The paper studies hyperbolic quotients of projection complexes and their actions.
problem Understanding the structure and properties of quotients of projection complexes.
method Analyzing the quotient of projection complexes by normal subgroups and studying the resulting actions.
result The quotient complex is δ-hyperbolic under certain conditions, and the quotient group is acylindrically hyperbolic.
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.
Classifies actions on complex space forms with Lagrangian orbits.
problem Classifying actions on complex space forms with Lagrangian orbits.
method Classifies holomorphic isometric actions on complex space forms.
result Only examples are Lagrangian affine subspace foliations of complex Euclidean spaces and Lagrangian horocycle foliations of complex hyperbolic spaces.
New method QMLE performs well in complex action spaces without policy gradients.
problem Why policy gradients outperform action-value methods in complex action spaces.
method QMLE framework for action-value methods based on three principles.
result QMLE performs comparably to policy gradient methods in complex action spaces.
Groups acting on CAT(0) cube complexes have hyperfinite boundary actions.
problem Hyperfiniteness of boundary actions for special groups.
method Proving hyperfiniteness for groups acting on CAT(0) cube complexes.
result Boundary actions of virtually special groups are hyperfinite.
The notion of a complex hyperpolar action on a symmetric space of non-compact type has recently been introduced as counterpart of a hyperpolar action on a symmetric space of compact type. In this paper, we construct examples of a complex hyperpolar action without singular orbit and investigate the geometry of the orbit…
We show that almost complex circle actions with exactly three fixed points do not exist in dimension 8 and present an infinite series of 6-dimensional manifolds possessing an almost complex circle action with exactly two fixed points.
Groups can act on 3-manifolds if their Cayley complex can embed in specific types of 3-manifolds.
problem Understanding group actions on 3-manifolds.
method Proving groups admit actions on 3-manifolds if their Cayley complexes can embed in specific 3-manifolds.
result Groups with embeddable Cayley complexes can act on four specific types of 3-manifolds.
Holomorphic actions on complex spaces for nilpotent groups.
problem Understanding polynomial actions on complex spaces for nilpotent groups.
method Explicit construction of biholomorphisms by polynomial maps.
result Simply connected nilpotent Lie groups are biholomorphic to Cn. We classify polar actions on complex hyperbolic spaces up to orbit equivalence.
Study Dolbeault cohomology on complex manifolds with torus action.
problem Describe Dolbeault cohomology of complex manifolds with torus action.
method Describe Dolbeault cohomology algebra of canonical foliation, provide dga model, prove Hodge decomposition.
result Hodge decomposition for basic Dolbeault cohomology proved.
Morse theory on complexes for CAT(0) groups.
problem Finding universal spaces for proper actions of CAT(0) groups.
method Equivariant discrete Morse theory on Vietoris-Rips complexes.
result Exhibited finite universal spaces for proper actions of all asymptotically CAT(0) groups.
Study shows Dolbeault cohomology is unchanged by complex Lie group actions.
problem Understanding how complex Lie group actions affect Dolbeault cohomology.
method Analyzes Dolbeault cohomology of compact complex manifolds with group actions.
result Induced action on Dolbeault cohomology is trivial.
The paper develops a local index formula for complex manifolds with C∗-action.
problem Analyzing the m-index on complex manifolds with C∗-action. method Applying the method of transversal heat kernel asymptotics.
result Obtained a local index formula for the m-index. We consider stochastic multi-armed bandit problems with complex actions over a set of basic arms, where the decision maker plays a complex action rather than a basic arm in each round. The reward of the complex action is some function of the basic arms' rewards, and the feedback observed may not necessarily be the rewa…
Develops theory of relatively geometric actions on CAT(0) cube complexes.
problem Understand actions of relatively hyperbolic groups on CAT(0) cube complexes.
method Introduces and studies relatively geometric actions, proving key results.
result Proves full relatively quasi-convex subgroups are convex compact.
