arXiv subjects

Joseph Palmer

Publications and source records attributed to Joseph Palmer.

16 recordsLinked to original sources

arXiv2025

Integrable systems with symmetries: toric, semitoric, and beyond

This article presents an overview of the theory of integrable systems with symmetries, focusing on toric systems, semitoric systems, and their classifications via decorated polygons. We discuss certain one-parameter families of integrable systems called semitoric families, and explain how deforming systems through controlled bifurcations in such families (and their generalizations) can be used to construct explicit semitoric systems with prescribed invariants. The first part of the paper serves as a quick introduction to integrable systems for newcomers to the field, such as graduate students, while the majority of the exposition surveys recent developments and technical details that will be of interest to experts. It closes with a look at future directions, including hypersemitoric systems and complexity one integrable systems.

math.SG↗

arXiv2024

Families of four-dimensional integrable systems with $S^1$-symmetries

The aim of this paper is to give new insights about families of integrable systems lifting a Hamiltonian $S^1$-space. Specifically, we study one-parameter families $(M^4,ω,F_t=(J,H_t))_{0 \leq t \leq 1}$ of systems with a fixed Hamiltonian $S^1$-space $(M,ω,J)$ and which are semitoric for certain values of the parameter $t$, with a focus on such families in which one singular point undergoes a Hamiltonian-Hopf bifurcation (also called nodal trade in the context of semitoric systems, and more generally almost toric fibrations). Beyond semitoric systems, we also study families containing hypersemitoric systems, and we investigate the local theory of a nodal trade.\\Building on and generalizing the ideas of a previous paper, we show how such families can be used to find explicit semitoric systems with certain desired invariants (bundled in the marked semitoric polygon). This allows us to make progress on the semitoric minimal model program by understanding and coming up with explicit systems for each strictly minimal type (i.e., those not admitting any toric or semitoric type blowdown). In order to obtain these systems, we develop strategies for constructing and understanding explicit examples of semitoric (and hypersemitoric) systems in general. One strategy we make use of is to start from a well-understood system (such as a toric system) and to explicitly induce Hamiltonian-Hopf bifurcations to produce focus-focus singular points. This is an expanded version of the technique used in the aforementioned previous paper, in order to apply it to semitoric systems which include non-trivial isotropy spheres in the underlying $S^1$-space (i.e., $\mathbb{Z}_k$-spheres), which occurs in several of the strictly minimal systems.\\In particular, we give an explicit one-parameter family of systems on $\mathbb{CP}^2$ which transitions between being of toric type, semitoric type, and hypersemitoric type depending on the value of the parameter. We study this system at each stage, computing the marked semitoric polygon of the semitoric system and determining several properties of the hypersemitoric system, including the existence of a unique flap and two parabolic orbits. Furthermore, we study the transitions between these stages.\\We also come up with new explicit semitoric systems on all Hirzebruch surfaces which, together with the previous systems and the systems already contained in the literature, gives an explicit model for every type of strictly minimal system. Moreover, we show how to obtain every strictly minimal system by applying sequences of alternating toric type blowups and blowdowns to simple explicit systems. In particular, we obtain that every strictly minimal semitoric polygon can be obtained from a semitoric system which is part of a family $(M,ω,F_t=(J,H_t))$ which is semitoric for all but a finite number of values of $t$, called a semitoric family.

math.SG↗

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

Packing Densities of Delzant and Semitoric Polygons

Exploiting the relationship between 4-dimensional toric and semitoric integrable systems with Delzant and semitoric polygons, respectively, we develop techniques to compute certain equivariant packing densities and equivariant capacities of these systems by working exclusively with the polygons. This expands on results of Pelayo and Pelayo-Schmidt. We compute the densities of several important examples and we also use our techniques to solve the equivariant semitoric perfect packing problem, i.e., we list all semitoric polygons for which the associated semitoric system admits an equivariant packing which fills all but a set of measure zero of the manifold. This paper also serves as a concise and accessible introduction to Delzant and semitoric polygons in dimension four.

math.SG↗

arXiv2023

The twisting index in semitoric systems

Semitoric integrable systems were symplectically classified by Pelayo and Vu Ngoc in 2009-2011 in terms of five invariants. Four of these invariants were already well-understood prior to the classification, but the fifth invariant, the so-called twisting index invariant, came as a surprise. Intuitively, the twisting index encodes how the structure in a neighborhood of a focus-focus fiber compares to the large-scale structure of the semitoric system and it was originally defined by comparing certain momentum maps. In the first half of the present paper, we produce several new formulations of the twisting index which give rise to dynamical, geometric, and topological interpretations. More specifically, we describe it in terms of differences of action variables, Taylor series, and homology cycles. In the second half of the paper, we compute the twisting index invariant of a specific family of systems with two focus-focus singular points (the so-called generalized coupled angular momenta), which is the first time that the twisting index has been computed for a system with more than one focus-focus point. Moreover, we also compute the terms of the Taylor series invariant up to second order. Since the other invariants of this family were already computed, this becomes the third family of semitoric systems for which all invariants are known, after the coupled spin oscillators and the coupled angular momenta.

