The study of semi-free actions on manifolds with specific orbit spaces.
problem Characterizing smooth, semi-free actions on manifolds with specific properties.
method Analyzing smooth, semi-free actions on closed, smooth, simply connected manifolds.
result Only connected sums of S3-bundles over S2 are simply connected 5-manifolds with certain semi-free circle actions. New Smith-Gysin sequence for non-semi-free actions without semi-free condition.
problem Constructing a Smith-Gysin sequence for non-semi-free actions.
method Developed a new Smith-Gysin sequence that includes an exotic term.
result Inclusion of an exotic term that depends on the subset MS1. In this paper we describe a method to establish when a symplectic manifold M with semi-free Hamiltonian S1-action is unique up to isomorphism (equivariant symplectomorphism). This will rely on a study of the symplectic topology of the reduced spaces. We prove that if the reduced spaces satisfy a rigidity conditi…
Constructs a Dirac-type operator for semi-free S1-actions on manifolds.
problem Defines an integer-valued signature for orbit spaces of semi-free S1-actions. method Develops induced Dirac-Schrödinger-type operators and shows their essential self-adjointness.
result The index of the constructed operator coincides with Lott's signature, under certain conditions.
Let M be a compact oriented simply-connected manifold of dimension at least 8. Assume M is equipped with a torsion-free semi-free circle action with isolated fixed points. We prove M has a perfect invariant Morse-Smale function. The major ingredient in the proof is a new cancellation theorem for the invariant Mor…
This paper contains several results concerning circle action on almost-complex and smooth manifolds. More precisely, we show that, for an almost-complex manifold M2mn(resp. a smooth manifold N4mn), if there exists a partition λ=(λ1,...,λu) of weight m such that the Chern number $(c_{λ_{1}}... c_{λ_{…
We show that, on the connected sum of complex projective planes, any toric LeBrun metric can be identified with a Joyce metric admitting a semi-free circle action through an explicit conformal equivalence. A crucial ingredient of the proof is an explicit connection form for toric LeBrun metrics.
We investigate U(1)-equivariant deformations of C. LeBrun's self-dual metric with torus action. We explicitly determine all U(1)-subgroups of the torus for which one can obtain U(1)-equivariant deformation that do not preserve semi-free U(1)-action. This gives many new self-dual metrics with U(1)-action which are not c…
Kawakubo and Uchida showed that, if a closed oriented 4k-dimensional manifold M admits a semi-free circle action such that the dimension of the fixed point set is less than 2k, then the signature of M vanishes. In this note, by using G-signature theorem and the rigidity of the signature operator, we generaliz…
We study topological T-duality for spaces with a semi-free S1−action with isolated fixed points. Physically, these correspond to spacetimes containing Kaluza-Klein monopoles. We demonstrate that the physical dyonic coordinate of such spaces has an analogue in our formalism. By analogy with the Dirac monopole, we stu…
Stable approach solves equivariant Hopf theorem for G-manifolds.
problem Describe homotopy classes of G-equivariant maps into a G-sphere.
method Equivariant stable homotopy theory with semi-free G-universe.
result Degrees of maps are characterized by congruences.
Constructs a Dirac-Schrödinger operator on Thom-Mather stratified spaces.
problem Computing an index for orbit spaces of semi-free S1-actions. method Pushing down an S1-invariant first order transversally elliptic operator to the quotient space. result The constructed operator's index coincides with Lott's signature.
Abstract: Necessary and sufficient conditions for circle actions on 4-manifolds with discrete fixed points.
problem Conditions for circle actions on 4-manifolds with discrete fixed points.
method Demonstrated pairs of integers that arise as weights of a circle action also arise as weights of a restriction of a T2-action. result Provided necessary and sufficient conditions for pairs of integers to arise as weights and Chern numbers of circle actions.
In this paper we introduce invariants of semi-free Hamiltonian actions of $S\sp 1$ on compact symplectic manifolds (which satisfy some technical conditions related to positivity) using the space of solutions to certain gauge theoretical equations. These equations generalize at the same time the vortex equations and the…
Study circle actions on unitary manifolds with discrete fixed points.
problem Understanding circle actions on compact unitary manifolds with discrete fixed points.
method Prove relationships between weights at fixed points and derive results regarding the first equivariant Chern class and Hirzebruch χy-genus. result Derive a multigraph encoding fixed point data, leading to new insights into unitary S1-manifolds. The study simplifies complex functions on surfaces using a special transformation.
problem Understanding functions with degenerate singularities on various surfaces.
method Established a 'normal form' for functions using a specific transformation.
result Any function in the class can be simplified to a 'simplest' Morse function through a transformation.
We construct metrics of positive scalar curvature on manifolds with circle actions. One of our main results is that there exist S1-invariant metrics of positive scalar curvature on every S1-manifold which has a fixed point component of codimension 2. As a consequence we can prove that there are non-invariant metr…
We construct geometric generators of the effective S1-equivariant Spin- (and oriented) bordism groups with two inverted. We apply this construction to the question of which S1-manifolds admit invariant metrics of positive scalar curvature. It turns out that, up to taking connected sums with several copies of the …
Przytycki and Sokolov proved that a three-manifold admits a semi-free action of the finite cyclic group of order p with a circle as the set of fixed points if and only if M is obtained from the three-sphere by surgery along a strongly p−periodic link L. Moreover, if the quotient three-manifold is an integral ho…
In this paper we study the topological T-dual of spaces with a non-free circle action mainly using the stack theory method of Bunke and co-workers \cite{Bunke1}. We first compare three formalisms for obtaining the Topological T-dual of a semi-free S1-space in a simple example. Then, we calculate the T-dual of genera…
Formula for Lefschetz number of knot branched covers.
problem Calculating Lefschetz numbers for knot branched covers.
method Analysis of Seiberg-Witten equations on 3-orbifolds, η-invariants.
result Formula for Lefschetz number of covering translations.
We explore a connection between the Finslerian area functional based on the Busemann-Hausdorff-volume form, and well-investigated Cartan functionals to solve Plateau's problem in Finsler 3-space, and prove higher regularity of solutions. Free and semi-free geometric boundary value problems, as well as the Douglas probl…
We formulate a refinement of SU(N) Chern-Simons theory on a three-manifold via the refined topological string and the (2,0) theory on N M5 branes. The refined Chern-Simons theory is defined on any three-manifold with a semi-free circle action. We give an explicit solution of the theory, in terms of a one-parameter refi…
Geometrically solves differentiating simplicial manifolds.
problem Differentiating simplicial manifolds.
method Establishes a normal form theorem, identifies a differentiating ideal, proves quotient semi-freeness, interprets as Chevalley-Eilenberg algebra of higher Lie algebroid.
result Introduces higher van Est map and proves van Est isomorphism theorem.
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.
AE-DQN learns to eliminate sub-optimal actions in complex RL environments.
problem Challenges in RL with many actions, especially redundancy.
method Combines DQN with AEN that predicts invalid actions.
result Significant speedup and robustness in games with many actions.
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.
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.
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.
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.