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Álvaro Pelayo

Publications and source records attributed to Álvaro Pelayo.

At least 19 recordsLinked to original sources
arXiv2026

The contrast between real and $p$-adic integrability

One of the most intriguing questions in $p$-adic symplectic geometry is to what extent it captures phenomena outside of real symplectic geometry. Since symplectic geometry has its origins in classical mechanics, and since integrable systems are essentially physical systems with many conserved quantities, it is natural to ask if considering the same system with coefficients in $\mathbb{R}$ and $\mathbb{Q}_p$ respectively, will lead us to see the same or different behaviors. Our first goal in this paper is prove that the $p$-adic and real local normal forms of the same system have no relation, that is, any combination of real and $p$-adic local normal forms is possible. This implies that $\mathbb{R}$ and $\mathbb{Q}_p$ capture fundamentally different information about the system, despite the fact that they both contain $\mathbb{Q}$ as a dense subset and that experiments can only measure rational quantities. Hence a $p$-adic version of symplectic geometry and Hamiltonian mechanics will be different from its real counterpart, at least from the point of view of integrable systems. Our second goal in this paper is to introduce techniques to compute these normal forms. We establish these results combining Galois theory of $p$-adic fields with analytic estimates.

math.SG↗
arXiv2026

Darboux's Theorem in $p$-adic symplectic geometry

We prove a non Archimedean Darboux's Theorem: any two symplectic forms on a $p$-adic analytic manifold are locally isomorphic. Understanding local problems such as the existence of flows or the normalization of singularities in the theory of integrable systems, is essential to understand the physics behind these systems. Our result tells us that the phase space defined by a $p$-adic manifold is locally standard, allowing us to concentrate on the equations defining the dynamics rather than on the space itself. Our proof uses a non Archimedean version of Moser's Path Method to push one symplectic form onto another one by a flow. A central technical contribution of the paper is the proof that the flow is given by a power series with non zero radius of convergence, which requires geometric analytic estimates and does not follow from algebraic considerations. The main difficulty of the paper is analytic: an ODE will not have a long-time solution even in a neighborhood of a fixed point but we are able to find a condition for a long-time solution to exist, which allows us to tweak the ODE in Moser's Path Method so that it satifies the condition. As a global application, we derive a classification of second-countable $p$-adic analytic symplectic manifolds in terms of $p$-adic volume, which generalizes a classical theorem of J-P. Serre.

math.SG↗
arXiv2026

$p$-adic symplectic geometry of integrable systems and Weierstrass-Williamson theory II

This paper is a sequel to arXiv:2501.14444, in which we shall give proofs of several results stated in arXiv:2501.14444 (Theorems D--L) which, for brevity and clarity, we postponed to this sequel paper. These results were the following: for any prime number $p$, first we show that every $2$-by-$2$ symmetric matrix with coefficients in $\mathbb{Q}_p$ can be reduced to a canonical form, and we give the exact numbers of families of normal forms with one parameter and of isolated normal forms, which depend on $p$. Then we make the same analysis for $4$-by-$4$ matrices. We also prove that, for higher size, the number of families of normal forms of matrices, even in the non-degenerate case, grows almost exponentially with the size. The paper can be read independently of arXiv:2501.14444 as we recall the statements of arXiv:2501.14444 that we shall prove here. The statements and proofs of the present paper are of an algebraic and arithmetical nature, and rely mainly on Galois theory of $p$-adic extension fields.

math.SG↗
arXiv2026

$p$-adic symplectic geometry of integrable systems and Weierstrass-Williamson theory I

The local symplectic theory of integrable systems is fundamental to understand their global theory, as well as the behavior near singularities of fundamental models from classical and quantum mechanics which are known to be integrable, such as the Jaynes-Cummings model and the coupled angular momenta. We establish the foundations of the local symplectic geometry of $p$-adic integrable systems on $4$-dimensional $p$-adic analytic symplectic manifolds, by classifiying all their possible local models. In order to do this we develop a new approach, of independent interest, to the theory of Weierstrass and Williamson concerning the diagonalization of real matrices by real symplectic matrices. We show that this approach can be generalized to $p$-adic matrices, leading to a classification of real $(2n)$-by-$(2n)$ matrices and of $p$-adic $2$-by-$2$ and $4$-by-$4$ matrix normal forms, including, up to dimension $4$, the classification in the degenerate case, for which the literature is limited even in the real case. A combination of these results and the Hardy-Ramanujan formula shows that both the number of $p$-adic matrix normal forms and the number of local models of $p$-adic integrable systems grow almost exponentially with their dimensions, in strong contrast with the real case. These results fit in a program, proposed a decade ago by Voevodsky, Warren and the second author, to develop a $p$-adic theory of integrable systems with the goal of later implementing it using proof assistants. In the present paper we shall prove all of the statements concerning integrable systems; their proofs rely on the $p$-adic analog of the Weierstrass and Williamson theory of matrices. The proofs concerning this new $p$-adic theory of matrices are of an algebraic and arithmetical nature, and we give them in the sequel paper, part II.