math.SG↗

arXiv2022

Semitoric families

Semitoric systems are a type of four-dimensional integrable system for which one of the integrals generates a global $S^1$-action; these systems were classified by Pelayo and Vu Ngoc in terms of five symplectic invariants. We introduce and study semitoric families, which are one-parameter families of integrable systems with a fixed $S^1$-action that are semitoric for all but finitely many values of the parameter, with the goal of developing a strategy to find a semitoric system associated to a given partial list of semitoric invariants. We also enumerate the possible behaviors of such families at the parameter values for which they are not semitoric, providing examples illustrating nearly all possible behaviors, which describes the possible limits of semitoric systems with a fixed $S^1$-action. Furthermore, we introduce natural notions of blowup and blowdown in this context, investigate how semitoric families behave under these operations, and use this to prove that each Hirzebruch surface admits a semitoric family with certain desirable invariants; these families are related to the semitoric minimal model program. Finally, we give several explicit semitoric families on the first and second Hirzebruch surfaces showcasing various possible behaviors of such families which include new semitoric systems.

math.SG↗

arXiv2021

Extending compact Hamiltonian $\mathbb{S}^1$-spaces to integrable systems with mild degeneracies in dimension four

Given any compact connected four dimensional symplectic manifold $(M,\omega)$ and smooth function $J\colon M\to \mathbb{R}$ which generates an effective $\mathbb{S}^1$-action, we show that there exists a smooth function $H\colon M\to\mathbb{R}$ such that $(M,\omega,(J,H))$ is a completely (Liouville) integrable system of a type we call hypersemitoric -- these are systems for which all singularities are non-degenerate, except possibly for a finite number of families of degenerate points of a relatively tame type called parabolic (also sometimes called cuspidal). Such an $(M,\omega,J)$ is often referred to as a Hamiltonian $\mathbb{S}^1$-space (classified by Karshon in 1999) and we call any integrable system of the form $(M,\omega,(J,H))$ an extension of $(M,\omega,J)$. Using this terminology, our main result is that any Hamiltonian $\mathbb{S}^1$-space can be extended to a hypersemitoric integrable system. We also show that there exist Hamiltonian $\mathbb{S}^1$-spaces for which any extension must include at least one degenerate singular point. Parabolic points are among the most common and natural degenerate points, and thus hypersemitoric systems are in this sense the `nicest' class of systems to which all Hamiltonian $\mathbb{S}^1$-spaces can be extended. We also prove several foundational results about these systems, such as the non-existence of loops of hyperbolic-regular points and some properties about their fibers.

math.SG↗

arXiv2019

Invariance of immersed Floer cohomology under Lagrangian surgery

We show that Floer cohomology of an immersed Lagrangian brane is invariant under smoothing of a self-intersection point if the quantum valuation of the weakly bounding cochain vanishes and the Lagrangian has dimension at least two. The chain-level map replaces the two orderings of the self-intersection point with meridianal and longitudinal cells on the handle created by the surgery, and uses a bijection between holomorphic disks developed by Fukaya-Oh-Ohta-Ono. Our result generalizes invariance of potentials for certain Lagrangian surfaces in Dimitroglou-Rizell--Ekholm--Tonkonog, and implies the invariance of Floer cohomology under mean curvature flow with this type of surgery, as conjectured by Joyce.

math.SG↗

arXiv2018

A family of compact semitoric systems with two focus-focus singularities

About 6 years ago, semitoric systems were classified by Pelayo & Vu Ngoc by means of five invariants. Standard examples are the coupled spin oscillator on $\mathbb{S}^2 \times \mathbb{R}^2$ and coupled angular momenta on $\mathbb{S}^2 \times \mathbb{S}^2$, both having exactly one focus-focus singularity. But so far there were no explicit examples of systems with more than one focus-focus singularity which are semitoric in the sense of that classification. This paper introduces a 6-parameter family of integrable systems on $\mathbb{S}^2 \times \mathbb{S}^2$ and proves that, for certain ranges of the parameters, it is a compact semitoric system with precisely two focus-focus singularities. Since the twisting index (one of the semitoric invariants) is related to the relationship between different focus-focus points, this paper provides systems for the future study of the twisting index.

