Study Spin(7)-manifolds with a 4-torus action using a symmetric matrix ansatz.
problem Characterize Spin(7)-manifolds with a 4-torus action. method Provide a Gibbons-Hawking type ansatz using a symmetric 4imes4-matrix of functions. result First known Spin(7)-manifolds with a rank 4 symmetry group and full holonomy. We consider G2-manifolds with an effective torus action that is multi-Hamiltonian for one or more of the defining forms. The case of T3-actions is found to be distinguished. For such actions multi-Hamiltonian with respect to both the three- and four-form, we derive a Gibbons-Hawking type ansatz giving the geometr…
Higher KdV flows on spaces of closed equicentroaffine plane curves are studied and it is shown that the flows are described as certain multi-Hamiltonian systems on the spaces. Multi-Hamiltonian systems describing higher mKdV flows are also given on spaces of closed Euclidean plane curves via the geometric Miura transfo…
We show that Plebanski's second heavenly equation, when written as a first-order nonlinear evolutionary system, admits multi-Hamiltonian structure. Therefore by Magri's theorem it is a completely integrable system. Thus it is an example of a completely integrable system in four dimensions.
This work extends locally conformal analysis to multi-Hamiltonian settings, providing new geometric structures and Hamiltonian dynamics.
problem Globalization problem in multi-Hamiltonian formalisms due to incompatibilities on chart overlaps.
method Investigation of locally conformally Nambu--Poisson and locally conformally generalized Poisson manifolds, constructing Hamiltonian-type evolution equations.
result Unified framework for classical, Nambu--Poisson, and generalized Poisson manifolds within a locally conformal context.
Study G2-manifolds from symplectic SU(3)-manifolds with T2-symmetry.
problem Understanding G2-manifolds from symplectic SU(3)-manifolds. method Cohomological lifting of multi-toric graphs.
result Compact part of G2-moment graph can be obtained cohomologically from the base. Study closed G2-structures with T3-symmetry, classifying them into types and deriving hypersymplectic structures.
problem Classify closed G2-structures with T3-symmetry and derive associated hypersymplectic structures.
method Decompose G2-structures into canonical forms, classify structures based on orbit isotropy, and derive hypersymplectic structures.
result Closed G2-structures with T3-symmetry are classified into two types, leading to specific hypersymplectic structures.
The Lagrangian representation of multi-Hamiltonian PDEs has been introduced by Y. Nutku and one of us (MVP). In this paper we focus on systems which are (at least) bi-Hamiltonian by a pair A1, A2, where A1 is a hydrodynamic-type Hamiltonian operator. We prove that finding the Lagrangian representation is equiv…
Examples show cyclic actions on 4-manifolds can't extend to Hamiltonian circle actions.
problem Cyclic group actions on 4-manifolds that trivialize homology cannot be extended to Hamiltonian circle actions.
method Holomorphic methods applied to combinatorial tools for circle actions.
result Cyclic actions on 4-manifolds that trivialize homology do not extend to Hamiltonian circle actions.
Reduces proper actions to simpler core actions for analysis.
problem Understanding properties of proper actions on manifolds.
method Extending Skjelbred and Straume's construction to non-compact groups, focusing on core of actions.
result Properties of proper actions are determined by simpler core actions.
A new method learns disentangled macro actions from sequences for reinforcement learning.
problem Curse of dimensionality in reinforcement learning action space.
method Autonomously learns disentangled factor representation of actions to generate macro actions.
result Higher scores in complex environments compared to other reinforcement learning algorithms.
Introduces Conditional Action Trees to simplify RL action spaces.
problem Challenges in RL with large, complex action spaces.
method Structures action spaces and reduces complexity through Conditional Action Trees.
result Demonstrates effectiveness in reducing action space and improving decision making.
Study polar actions on Damek-Ricci spaces, proving existence and finding examples.
problem Characterize polar actions on Damek-Ricci spaces.
method Prove criteria for isometric actions to be polar, find examples, and classify actions.
result Non-trivial polar actions exist on all Damek-Ricci spaces.
New RL algorithm tackles complex discrete action spaces.
problem Challenges in applying on-policy RL in high-dimensional discrete action spaces.
method Action-value critic, correlated actions, gradient sparsification.
result Empirically outperforms related on-policy algorithms.
Finite group actions on smooth 3-manifolds can be smoothed.
problem Finite group actions on smooth 3-manifolds.
method Uniform limit of smooth actions.
result Every continuous action of a finite group on a smooth 3-manifold is a uniform limit of smooth actions.