math.SG↗
arXiv2025

Group actions on $p$-adic symplectic manifolds

Let $p$ be a prime number. We introduce symplectic actions of $p$-adic analytic Lie groups on $p$-adic symplectic manifolds. Then we show that any $p$-adic symplectic action $G\times(M,ω)\to(M,ω)$ has a momentum map $μ:M\to\mathfrak{g}^*$, and that a proper $p$-adic symplectic action is Hamiltonian if and only if every orbit is isotropic. We conclude by defining $p$-adic symplectic toric manifolds, by analogy with the real case.

math.SG↗
arXiv2025

$p$-adic angular momentum coupling in symplectic geometry

The coupled angular momentum is an integrable system with two degrees of freedom which is fundamental in physics and the theory of integrable systems. It is obtained by coupling two angular momenta. We construct a $p$-adic analog of this system for any prime number $p$ and describe its symplectic normal forms at the critical points. This analog has a rich singularity theory with up to thirteen non-equivalent symplectic normal forms, which stands in contrast with the real case where there are exactly three normal forms.

math.SG↗
arXiv2025

Moduli spaces of Delzant polytopes and symplectic toric manifolds

This paper introduces modern geometric combinatorial technology from the theory of triangulations in order to derive results in toric symplectic geometry. In the main part of the paper we prove a number of properties of the space $\mathcal{D}(n)$ of $n$-dimensional Delzant polytopes. Two highlights are the construction of examples showing that, in contrast with the classical work of Oda in dimension $2$, no classification of combinatorially minimal Delzant polytopes can be expected in dimension $3$ or higher, and a proof that the space of $n$-dimensional Delzant polytopes is path-connected. Our proof of the latter is based on the fact that every rational fan can be refined to a unimodular fan, which is a standard technique used for resolution of singularities of toric varieties. In the last part of the paper, using the Delzant correspondence, these results allow us to answer several open questions concerning the moduli space $\mathcal{M}(n)$ of symplectic toric manifolds of dimension $2n$, since this space is isometric to the space of Delzant polytopes. Our results imply that no classification of minimal models of symplectic toric manifolds is plausible in dimension $6$ or higher, which answers in the negative a long-standing folklore question originating in Oda's work (1978).

math.SG↗
arXiv2025

Rigidity and flexibility in $p$-adic symplectic geometry

Let $n\ge 2$ be an integer and let $p$ be a prime number. We prove that the analog of Gromov's non-squeezing theorem does not hold for $p$-adic embeddings: for any $p$-adic absolute value $R$, the entire $p$-adic space $(\mathbb{Q}_p)^{2n}$ is symplectomorphic to the $p$-adic cylinder $\mathrm{Z}_p^{2n}(R)$ of radius $R$, showing a degree of flexibility which stands in contrast with the real case. However, some rigidity remains: we prove that the $p$-adic affine analog of Gromov's result still holds. We will also show that in the non-linear situation, if the $p$-adic embeddings are equivariant with respect to a torus action, then non-squeezing holds, which generalizes a recent result by Figalli, Palmer and the second author. This allows us to introduce equivariant $p$-adic analytic symplectic capacities, of which the $p$-adic equivariant Gromov width is an example.

math.SG↗
arXiv2024

The $p$-adic Jaynes-Cummings model in symplectic geometry

The notion of classical $p$-adic integrable system on a $p$-adic symplectic manifold was proposed by Voevodsky, Warren and the second author a decade ago in analogy with the real case. In the present paper we introduce and study, from the viewpoint of symplectic geometry and topology, the basic properties of the $p$-adic version of the classical Jaynes-Cummings model. The Jaynes-Cummings model is a fundamental example of integrable system going back to the work of Jaynes and Cummings in the 1960s, and which applies to many physical situations, for instance in quantum optics and quantum information theory. Several of our results depend on the value of $p$: the structure of the model depends on the class of the prime $p$ modulo $4$ and $p=2$ requires special treatment.

math.SG↗
arXiv2024

Ewald's Conjecture and integer points in algebraic and symplectic toric geometry

We solve several open problems concerning integer points of polytopes arising in symplectic and algebraic geometry. In this direction we give the first proof of a broad case of Ewald's Conjecture (1988) concerning symmetric integral points of monotone lattice polytopes in arbitrary dimension. We also include an asymptotic quantitative study of the set of points appearing in Ewald's Conjecture. Then we relate this work to the problem of displaceability of orbits in symplectic toric geometry. We conclude with a proof for the $2$-dimensional case, and for a number of cases in higher dimensions, of Nill's Conjecture (2009), which is a generalization of Ewald's conjecture to smooth lattice polytopes. Along the way the paper introduces two new classes of polytopes which arise naturally in the study of Ewald's Conjecture and symplectic displaceability: neat polytopes, which are related to Oda's Conjecture, and deeply monotone polytopes.

