arXiv research

Study extends Kobayashi's method to non-reductive subgroups for homogeneous spaces.

The paper extends invariant theory to non-compact and non-reductive actions, classifying four regimes.

It is shown that, in the Gromov space of isometry classes of pointed proper metric spaces, the equivalence relations defined by existence of coarse quasi-isometries or being at finite Gromov-Hausdorff distance, cannot be reduced to the equivalence relation defined by any Polish action.

A method, due to Élie Cartan, is used to give an algebraic classification of the non-reductive homogeneous pseudo-Riemannian manifolds of dimension four. Only one case with Lorentz signature can be Einstein without having constant curvature, and two cases with (2,2) signature are Einstein of which one is Ricci-flat. If…

We introduce an analogue in hyperkahler geometry of the symplectic implosion, in the case of SU(n) actions. Our space is a stratified hyperkahler space which can be defined in terms of quiver diagrams. It also has a description as a non-reductive geometric invariant theory quotient.

We classify non-reductive four-dimensional homogeneous conformally Einstein manifolds.

We classify totally geodesic and parallel hypersurfaces of four-dimensional non-reductive homogeneous pseudo-Riemannian manifolds.

It is known that the automorphism group of a K-polystable Fano manifold is reductive. Codogni and Dervan construct a canonical filtration of the section ring, called Loewy filtration, and conjecture that the Loewy filtration destabilizes any Fano variety with non-reductive automorphism group. In this note, we give a co…

We study the K-stability of a polarised variety with non-reductive automorphism group. We associate a canonical filtration of the co-ordinate ring to each variety of this kind, which destabilises the variety in several examples which we compute. We conjecture this holds in general. This is an algebro-geometric analogue…

For two positive integers m and n, we let ${\mathcal P}_n$ be the open convex cone in ${\mathbb R}^{n(n+1)/2}$ consisting of positive definite n x n real symmetric matrices and let ${\mathbb R}^{(m,n)}$ be the set of all m x n real matrices. In this article, we investigate differential operators on the non-reductive ma…

We represent algebraic curves via commuting matrix polynomials. This allows us to show that the Hilbert scheme of cohomologically stable twisted rational curves of degree $d$ in ${\Bbb P}^3\backslash {\Bbb P}^1$ is isomorphic to a complexified hyperkähler quotient of an open subset of a vector space by a non-reductive …

Develops a new framework for generalized Ricci flow on Lie groups.

A subalgebra of a Lie algebra $\mathfrak{h}\subset\mathfrak{g}$ determines $\mathfrak{h}$-representation $ρ$ on $\mathfrak{m}=\mathfrak{g}/\mathfrak{h}$. In this note we discuss how to reconstruct $\mathfrak{g}$ from $(\mathfrak{h},\mathfrak{m},ρ)$. In other words, we find all the ingredients for building non-reductive…

Develops a new geometric framework for quantum metrics.

Ordinarily, quiver varieties are constructed as moduli spaces of quiver representations in the category of vector spaces. It is also natural to consider quiver representations in a richer category, namely that of vector bundles on some complex variety equipped with a fixed sheaf that twists the morphisms. Representatio…

We embed polarised orbifolds with cyclic stabiliser groups into weighted projective space via a weighted form of Kodaira embedding. Dividing by the (non-reductive) automorphisms of weighted projective space then formally gives a moduli space of orbifolds. We show how to express this as a reductive quotient and so a GIT…

Constructs geometric models for moduli spaces of Higgs bundles over Riemann sphere.

Study calculates Ricci bounds for special Fano manifolds.

We give examples of symplectic actions of a cyclic group, inducing a trivial action on homology, on four-manifolds that admit Hamiltonian circle actions, and show that they do not extend to Hamiltonian circle actions. Our work applies holomorphic methods to extend combinatorial tools developed for circle actions to stu…

Reduces proper actions to simpler core actions for analysis.

Introduces Conditional Action Trees to simplify RL action spaces.

One problem in the application of reinforcement learning to real-world problems is the curse of dimensionality on the action space. Macro actions, a sequence of primitive actions, have been studied to diminish the dimensionality of the action space with regard to the time axis. However, previous studies relied on human…

Study polar actions on Damek-Ricci spaces, proving existence and finding examples.

Reduction principles for proper actions on smooth manifolds.

Totally geodesic sections found in polar actions.

Simplifies large action space bandits by selecting representative actions.

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…

Classifies totally geodesic submanifolds and polar actions on Stiefel manifolds.

Conditions for reducing quasi-actions to tree actions and group properties.

We identify action representations from video data, proving their statistical benefits.

Study properties of orbits of Hermann actions without commutability assumptions.

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.

A new method learns action representations for reinforcement learning.

The paper studies curvatures and austere properties of orbits in symmetric spaces.

New reinforcement learning framework for adapting to new actions.

Defines and computes a generalized spectral action for Lorentz warped products.

The paper studies acylindrical actions on trees and proves acylindrical hyperbolicity of Baumslag-Solitar groups.

Classifies polar actions on 3D homogeneous spaces.

UTE improves reinforcement learning by measuring action uncertainty, enhancing policy learning efficiency.

Proper actions on bornological spaces are characterized with compatible coarse structures.

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…

Reinforcement learning (RL) in discrete action space is ubiquitous in real-world applications, but its complexity grows exponentially with the action-space dimension, making it challenging to apply existing on-policy gradient based deep RL algorithms efficiently. To effectively operate in multidimensional discrete acti…

The study examines how perturbations of lattice actions on group boundaries behave.

Non-proper surface group action on product of trees found.

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 free circle actions on specific 7-manifolds with positive Ricci curvature.

Study of symplectomorphisms on ruled surfaces under circle actions.