arXiv subjects

Miguel Abreu

Publications and source records attributed to Miguel Abreu.

At least 19 recordsLinked to original sources

arXiv2026

Multiplicity of closed Reeb orbits on contact manifolds with periodic equivariant symplectic homology

We consider closed contact manifolds $(M,ξ)$ with periodic positive equivariant symplectic homology. This is a very large class of contact manifolds and, to the best of our knowledge, includes all currently known examples admitting Reeb flows with finitely many closed orbits for which this homology is well defined. Under weak and homologically natural index assumptions on a non-degenerate contact form $α$ on $M$, we establish a sharp lower bound $r_M$ for the number of simple closed Reeb orbits of $α$. The quantity $r_M$ is completely defined in terms of the positive equivariant symplectic homology of $M$. Moreover, we show that this bound is attained if and only if $α$ is lacunary, i.e., the Conley-Zehnder indices of all closed orbits have the same parity. Consequently, the invariant $r_M$ admits a clean dynamical characterization: whenever $M$ admits a non-degenerate lacunary contact form, $r_M$ equals the number of its simple closed Reeb orbits and is therefore independent of the choice of such a form. In particular, in the lacunary case the number of such orbits is a contact invariant completely determined by Floer theory. We compute $r_M$ for a broad class of examples, including several prequantizations of symplectic orbifolds, and show that in this case $r_M=\dim H_*(M/S^1;\mathbb{Q})$, thereby giving a topological characterization of the invariant. Motivated by these results, we conjecture that any contact form with finitely many closed Reeb orbits is necessarily non-degenerate and lacunary, and that the underlying contact manifold is a prequantization of this type.

math.SG↗

arXiv2026

Contact flexibility and rigidity for toric Gorenstein prequantizations and Ehrhart theory of toric diagrams

Gorenstein toric contact manifolds are good toric contact manifolds with zero first Chern class that are completely determined by certain integral convex polytopes called toric diagrams. The Ehrhart polynomial of these toric diagrams determines and is determined by the contact Betti numbers of the corresponding contact manifolds, i.e. the dimension of their cylindrical contact homology in eachdegree. In this paper we look into the following natural question: to what extent do these contact invariants determine the Gorenstein toric contact manifold? Flexibility is the norm and we illustrate it with the family of Gorenstein toric contact manifolds that arise as the prequantization of monotone iterated ${\mathbb P}^1$-bundles, i.e. monotone Bott manifolds. In each dimension, the Ehrhart polynomial of their toric diagrams is equal to the Ehrhart polynomial of the cross-polytope, corresponding to the monotone prequantization of ${\mathbb P}^1 \times \cdots \times {\mathbb P}^1$, and we describe the unimodular classification of these toric diagrams. On the rigidity side, we will show that the primitive prequantization of ${\mathbb P}^1 \times \cdots \times {\mathbb P}^1$ is rigid, i.e. completely determined as a Gorenstein toric contact manifold by its contact Betti numbers. More precisely, in each dimension, we show that its toric diagram, which we name small cross-polytope, is the unique toric diagram with its particular Ehrhart polynomial. We will also prove a rigidity result for a family of Gorenstein toric contact manifolds that arise as the primitive prequantization of monotone ${\mathbb P}^1$-bundles over ${\mathbb P}^{n-1}$.

math.SG↗

arXiv2026

Periodic orbits of non-degenerate lacunary contact forms on prequantization bundles

A non-degenerate contact form is lacunary if the indexes of every contractible periodic Reeb orbit have the same parity. To the best of our knowledge, every contact form with finitely many periodic orbits known so far is non-degenerate and lacunary. We show that every non-degenerate lacunary contact form on a suitable prequantization of a closed symplectic manifold $B$ has precisely $r_B$ contractible closed orbits, where $r_B=\dim \mathrm{H}_*(B;{\mathbb Q})$. Examples of such prequantizations include the standard contact sphere, the unit cosphere bundle of a compact rank one symmetric space (CROSS) and many others. We also consider some prequantizations of orbifolds, like lens spaces and the unit cosphere bundle of lens spaces, and obtain multiplicity results for these prequantizations.

math.SG↗

arXiv2025

Designing a skilled soccer team for RoboCup: exploring skill-set-primitives through reinforcement learning