Extends T-duality to non-principal torus actions with elliptic tangent bundles.
problem Classifying and understanding non-principal torus actions with singularities.
method Introduces elliptic tangent bundle to control singularities, uses it to define connections and transport generalized complex structures via T-duality.
result New insights into the classification of torus actions and transport of generalized complex structures.
FSQ algorithm extends Q-learning to continuous actions with linear complexity.
problem Extending Q-learning to continuous action spaces with linear complexity.
method Discretization of the action space to maintain linear complexity.
result FSQ algorithm achieves linear complexity in the discretized problem.
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…
Survey on finite group actions on CW-complexes homotopy to spheres.
problem Understanding finite group actions on CW-complexes homotopy equivalent to spheres.
method Survey of extensive literature on finite G-CW-complexes homotopy equivalent to spheres. result Finite G-CW-complexes homotopy equivalent to spheres have finite group actions. Paper finds at least 6 fixed points for a specific circle action on a 10D manifold.
problem Finding the minimum number of fixed points for a circle action on a 10D almost complex manifold.
method Established a lower bound by showing the non-existence of a circle action with 4 fixed points.
result There are at least 6 fixed points for a circle action on a 10D compact almost complex manifold.
Characterizes geometric actions on graphs with flexible stabilizers.
problem Understanding geometric actions on flexible stabilizers.
method Defining generalized fine actions and proving relative quasi-convexity criteria.
result Characterizes Bowditch boundary points in relatively geometric actions.
New RL method reduces sample complexity for large state-action spaces.
problem Handling large state-action spaces in RL with general Q-functions.
method Nonparametric Q-learning using kernel ridge regression.
result Sample complexity is order optimal with respect to ε and kernel complexity.
The action dimension of a discrete group G is the minimum dimension of contractible manifold that admits a proper G-action. We compute the action dimension of the direct limit of a simple complex of groups for several classes of examples including: 1) Artin groups, 2) graph products of groups, and 3) fundamental gr…
Study finite deformations from heterotic superpotential, leading to new complex effective action.
problem Finite deformations of the Hull--Strominger system.
method Expanding the heterotic superpotential around a supersymmetric vacuum, identifying complex coordinates, and using Maurer--Cartan equation.
result Generalizes complex effective action of Kodaira--Spencer and holomorphic Chern--Simons theory, with a supersymmetric locus described by an L3 algebra. 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 the notion of maximal actions of compact tori on smooth manifolds and study compact connected complex manifolds equipped with maximal actions of compact tori. We give a complete classification of such manifolds, in terms of combinatorial objects, which are triples (Δ,h,G) of n…
New submanifolds found in toric manifolds with specific actions.
problem Understanding submanifolds in toric manifolds with complex subtorus actions.
method Analyzing the closure of a complex subtorus in a toric manifold and its Hamiltonian action.
result The image of the moment map for the Hamiltonian subtorus action coincides with the image of the Delzant polytope.
In the present paper we introduce the notion of complex asystatic Hamiltonian action on a Kähler manifold. In the algebraic setting we prove that if a complex linear group G acts complex asystatically on a Kähler manifold then the G-orbits are spherical. Finally we give the complete classification of complex asysta…
Study fixed-point sets of S1-actions on quaternionic manifolds.
problem Characterize fixed-point sets and compatible complex structures on quaternionic manifolds.
method Analyze fixed-point sets and derive equations involving first Chern classes.
result Conditions for the existence of hypercomplex structures on quaternionic manifolds.
We show that if a holomorphic n dimensional compact torus action on a compact connected complex manifold of complex dimension n has a fixed point then the manifold is equivariantly biholomorphic to a smooth toric variety.
Study quotients of curve complex actions by mapping class group.
problem Understanding actions of mapping class group on curve complex quotients.
method Cone off uniformly quasi-convex subspaces to form symmetric curve sets, non-maximal train track sets, and compression body disc sets. Analyze actions of mapping class group on these quotients.
result Actions of mapping class group on quotients are strongly WPD, non-elementary, and have infinite diameter.