math.CO↗
arXiv2023

Semitoric systems of non-simple type

Within integrable systems, the class of so called "semitoric" integrable systems in dimension four has attracted a lot of attention in recent years, especially since fundamental examples from classical and quantum mechanics have been identified as semitoric by different groups of researchers. Several of these examples, however, show a particular trait not included in the original theory, that is, the presence of multiple (i.e. two or more) rank zero isolated singularities in the same energy-momentum level sets. Systems with this property are called non-simple. This paper extends the original theory of Pelayo and Vũ Ngoc to non-simple systems.

math.SG↗
arXiv2023

The structure of monotone blow-ups in symplectic toric geometry and a question of McDuff

Monotone polytopes, also known as smooth reflexive polytopes, are the polytopes associated to monotone symplectic toric manifolds and Gorenstein Fano toric varieties. We first show that the only monotone polytopes admitting blow-ups at vertices are the simplex and the result of a codimension-two blow-up in it (this is the polyhedral version of a result of Bonavero from 2002). Then we show that the $n$-simplex admits disjoint blow-ups at faces if and only if the faces are disjoint and have dimensions adding up to $n-1$ or $n-2$. These results answer a question posed by Dusa McDuff in 2011.

math.SG↗
arXiv2023

The random Arnold Conjecture: a new probabilistic Conley-Zehnder Theory for symplectic maps

We take the first steps to develop Conley-Zehnder Theory, as conjectured by Arnold, in the world of probability. As far as we know, this paper provides the first probabilistic theorems about the density of fixed points of symplectic twist maps in dimensions greater than $2$. In particular we will show that, when the analogue conditions to classical Conley-Zehnder theory hold, quasiperiodic symplectic twist maps have infinitely many fixed points almost surely. The paper contains also a number of theorems which go well beyond the quasiperiodic case.

math.DS↗
arXiv2022

Vu Ngoc's Conjecture on focus-focus singular fibers with multiple pinched points

We classify, up to fiberwise symplectomorphisms, a saturated neighborhood of a singular fiber of an integrable system (which is proper onto its image and has connected fibers) containing $k > 1$ focus-focus critical points. Our result shows that there is a one-to-one correspondence between such neighborhoods and $k$ formal power series, up to a $(\mathbb{Z}_2 \times D_k)$-action, where $D_k$ is the $k$-th dihedral group. The $k$ formal power series determine the dynamical behavior of the Hamiltonian vector fields associated to the components of the momentum map on the symplectic manifold $(M,ω)$ near the singular fiber containing the $k$ focus-focus critical points. This proves a conjecture of San Vu Ngoc from 2002.

math.SG↗
arXiv2020

Symplectic invariants of semitoric systems and the inverse problem for quantum systems

Simple semitoric systems were classified about ten years ago in terms of a collection of invariants, essentially given by a convex polygon with some marked points corresponding to focus-focus singularities. Each marked point is endowed with labels which are symplectic invariants of the system. We will review the construction of these invariants, and explain how they have been generalized or applied in different contexts. One of these applications concerns quantum integrable systems and the corresponding inverse problem, which asks how much information of the associated classical system can be found in the spectrum. An approach to this problem has been to try to compute invariants in the spectrum. We will explain how this has been recently achieved for some of the invariants of semitoric systems, and discuss an open question in this direction.

math.SG↗
arXiv2018

Symplectic geometry and spectral properties of classical and quantum coupled angular momenta

We give a detailed study of the symplectic geometry of a family of integrable systems obtained by coupling two angular momenta in a non trivial way. These systems depend on a parameter t $\in$ [0, 1] and exhibit different behaviors according to its value. For a certain range of values, the system is semitoric, and we compute some of its symplectic invariants. Even though these invariants have been known for almost a decade, this is to our knowledge the first example of their computation in the case of a non-toric semitoric system on a compact manifold (the only invariant of toric systems is the image of the momentum map). In the second part of the paper we quantize this system, compute its joint spectrum, and describe how to use this joint spectrum to recover information about the symplectic invariants.

math-ph↗
arXiv2018

Classifying Toric and Semitoric Fans by Lifting Equations from ${\rm SL}_2({\mathbb Z})$

We present an algebraic method to study four-dimensional toric varieties by lifting matrix equations from the special linear group ${\rm SL}_2({\mathbb Z})$ to its preimage in the universal cover of ${\rm SL}_2({\mathbb R})$. With this method we recover the classification of two-dimensional toric fans, and obtain a description of their semitoric analogue. As an application to symplectic geometry of Hamiltonian systems, we give a concise proof of the connectivity of the moduli space of toric integrable systems in dimension four, recovering a known result, and extend it to the case of semitoric integrable systems with a fixed number of focus-focus points and which are in the same twisting index class. In particular, we show that any semitoric system with precisely one focus-focus singular point can be continuously deformed into a system in the same isomorphism class as the Jaynes-Cummings model from optics.

math.SG↗