The RoboCup 3D Soccer Simulation League serves as a competitive platform for showcasing innovation in autonomous humanoid robot agents through simulated soccer matches. Our team, FC Portugal, developed a new codebase from scratch in Python after RoboCup 2021. The team's performance relies on a set of skills centered around novel unifying primitives and a custom, symmetry-extended version of the Proximal Policy Optimization algorithm. Our methods have been thoroughly tested in official RoboCup matches, where FC Portugal has won the last two main competitions, in 2022 and 2023. This paper presents our training framework, as well as a timeline of skills developed using our skill-set-primitives, which considerably improve the sample efficiency and stability of skills, and motivate seamless transitions. We start with a significantly fast sprint-kick developed in 2021 and progress to the most recent skill set, including a multi-purpose omnidirectional walk, a dribble with unprecedented ball control, a solid kick, and a push skill. The push addresses low-level collision scenarios and high-level strategies to increase ball possession. We address the resource-intensive nature of this task through an innovative multi-agent learning approach. Finally, we release the team's codebase to the RoboCup community, providing other teams with a robust and modern foundation upon which they can build new features.

cs.RO↗

arXiv2025

Addressing imperfect symmetry: A novel symmetry-learning actor-critic extension

Symmetry, a fundamental concept to understand our environment, often oversimplifies reality from a mathematical perspective. Humans are a prime example, deviating from perfect symmetry in terms of appearance and cognitive biases (e.g. having a dominant hand). Nevertheless, our brain can easily overcome these imperfections and efficiently adapt to symmetrical tasks. The driving motivation behind this work lies in capturing this ability through reinforcement learning. To this end, we introduce Adaptive Symmetry Learning (ASL), a model-minimization actor-critic extension that addresses incomplete or inexact symmetry descriptions by adapting itself during the learning process. ASL consists of a symmetry fitting component and a modular loss function that enforces a common symmetric relation across all states while adapting to the learned policy. The performance of ASL is compared to existing symmetry-enhanced methods in a case study involving a four-legged ant model for multidirectional locomotion tasks. The results show that ASL can recover from large perturbations and generalize knowledge to hidden symmetric states. It achieves comparable or better performance than alternative methods in most scenarios, making it a valuable approach for leveraging model symmetry while compensating for inherent perturbations.

cs.LG↗

arXiv2024

Symmetric periodic Reeb orbits on the sphere

A long standing conjecture in Hamiltonian Dynamics states that every contact form on the standard contact sphere $S^{2n+1}$ has at least $n+1$ simple periodic Reeb orbits. In this work, we consider a refinement of this problem when the contact form has a suitable symmetry and we ask if there are at least $n+1$ simple symmetric periodic orbits. We show that there is at least one symmetric periodic orbit for any contact form and at least two symmetric closed orbits whenever the contact form is dynamically convex.

math.SG↗

arXiv2023

Contact invariants of Q-Gorenstein toric contact manifolds, the Ehrhart polynomial and Chen-Ruan cohomology

Q-Gorenstein toric contact manifolds provide an interesting class of examples of contact manifolds with torsion first Chern class. They are completely determined by certain rational convex polytopes, called toric diagrams, and arise both as links of toric isolated singularities and as prequantizations of monotone toric symplectic orbifolds. In this paper we show how the cylindrical contact homology invariants of a Q-Gorenstein toric contact manifold are related to (i) the Ehrhart (quasi-)polynomial of its toric diagram; (ii) the Chen-Ruan cohomology of any crepant toric orbifold resolution of its corresponding toric isolated singularity; (iii) the Chen-Ruan cohomology of any monotone toric symplectic orbifold base that gives rise to it through prequantization.

math.SG↗

arXiv2022

Dynamical implications of convexity beyond dynamical convexity

We establish sharp dynamical implications of convexity on symmetric spheres that do not follow from dynamical convexity. It allows us to show the existence of elliptic and non-hyperbolic periodic orbits and to furnish new examples of dynamically convex contact forms, in any dimension, that are not equivalent to convex ones via contactomorphisms that preserve the symmetry. Moreover, these examples are $C^1$-stable in the sense that they are actually not equivalent to convex ones via contactomorphisms that are $C^1$-close to those preserving the symmetry. We also show the multiplicity of symmetric non-hyperbolic and symmetric (not necessarily non-hyperbolic) closed Reeb orbits under suitable pinching conditions.