If X is a connected complex manifold with dX=2 that admits the holomorphic and transitive action of a (connected) Lie group G, then the action extends to an action of the complexification G^ of G on X except when either the unit disk or else a strictly pseudoconcave homogeneous complex manifold is i…
Growing action spaces accelerates learning in complex tasks.
problem Efficient learning in tasks with large combinatorial action spaces.
method Curriculum of progressively growing action spaces, off-policy reinforcement learning, data transfer.
result Efficacy demonstrated in control tasks and large-scale StarCraft tasks.
Classifies symplectic torus actions up to equivariant symplectomorphism.
problem Classifying symplectic torus actions up to equivariant symplectomorphism.
method Classification theorems based on Duistermaat and Pelayo's work on symplectic torus actions with coisotropic orbits.
result Every almost isotropy-maximal symplectic torus action is equivariantly diffeomorphic to a product of a symplectic toric manifold and a torus.
Study on Einstein metrics on complex projective spaces with specific group actions.
problem Finding Einstein metrics invariant under cohomogeneity one Lie group actions.
method Analyzing Einstein equation for diagonal invariant metrics under five Takagi models.
result Nonexistence of smooth globally defined invariant Einstein metrics in four models, necessary condition in the fifth.
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.
Non-uniform lattices in PU(n,1) cannot geometrically act on CAT(0) cube complexes.
problem Non-uniform lattices in PU(n,1) cannot geometrically act on CAT(0) cube complexes.
method Proving non-uniform lattices in PU(n,1) cannot geometrically act on CAT(0) cube complexes.
result Non-uniform lattices in PU(n,1) cannot geometrically act on CAT(0) cube complexes.
MP-DQN improves deep Q-learning for complex action spaces.
problem Learning with discrete actions and continuous parameters in reinforcement learning.
method Multi-pass deep Q-networks (MP-DQN) to handle parameterised actions.
result Significantly outperforms P-DQN and other methods in data efficiency and policy performance.
The main result of the paper is the complete classification of the compact connected Lie groups acting coisotropically on complex Grassmannians. This is used to determine the polar actions on the same manifolds.
Let M be a manifold homotopy equivalent to the complex projective space $\C P^m$. Petrie conjectured that M has standard total Pontrjagin class if M admits a non-trivial action by S1. We prove the conjecture for m<12 under the assumption that the action extends to a nice Pin(2)-action with fixed point. The…
Compact complex manifolds with specific group actions are conformally flat.
problem Compact complex manifolds with invariant conformal holomorphic structures.
method Study of manifolds with transitive and essentially acting complex semi-simple Lie groups.
result If a complex semi-simple Lie group acts transitively and essentially, the manifold is conformally flat.
Consider an effective Hamiltonian torus action T×M→M on a topologically twisted,generalized complex manifold M of dimension 2n. We prove that the rank(T)≤n−2 and that the topological twisting survives Hamiltonian reduction. We then construct a large new class of such actions satisfying $rank(T) =…
It has long been known that differential forms on complex manifolds can be decomposed under the action of the complex structure to give the Dolbeault complex. This paper presents an analogous double complex for quaternionic manifolds using the fact that the cotangent space is isomorphic to a quaternionic vector space. …
Probabilistic model for exhaustion in infinite-genus curve complexes.
problem Action rigidity in infinite-genus curve complexes.
method Costa and Farber's model for random simplicial complexes.
result Probabilistic evidence for exhaustion via rigid expansions.
The author proved that if the circle acts symplectically on a compact, connected symplectic manifold M with three fixed points, then M is equivariantly symplectomorphic to some standard action on CP2. In this paper, we extend the result to a circle action on an almost complex manifold; if the circle act…
The paper studies quaternionic structures on GKM graphs and their relation to torus actions on quaternionic projective spaces.
problem Understanding quaternionic structures on GKM graphs and their implications for torus actions.
method Introducing quaternionic structures on GKM graphs and analyzing their properties in the context of torus actions.
result Abstract GKM graphs with specific 2-face structures correspond to torus actions on quaternionic projective spaces or Grassmannians.
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.