Reduction principles for proper actions on smooth manifolds.
problem Proper actions on smooth manifolds and their properties.
method Exhibit constructions and prove reduction principles for proper actions.
result Reduction principles hold for proper actions, polar actions, and copolarity.
Totally geodesic sections found in polar actions.
problem Understanding sections of polar actions on Riemannian manifolds.
method Elementary proof of a folklore result.
result Sections of polar actions are totally geodesic.
Simplifies large action space bandits by selecting representative actions.
problem Efficiently managing large action spaces with correlated outcomes.
method Random sampling and solving of bandit instances to identify representative actions.
result The algorithm selects a smaller set of representative actions that perform nearly as well as the full action space.
We study isometric actions on Riemannian symmetric spaces of noncompact type which are induced by reductive algebraic subgroups of the isometry group. We show that for such an action there exists a corresponding isometric action on a dual compact symmetric space, which reflects many properties of the original action. F…
The study proves conditions for symplectic torus actions on manifolds with non-contractible orbits.
problem Conditions for symplectic torus actions with non-contractible orbits.
method Analyzes symplectic torus actions on manifolds, proving conditions for Hamiltonian actions and orbit properties.
result Symplectic Tn−1 actions with non-contractible orbits are not Hamiltonian unless the orbits are contractible. Classifies totally geodesic submanifolds and polar actions on Stiefel manifolds.
problem Classifying totally geodesic submanifolds and polar actions on Stiefel manifolds.
method Classification through polar actions and cohomogeneity-one actions.
result Classification of orbits of polar actions on Stiefel manifolds.
Conditions for reducing quasi-actions to tree actions and group properties.
problem Conditions for reducing quasi-actions to tree actions.
method Reduction to cobounded isometric actions on trees.
result Groups with quasi-orbits quasi-isometric to trees are virtually free.
New stability theorem for non-hyperbolic group actions.
problem Structural stability of non-hyperbolic group actions.
method Introducing 'meandering hyperbolicity' for group actions on geodesic metric spaces.
result Meandering-hyperbolic actions are structurally stable.
We identify action representations from video data, proving their statistical benefits.
problem Identifying latent action policies from video data.
method Entropy-regularized LAPO objective, formalizing desiderata for action representations.
result Entropy-regularized LAPO identifies action representations satisfying desiderata under suitable conditions.
New method learns action representations for better reinforcement learning.
problem Lack of structured action representations in reinforcement learning.
method Decomposes policy into action representation and action transformation components.
result Action representations improve generalization in large action spaces.
Study properties of orbits of Hermann actions without commutability assumptions.
problem Investigate geometric properties of orbits of Hermann actions.
method Compute the second fundamental form and provide conditions for weak reflection and aridity.
result Sufficient conditions for weak reflection and aridity of orbits of Hermann action.
A new method learns action representations for reinforcement learning.
problem Efficient action-value estimation in reinforcement learning.
method Action hypergraph networks framework for learning action representations.
result Hypergraph Q-networks show effectiveness on various domains.
We classify the polar actions on the complex hyperbolic plane up to orbit equivalence. Apart from the trivial and transitive polar actions, there are five polar actions of cohomogeneity one and four polar actions of cohomogeneity two.
The paper studies curvatures and austere properties of orbits in symmetric spaces.
problem Analyzing curvatures and austere properties of orbits in symmetric spaces.
method Using Hermann actions and hyperpolar properties, the paper derives explicit formulas for principal curvatures and conditions for orbits to be austere.
result The paper provides conditions for orbits to be austere and extends previous results to a larger class of infinite-dimensional submanifolds.
New reinforcement learning framework for adapting to new actions.
problem Making reinforcement learning agents adaptable to new actions without retraining.
method Two-stage framework: infer action representations first, then train a flexible policy.
result Agents can make decisions from new action sets without retraining.
AQL uses amortized inference to handle high-dimensional action spaces in Q-learning.
problem Difficulty in maximizing over large action spaces in Q-learning.
method Replace expensive maximization over all actions with a maximization over a small subset sampled from a learned proposal distribution.
result AQL outperforms existing methods on continuous control tasks with up to 21 dimensional actions.
The paper studies acylindrical actions on trees and proves acylindrical hyperbolicity of Baumslag-Solitar groups.
problem Exploring acylindrical actions on trees and their properties.
method Demonstrates criteria for preserving acylindrical hyperbolicity and analyzes the outer automorphism group of Baumsligar-Solitar groups.
result Proves acylindrical hyperbolicity of non-solvable Baumsligar-Solitar groups.