math.SG↗

arXiv2022

Learning Hybrid Locomotion Skills -- Learn to Exploit Residual Dynamics and Modulate Model-based Gait Control

This work aims to combine machine learning and control approaches for legged robots, and developed a hybrid framework to achieve new capabilities of balancing against external perturbations. The framework embeds a kernel which is a fully parametric closed-loop gait generator based on analytical control. On top of that, a neural network with symmetric partial data augmentation learns to automatically adjust the parameters for the gait kernel and to generate compensatory actions for all joints as the residual dynamics, thus significantly augmenting the stability under unexpected perturbations. The performance of the proposed framework was evaluated across a set of challenging simulated scenarios. The results showed considerable improvements compared to the baseline in recovering from large external forces. Moreover, the produced behaviours are more natural, human-like and robust against noisy sensing.

cs.RO↗

arXiv2022

On contact invariants of non-simply connected Gorenstein toric contact manifolds

The first two authors showed in~\cite{AM1} how the Conley-Zehnder index of any contractible periodic Reeb orbit of a non-degenerate toric contact form on a good toric contact manifold with zero first Chern class, i.e. a Gorenstein toric contact manifold, can be explicitly computed using moment map data. In this paper we show that the same explicit method can be used to compute Conley-Zehnder indices of non-contractible periodic Reeb orbits. Under appropriate conditions, the (finite) number of such orbits in a given free homotopy class and with a given index is a contact invariant of the underlying contact manifold. We compute these invariants for two sets of examples that satisfy these conditions: $5$-dimensional contact manifolds that arise as unit cosphere bundles of $3$-dimensional lens spaces, and $2n+1$-dimensional Gorenstein contact lens spaces. As applications, we will see that these invariants can be used to show that diffeomorphic lens spaces might not be contactomorphic and that there are homotopy classes of diffeomorphisms of some lens spaces that do not contain any contactomorphism. Following a suggestion by one referee, we will also see that this type of applications can be proved alternatively by looking at the total Chern class of these canonical contact structures on lens spaces.

math.SG↗

arXiv2021

A CPG-Based Agile and Versatile Locomotion Framework Using Proximal Symmetry Loss

Humanoid robots are made to resemble humans but their locomotion abilities are far from ours in terms of agility and versatility. When humans walk on complex terrains, or face external disturbances, they combine a set of strategies, unconsciously and efficiently, to regain stability. This paper tackles the problem of developing a robust omnidirectional walking framework, which is able to generate versatile and agile locomotion on complex terrains. The Linear Inverted Pendulum Model and Central Pattern Generator concepts are used to develop a closed-loop walk engine, which is then combined with a reinforcement learning module. This module learns to regulate the walk engine parameters adaptively, and generates residuals to adjust the robot's target joint positions (residual physics). Additionally, we propose a proximal symmetry loss function to increase the sample efficiency of the Proximal Policy Optimization algorithm, by leveraging model symmetries and the trust region concept. The effectiveness of the proposed framework was demonstrated and evaluated across a set of challenging simulation scenarios. The robot was able to generalize what it learned in unforeseen circumstances, displaying human-like locomotion skills, even in the presence of noise and external pushes.

cs.RO↗

arXiv2021

Robust Biped Locomotion Using Deep Reinforcement Learning on Top of an Analytical Control Approach

This paper proposes a modular framework to generate robust biped locomotion using a tight coupling between an analytical walking approach and deep reinforcement learning. This framework is composed of six main modules which are hierarchically connected to reduce the overall complexity and increase its flexibility. The core of this framework is a specific dynamics model which abstracts a humanoid's dynamics model into two masses for modeling upper and lower body. This dynamics model is used to design an adaptive reference trajectories planner and an optimal controller which are fully parametric. Furthermore, a learning framework is developed based on Genetic Algorithm (GA) and Proximal Policy Optimization (PPO) to find the optimum parameters and to learn how to improve the stability of the robot by moving the arms and changing its center of mass (COM) height. A set of simulations are performed to validate the performance of the framework using the official RoboCup 3D League simulation environment. The results validate the performance of the framework, not only in creating a fast and stable gait but also in learning to improve the upper body efficiency.

