 0, the spectra converge to a polytope. The spectra lie in a plane, so they correspond to a four-dimensional integrable system with two degrees of freedom.
Theorem 3.5 (Charles-Pelayo-Vũ Ngo . c). Let (M, ω, µ : M → R n ) be a toric integrable system equipped with a prequantum bundle L and a compatible complex structure j. Let T 1 , . . . , T n be commuting Toeplitz operators of T (M, L, j) whose principal symbols are the components of µ. In other words, one can recover the classical system from the limit of the joint spectrum.
The proof of this theorem combines geometric and algebraic methods, following the framework introduced in Duistermaat-Pelayo [21], with semiclassical analytic techniques inspired by recent work of Charles [12,13].
3.4. Existence of quantization. The quantum system is the data for us; right now we are not interested in the question of whether one can find, or how to find, such operators -that is a different question. However, in the case of toric systems, we could address this question, as a consequence of a global normal form theorem we proved.
Our conclusion was that if (M, ω) is a prequantizable compact connected symplectic manifold of dimension 2n equipped with a toric integrable system given by a momentum map µ := (µ 1 , . . . , µ n ) : M → R n of a Hamiltonian n-torus action on M , then for any Toeplitz quantization (H k ) k∈N * of M there exists a quantization of µ, i.e., a set of n Toeplitz commuting operators whose principal symbols are µ 1 , . . . , µ n . In the general analytic case, the obstruction has been found by Garay.

Other projects
4.1. Algorithmic version of spectral goal and proof strategy. Given the semiclassical joint spectrum of a quantum system, we would like to have an algorithm that implements the strategy of proof of the inverse spectral conjecture to first compute the symplectic invariants from the spectrum (Step 1) and then constructs the system from the invariants (Step 2). In the case of toric systems (for which the inverse spectral conjecture is a theorem), the only invariant is (by Delzant's theorem) the polytope and the recipe to recover this polytope from the spectrum is already described in Theorem 3.5: take the Hausdorff limit of the sequence of spectra; this gives Step 1.
It is not completely clear how to implement an algorithm to carry out Step 2, which is Delzant's construction (an explicit construction by symplectic reduction on some complex space C N ). From the point of view of applications, it may be enough to extract information from the polytope about M and give a list of its properties (number of fixed points, symplectic volume, etc.) rather than try to give a construction of the toric system, which may be of little use. The key point is that all the information about the toric system is in the polytope, in the same way that all the information about a semitoric system is in the list of our five symplectic invariants.

4.2.
Type theory and Univalent Foundations. The formalization of mathematics in type theory, and in particular in the Coq system, in the language and setting of Voevodsky's Univalent Foundations [58], and using Voevodsky's libraries is a project that one could view as a modern version of Bourbaki. The Univalent Foundations are constructive and new mathematics is likely to emerge from the necessity to formulate various components of the integrable systems theory constructively.
The goal is to formalize the theory of finite dimensional integrable systems. We are suggesting that this is a promising way to think about the organization of the theory of integrable systems, which should facilitate progress in the field in the future. In particular, one expects to create novel mathematics by means of this formalization. In short, the idea of the Voevodsky's program is to develop mathematics within the world of homotopy types. Voevodsky's program uses deep insights and tools from homotopy theory. The project ofÁ. Pelayo, M.A. Warren, and V. Voevodsky has the following steps: i) formalize some basic p-adic analysis (a preliminary file dealing with this case may be found in [47]); ii) to formalize p-adic integrable systems. In fact first one needs to rigorously give the "correct" formulation of p-adic integrable systems. This is a very new and interesting topic on its own (separately from type theory and Coq), as a problem about integrable systems. In fact, it is non-trivial to describe a useful notion of p-adic integrable system. iii) continue to formalize general integrable systems. We give three reasons for which we believe one should be interested in this formalization: a) Creation of Algorithms: because everything is done in a system that has good algorithmic properties (assuming that Voevodsky's conjecture regarding the constructive status of the Univalence Axiom holds), one should be able to extract good algorithms from the proofs. This can be extraordinarily useful, because we may potentially be able to get explicit information when otherwise we would only have existence results; b) Numerical information about experiments ("Applied Spectral Goal"): this section should have applications to Section 4.1. The construction of algorithms in a), should help with the following outstanding problem, of which we discussed an exact version earlier in this paper: given numerical spectral data about a quantum integrable system (coming from an experiment), extract (optimizing the algorithms obtained as proof terms in (a)) an algorithm to reconstruct the classical integrable system. c) New theoretical insights: by virtue of the fact that the project involves formalizing mathematics in the world of homotopy, the notions formalized can be seen in many cases as "up to homotopy versions" of the original notions, and therefore are more general. As for example derived algebraic geometry suggests, it is often mathematically more useful to work with mathematical structures up to homotopy than it is to work with their strict analogues. The hope is therefore that this project will result in the creation of novel mathematics and not just the formalization of known mathematics.