The study examines largest hyperbolic actions in groups and finds many do not exist.
problem Identifying largest hyperbolic actions in groups.
method Analysis of equivalence classes of cobounded actions on hyperbolic metric spaces.
result Many families of groups, including 3-manifold groups and mapping class groups, do not have largest hyperbolic actions.
Defines and computes a generalized spectral action for Lorentz warped products.
problem Computing spectral actions for Lorentz warped products.
method Defines and computes the bimetric spectral Einstein-Hilbert action for Lorentz warped products.
result Derives a Kastler-Kalau-Walze type theorem for Lorentz warped products.
UTE improves reinforcement learning by measuring action uncertainty, enhancing policy learning efficiency.
problem Degrading performance of action repetition in reinforcement learning, especially with sub-optimal actions.
method UTE uses ensemble methods to measure uncertainty during action extension, allowing strategic exploration or certainty.
result UTE outperforms existing action repetition algorithms, significantly enhancing policy learning efficiency.
Classifies polar actions on 3D homogeneous spaces.
problem Classifying polar isometric actions on 3D homogeneous spaces.
method Orbit equivalence classification and study of cohomogeneity one actions.
result Classification of extrinsically homogeneous surfaces and orbit foliations.
Proper actions on bornological spaces are characterized with compatible coarse structures.
problem Characterizing proper actions on bornological spaces.
method Proving the existence of compatible coarse structures for proper actions.
result Bornological spaces admit compatible coarse structures for proper actions.
Learning how to act when there are many available actions in each state is a challenging task for Reinforcement Learning (RL) agents, especially when many of the actions are redundant or irrelevant. In such cases, it is sometimes easier to learn which actions not to take. In this work, we propose the Action-Elimination…
We analyze polar actions on Hermitian and quaternion-Kähler symmetric spaces of compact type. For complex integrable polar actions on Hermitian symmetric spaces of compact type we prove a reduction theorem and several corollaries concerning the geometry of these actions. The results are independent of the classificatio…
The study examines how perturbations of lattice actions on group boundaries behave.
problem Understanding how perturbations of lattice actions on group boundaries affect semi-conjugacy.
method Analyzes continuous factorization of perturbed actions onto original actions by semi-conjugacy.
result Perturbations of lattice actions on group boundaries can be C0 semi-conjugate or not. Study shows mean action of periodic orbits in annuli is bounded by their Calabi invariant.
problem Understanding the average distortion of periodic orbits in area-preserving annuli.
method Analyzes action functions and Calabi invariants of diffeomorphisms near annulus boundaries.
result Infimum of mean action of periodic orbits is bounded by their Calabi invariant.
Non-proper surface group action on product of trees found.
problem Proper surface group action on product of trees proposed, but not proper.
method Demonstrated non-properness of the surface group action.
result Surface group action on product of trees is not proper.
We consider actions of Z^k, k \ge 2, by Anosov diffeomorphisms which are uniformly quasiconformal on each coarse Lyapunov distribution. These actions generalize Cartan actions for which coarse Lyapunov distributions are one-dimensional. We show that, under certain non-resonance assumptions on the Lyapunov exponents, a …
Study of symplectomorphisms on ruled surfaces under circle actions.
problem Homotopy type of equivariant symplectomorphisms on rational ruled surfaces.
method Analysis of action on compatible and invariant almost complex structures, use of Delzant's and Karshon's classifications.
result Equivariant symplectomorphisms are homotopy equivalent to tori or their pushout.
Study free circle actions on specific 7-manifolds with positive Ricci curvature.
problem Classify and construct metrics on manifolds with a specific cohomology ring.
method Classify manifolds, construct free circle actions, and find positive Ricci curvature metrics.
result Almost all manifolds admit infinitely many non-equivalent free circle actions with positive Ricci curvature.
We consider a simple instance of action up to homotopy. More precisely, we consider strict actions of DGLAs in degrees -1 and 0 on degree 1 NQ-manifolds. In a more conventional language this means: strict actions of Lie algebra crossed modules on Lie algebroids. When the action is strict, we show that it integrates to …
Recent work has shown that reinforcement learning (RL) is a promising approach to control dynamical systems described by partial differential equations (PDE). This paper shows how to use RL to tackle more general PDE control problems that have continuous high-dimensional action spaces with spatial relationship among ac…
We introduce and systematically study the concept of a growth tight action. This generalizes growth tightness for word metrics as initiated by Grigorchuk and de la Harpe. Given a finitely generated, non-elementary group G acting on a G--space X, we prove that if G contains a strongly contracting eleme…