cs.RO↗

arXiv2020

On the mean Euler characteristic of Gorenstein toric contact manifolds

We prove that the mean Euler characteristic of a Gorenstein toric contact manifold, i.e. a good toric contact manifold with zero first Chern class, is equal to half the normalized volume of the corresponding toric diagram and give some applications. A particularly interesting one, obtained using a result of Batyrev and Dais, is the following: twice the mean Euler characteristic of a Gorenstein toric contact manifold is equal to the Euler characteristic of any crepant toric symplectic filling, i.e. any toric symplectic filling with zero first Chern class.

math.SG↗

arXiv2020

Two closed orbits for non-degenerate Reeb flows

We prove that every non-degenerate Reeb flow on a closed contact manifold $M$ admitting a strong symplectic filling $W$ with vanishing first Chern class carries at least two geometrically distinct closed orbits provided that the positive equivariant symplectic homology of $W$ satisfies a mild condition. Under further assumptions, we establish the existence of two geometrically distinct closed orbits on any contact finite quotient of $M$. Several examples of such contact manifolds are provided, like displaceable ones, unit cosphere bundles, prequantization circle bundles, Brieskorn spheres and toric contact manifolds. We also show that this condition on the equivariant symplectic homology is preserved by boundary connected sums of Liouville domains. As a byproduct of one of our applications, we prove a sort of Lusternik-Fet theorem for Reeb flows on the unit cosphere bundle of not rationally aspherical manifolds satisfying suitable additional assumptions.

math.SG↗

arXiv2017

Dynamical convexity and elliptic periodic orbits for Reeb flows

A long-standing conjecture in Hamiltonian Dynamics states that the Reeb flow of any convex hypersurface in $\mathbb{R}^{2n}$ carries an elliptic closed orbit. Two important contributions toward its proof were given by Ekeland in 1986 and Dell'Antonio-D'Onofrio-Ekeland in 1995 proving this for convex hypersurfaces satisfying suitable pinching conditions and for antipodal invariant convex hypersurfaces respectively. In this work we present a generalization of these results using contact homology and a notion of dynamical convexity first introduced by Hofer-Wysocki-Zehnder for tight contact forms on $S^3$. Applications include geodesic flows under pinching conditions, magnetic flows and toric contact manifolds.

math.SG↗

arXiv2016

Multiplicity of periodic orbits for dynamically convex contact forms

We give a sharp lower bound for the number of geometrically distinct contractible periodic orbits of dynamically convex Reeb flows on prequantizations of symplectic manifolds that are not aspherical. Several consequences of this result are obtained, like a new proof that every bumpy Finsler metric on $S^n$ carries at least two prime closed geodesics, multiplicity of elliptic and non-hyperbolic periodic orbits for dynamically convex contact forms with finitely many geometrically distinct contractible closed orbits and precise estimates of the number of even periodic orbits of perfect contact forms. We also slightly relax the hypothesis of dynamical convexity. A fundamental ingredient in our proofs is the common index jump theorem due to Y. Long and C. Zhu.

math.SG↗

arXiv2015

Toric constructions of monotone Lagrangian submanifolds in $\mathbb{CP}^2$ and $\mathbb{CP}^1 \times \mathbb{CP}^1$

We extract from a toric model of the Chekanov-Schlenk exotic torus in $\mathbb{CP}^2$ methods of construction of Lagrangian submanifolds in toric symplectic manifolds. These constructions allow for some control of the monotonicity. We recover this way some known monotone Lagrangians in the toric symplectic manifolds $\mathbb{CP}^2$ and $\mathbb{CP}^1 \times \mathbb{CP}^1$ as well as new examples.

math.SG↗

arXiv2013

Displacing Lagrangian toric fibers by extended probes

In this paper we introduce a new way of displacing Lagrangian fibers in toric symplectic manifolds, a generalization of McDuff's original method of probes. Extended probes are formed by deflecting one probe by another auxiliary probe. Using them, we are able to displace all fibers in Hirzebruch surfaces except those already known to be nondisplaceable, and can also displace an open dense set of fibers in the weighted projective space P(1,3,5) after resolving the singularities. We also investigate the displaceability question in sectors and their resolutions. There are still many cases in which there is an open set of fibers whose displaceability status is unknown.

math.SG↗