Perspective of this paper and closing remarks
Not only the field of integrable systems is vast on its own right, but also it is rare to find an area of mathematics in which they do not occupy a privileged position: from symplectic and algebraic geometry to representation theory, microlocal analysis, spectral theory, partial differential equations, numerical analysis, etc. 5.1. Symplectic theory: perspective and connections. Although this paper is not a survey, to help the interested gain some perspective, we have included in the references a small selection of the important contributions to integrable Hamiltonian systems which are most directly relevant to this paper. We also try to briefly outline a few connections to other works, without claiming any completeness.
5.1.1. Glimpse of references. For the reader's aide we have cited: (i) the essential works on which the ideas proposed in this paper are based; (ii) works in progress on open problems outlined in the paper; (iii) a few works that give perspective on the current paper and the subjects on which it touches. See our articles [50,14], and the references therein, for further information on related works and other motivations. The first few sections of [50] provide a brief review of several fundamentals results of the theory of finite dimensional integrable systems due to Arnold, Atiyah, Carathéodory, Darboux, Delzant, Duistermaat, Dufour, Eliasson, Guillemin, Liouville, Mineur, Molino, Sternberg, Toulet and N. T. Zung, among others, which have been key ingredients in the most recent developments outlined in the present paper.
Finally, we would like to point out that in the particular case that M is a cotangent bundle, fundamental work on integrable systems was done by Hitchin [33]. The work of Lax [40] has also been extremely influential. 5.1.2. Atiyah, Delzant, Guillemin, Kirwan, and Sternberg theory. As indicated in Section 2.2.1, the authors' intuition on integrable systems has been guided by a remarkable theory mostly developed in the 70s and 80s by Kostant, Atiyah [3], Guillemin-Sternberg [31], Delzant [19] and Kirwan [37] in the context of Hamiltonian torus actions.
The construction of the convex polytope ∆ in the Atiyah and Guillemin-Sternberg Theorem (Theorem 2.3) was the motivation and driving force behind the construction of the "polygon invariant": item (3) in Theorem 2.21. Indeed, Hamiltonian n-torus actions on symplectic 2n-manifolds form an important class of completely integrable systems, with well-behaved singularities, usually referred to as toric systems.
Delzant built on Theorem 2.3 to give a classification of toric systems. His theorem, which in dimension 4 is a particular case of Theorems 2.21, 2.24, was a main motivation for these results.

5.1.3.
Kolmogorov-Arnold-Moser theory. One motivation to study integrable systems comes from Kolmogorov-Arnold-Moser (KAM) theory. Since integrable systems are "solvable" in a precise sense, one expects to find valuable information about the behavior of dynamical systems that are obtained by small perturbations of them, and then the powerful KAM theory comes into play to deal with the properties of the perturbations (persistence of quasi-periodic motions).
An important limitation of KAM techniques is that they require the unperturbed integrable system to be written in action-angle coordinates. Having global action-angle coordinates is very exceptional; thus, most applications of KAM are limited to neighborhoods of regular Lagrangian tori.
Meanwhile, it has become clear that singularities of integrable systems may enter KAM theory : see for instance [65] where the various Kolmogorov conditions are deduced from the existence of hyperbolic or focus-focus singularities. This opens the way to a global KAM theory. Recent results of [7] show how to construct Cantor sets of invariant tori near focus-focus singularities, and thus define Hamiltonian monodromy for the perturbed system. It is tempting to apply these ideas to semitoric systems, and see which parts of the integral affine structure (and thus, which symplectic invariants of Section 2.2) survive the perturbation.
If this can be achieved, the quantum counterpart becomes fascinating: the reminiscence of the integral affine structure should have an effect on the spectrum of the perturbed quantum Hamiltonian. What makes it mysterious is that one doesn't have any joint spectrum in the sense of Section 3 for the perturbed Hamiltonian. This issue might perhaps be overcome by switching to non-selfadjoint perturbations as in [34].