math.SG↗

arXiv2018

Invariance of Immersed Floer cohomology under Maslov flows

We show that immersed Lagrangian Floer cohomology in compact rational symplectic manifolds is invariant under Maslov flows such as coupled mean curvature/Kaehler-Ricci flow in the sense of Smoczyk as a pair of self-intersection points is born or dies at a self-tangency, using results of Ekholm-Etnyre-Sullivan. This proves part of a conjecture of Joyce. We give a lower bound on the time for which the Floer cohomology is invariant under the (forward or backwards) flow, if it exists. This post-publication has an erratum written jointly with Hadi Azizi, which fills in a missing case in the proof Lemma 7.9 (b).

math.SG↗

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↗

arXiv2017

Moduli spaces of semitoric systems

Recently Pelayo-Vũ Ngoc classified semitoric integrable systems in terms of five symplectic invariants. Using this classification we define a family of metrics on the space of semitoric integrable systems. The resulting metric space is incomplete and we construct the completion.

math.SG↗

arXiv2016

Minimal models of compact symplectic semitoric manifolds

A symplectic semitoric manifold is a symplectic $4$-manifold endowed with a Hamiltonian $(S^1 \times \mathbb{R})$-action satisfying certain conditions. The goal of this paper is to construct a new symplectic invariant of symplectic semitoric manifolds, the helix, and give applications. The helix is a symplectic analogue of the fan of a nonsingular complete toric variety in algebraic geometry, that takes into account the effects of the monodromy near focus-focus singularities. We give two applications of the helix: first, we use it to give a classification of the minimal models of symplectic semitoric manifolds, where "minimal" is in the sense of not admitting any blowdowns. The second application is an extension to the compact case of a well known result of Vũ Ngoc about the constraints posed on a symplectic semitoric manifold by the existence of focus-focus singularities. The helix permits to translate a symplectic geometric problem into an algebraic problem, and the paper describes a method to solve this type of algebraic problem.

math.SG↗

arXiv2015

Symplectic $G$-capacities and integrable systems

For any Lie group $G$, we construct a $G$-equivariant analogue of symplectic capacities and give examples when $G = \mathbb{T}^k\times\mathbb{R}^{d-k}$, in which case the capacity is an invariant of integrable systems. Then we study the continuity of these capacities, using the natural topologies on the symplectic $G$-categories on which they are defined.

math.SG↗

arXiv2014

Metrics and convergence in the moduli spaces of maps

We provide a general framework to study convergence properties of families of maps. For manifolds $M$ and $N$ where $M$ is equipped with a volume form $\mathcal{V}$ we consider families of maps in the collection $\{(ϕ, B) : B \subset M, ϕ:B \rightarrow N\text{ with both measurable}\}$ and we define a distance function $\mathcal{D}$ similar to the $L^1$ distance on such a collection. The definition of $\mathcal{D}$ depends on several parameters, but we show that the properties and topology of the metric space do not depend on these choices. In particular we show that the metric space is always complete. After exploring the properties of $\mathcal{D}$ we shift our focus to exploring the convergence properties of families of such maps.

math.DG↗

arXiv2013

On the structure group of a decomposable model space

We study the structure group of a canonical algebraic curvature tensor built from a symmetric bilinear form, and show that in most cases it coincides with the isometry group of the symmetric form from which it is built. Our main result is that the structure group of the direct sum of such canonical algebraic curvature tensors on a decomposable model space must permute the subspaces $V_i$ on which they are defined. For such an algebraic curvature tensor, we show that if the vector space $V$ is a direct sum of subspaces $V_1$ and $V_2$, the corresponding structure group decomposes as well if $V_1$ and $V_2$ are invariant of the action of the structure group on $V$. We determine the freedom one has in permuting these subspaces, and show these subspaces are invariant if $\dim V_1 \neq \dim V_2$ or if the corresponding symmetric forms defined on those subspaces have different (but not reversed) signatures, so that in this situation, only the trivial permutation is allowable. We exhibit a model space that realizes the full permutation group, and, with exception to the balanced signature case, show the corresponding structure group is isomorphic to the wreath product of the structure group of a given symmetric bilinear form by the symmetric group. Using these results, we conclude that the structure group of any member of this family is isomorphic to a direct product of wreath products of pseudo-orthogonal groups by certain subgroups of the symmetric group. Finally, we apply our results to two families of manifolds to generate new isometry invariants that are not of Weyl type.

math.GR↗