Fomenko School theory.
In the 1980s and 1990s, the Fomenko school developed a far reaching Morse theory for regular energy surfaces of integrable systems, which is related, and serves as an inspiration, for several of the problems (for instance Problems 2.33, 2.34, 2.38, 2.39) discussed in Section 2.8.
One of our motivations to study integrable systems comes from the theory of singularities of fibrations ξ : M → R n that they developed [5]. They gave an extensive treatment of the topological properties of integrable systems viewed as fibrations over R n , and their work has exerted a decisive influence on the problems presented in Section 2.8 of the present paper. 5.1.5. Singular affine structures. A semitoric system as in Definition 2.16 gives rise to a torus fibration with singularities, and its base space becomes endowed with a singular integral affine structure. Singular affine structures are of key importance in various parts of symplectic topology, mirror symmetry, and algebraic geometry -for example they play a central role in the work of Gross and Siebert [29,30], Kontsevich and Soibelman [38], and Symington [56].
In the semitoric case, the singular affine structure is encoded in the first three invariants/ingredients in Theorems 2.21, 2.24: (1) number of singularities: the number of focus-focus singularities m f ; (2) singular foliation type: a formal Taylor series S(X, Y ) at each focus-focus singularity; (3) polygon invariant: a class of polygons equipped with m f oriented vertical lines (see Figure  10  5.1.7. Mechanics. As described in Sections 1 and 2, an important motivation for the present paper has been to provide a unified framework to study integrable systems. Many integrable systems of fundamental importance arise in classical mechanics: for instance, the coupled spin-oscillator (also called the Jaynes-Cummings model), the spherical pendulum, the two-body problem, the Lagrange top, the three wave interaction (see Section 2.1 for more details about these systems). In fact, more generally, integrable systems can also be found in the theory of geometric phases, rigid body systems, elasticity theory and plasma physics, and have been extensively studied by many authors. Integrable systems appear in numerous contexts, which we have not explored in the present paper. N. Reshetikhin's survey lectures on aspects of classical and quantum integrability [52] are an excellent source for related important developments; the lectures, among other contributions, outline the relation between solvable models in statistical mechanics, and classical and quantum integrable spin chains, which are closely related to the spin-oscillator example in Section 2.1.

5.2.
Spectral theory: perspective and connections. This part of the program outlined in this paper combines geometric ideas in the complex and symplectic settings with microlocal analytic methods dealing with semi-classical pseudodifferential and Toeplitz operators.

5.2.1.
Isospectrality for integrable systems. The central theme surrounding the "Spectral Goal for Integrable Systems" (Sections 1 and 3) is the isospectrality question for quantum integrable systems. In other words, in any finite dimension: does the semiclassical joint spectrum of a quantum toric integrable system, given by a sequence of commuting Toeplitz operators acting on quantum Hilbert spaces, determines the classical system given by the symplectic manifold and Poisson commuting functions, up to symplectic isomorphisms? This type of symplectic isospectral problem belongs to the realm of classical questions in inverse spectral theory and microlocal analysis, going back to pioneer works of Colin de Verdière on cotangent bundles [18,17] and Guillemin-Sternberg [32] in the 1970s and 1980s.

5.2.2.
Microlocal analysis of integrable systems. The notion of a quantum integrable system (Definition 3.1), as a maximal set of commuting quantum observables, dates back to the early quantum mechanics, to the works of Bohr, Sommerfeld and Einstein [22].
However, the most basic results in the symplectic theory of classical integrable systems like Darboux's theorem or action-angle variables could not be used in Schrödinger's quantum setting at that time because they make use of the analysis of differential (or pseudodifferential) operators in phase space, known now as microlocal analysis, which was developed only in the 1960s.
Effective models in quantum mechanics often require a compact phase space, and thus cannot be treated using pseudodifferential calculus. For instance the natural classical limit of a quantum spin is a symplectic sphere. The study of quantum action-angle variables in the case of compact symplectic manifolds treated in this paper was started by Charles [12], using the theory of Toeplitz operators.

5.2.3.
Kac's question. The Spectral Goal (Sections 1 and 3) fits in the framework of "isospectral questions": what is the relation between two operators that have the same spectrum? The question of isospectrality has been considered by many authors in different contexts, and may be traced back to a more general question of H. Weyl. In the case of the Riemannian Laplacian, the question is perhaps most famous thanks to Kac's article [35] (who attributes the question to S. Bochner), which also popularized the phrase: "can one hear the shape of a drum?".

Conclusion.
In this paper we have suggested some ideas to work towards developing the symplectic and spectral theory of finite dimensional integrable Hamiltonian systems. The paper has laid out many open problems. This paper has the authors' articles [48,49] as reference points, and it describes several somewhat disjoint paths in which one can make contributions to the topics at hand by dealing with the difficulties one at a time. We expect that each of these disjoint paths will involve substantial challenges. However, we believe that the overall strategy stands a reasonable chance of success, by what we mean that it should lead to improving our knowledge of the symplectic and spectral aspects of integrable systems significantly.
We expect that further information about the program described in this paper will be available in the authors' websites listed in this paper.