Simplified conditions for GO metrics in homogeneous manifolds.
problem Determine G-GO metrics in compact homogeneous manifolds.
method Simplified conditions for GO metrics based on equivalent isotropy submodules.
result Algebraic conditions for GO metrics in homogeneous manifolds.
In this paper, we consider a CscK metric defined away from divisor and with metric upper bound and lower bound going to zero in certain rate. And we'll prove that this "nicely" behaved metric is a smooth CscK metric across the divisor.
A homogeneous Riemannian space (M=G/H,g) is called a geodesic orbit space (shortly, GO-space) if any geodesic is an orbit of one-parameter subgroup of the isometry group G. We study the structure of compact GO-spaces and give some sufficient conditions for existence and non-existence of an invariant metric g wit…
In this short note, we show that the negative curvature is preserved in the deformation of hyperbolic warped product metrics under Ricci flow. It is also showed that the flow converges to a flat metric as time going to infinity.
New families of non-singular geodesic orbit nilmanifolds discovered.
problem Classifying non-singular geodesic orbit nilmanifolds.
method Complete classification through analysis of nilmanifolds.
result New families of non-singular GO nilmanifolds with dimensions 14 and 15.
In Finsler geometry, we use calculus to study the geometry of regular inner metric spaces. In this note I will briefly discuss various curvatures and their geometric meanings from the metric geometry point of view, without going into the forest of tensors.
We consider the moduli space of the extremal Kähler metrics on compact manifolds. We show that under the conditions of two-sided total volume bounds, L2n-norm bounds on $\Riem$, and Sobolev constant bounds, this Moduli space can be compactified by including (reduced) orbifolds with finitely many singularities…
Study semi-invariant metrics in hydrodynamics for better understanding of fluid motion.
problem Understanding fluid motion using semi-invariant Riemannian metrics.
method Geometric approach via geodesic initial value problem on diffeomorphism groups.
result Local and some global well-posedness results for geodesic initial value problem.
We give bounds on the first non-zero eigenvalue of the scalar Laplacian for both the Page and the Chen-LeBrun-Weber Einstein metrics. One notable feature is that these bounds are obtained without explicit knowledge of the metrics or numerical approximation to them. Our method also allows the calculation of the invarian…
Study nondifferentiable metrics in general relativity, resolving causality issues and limits evolution scenarios.
problem Causality issues and evolution scenarios in black hole interiors with closed timelike geodesics.
method Method of equivalence on Courant algebroids to derive new differential invariants.
result Resolved causality issues and limited evolution scenarios for gravitational collapse.
Researchers compactify magnetic monopole moduli spaces, revealing new metric properties.
problem Understanding the moduli space of SU(2) magnetic monopoles.
method Constructing a partial compactification of the moduli space, M_k, and analyzing the hyperKahler metric.
result The leading term of the asymptotic metric generalizes known results for monopoles of charge 1.
The flag curvature is a natural extension of the sectional curvature in Riemannian geometry, and the S-curvature is a non-Riemannian quantity which vanishes for Riemannian metrics. There are (incomplete) non-Riemannian Finsler metrics on an open subset in R^n with negative flag curvature and constant S-curvature. In th…
Study proves rigidity of marked length spectra in contracting group actions.
problem Rigidity of marked length spectra in contracting group actions.
method Unified approach using the Extension Lemma and metric geometry.
result Orbit map is a rough isometry if marked length spectra match.
Study on triaxial Bianchi IX metrics and Ricci flow singularity.
problem Understanding singularity formation in triaxial Bianchi IX metrics under Ricci flow.
method Investigates sufficient conditions for Type I singularity in triaxial Bianchi IX metrics on foliated manifolds.
result Generalizes previous studies on rotationally symmetric and triaxial metrics.
Study existence of conformal metrics with specific curvature properties on compact manifolds.
problem Existence of conformal metrics with constant scalar curvature and boundary mean curvature.
method Proving existence through specific cases and sequences of metrics.
result Existence of conformal metrics in various cases, including positive Yamabe constant.
Smooth HP^2 bundle over S^4 with nontrivial A-genus found.
problem Existence of Riemannian metrics with positive sectional curvature.
method Explained existence of a smooth HP^2 bundle over S^4 with nontrivial A-genus.
result Existence of a smooth HP^2 bundle over S^4 with nontrivial A-genus.
We prove that admissible functions for Fubini-Study metrics on the complex projective space PmC, of complex dimension m, invariant by a convenient automorphisms group, are lower bounded by a function going to minus infinity on the boundary of usual charts of PmC. A similar lower bound holds on some projecti…
We go further on the study of harmonicity for almost contact metric structures already initiated by Vergara-Diaz and Wood. By using the intrinsic torsion, we characterise harmonic almost contact metric structures in several equivalent ways and show conditions relating harmonicity and classes of almost contact metric st…
Adversarial policies beat superhuman Go AI systems.
problem Vulnerability of superhuman AI systems to adversarial attacks.
method Training adversarial policies to trick KataGo into making blunders.
result Adversarial policies achieve >97% win rate against KataGo at superhuman settings.
A new unbiased Hessian estimator for expectation-based objectives.
problem Estimating Hessian for objectives with non-reparameterizable nodes.
method GO Hessian estimator for expectation-based objectives.
result GO Hessian provides unbiased and low-variance estimation of Hessian.
New RL approach speeds up training across tasks.
problem Training reinforcement learning models quickly and efficiently.
method Combines planning quasi-metric and task-specific aimers.
result Achieves multiple-fold speed-up on bit-flip and robotic arm tasks.
Study of metric spaces and group actions using Vietoris-Rips and Čech complexes.
problem Understanding the homotopy type of quotient spaces under group actions.
method Intermediate scale parameters for Vietoris-Rips and Čech complexes.
result First scale parameter where homotopy type of projective spaces changes.
We have shown that the Beltrami Theorem in Riemannian geometry is still true for square metrics if the dimension n≥3, namely, an n(≥3)-dimensional square metric is locally projectively flat if and only if it is of scalar flag curvature. In this paper, we go on with the study of the Beltrami Theorem for a larg…
Gluing theorem for collapsing warped-QAC Calabi-Yau manifolds verified.
problem Behavior of warped-QAC Calabi-Yau metrics on affine quadrics.
method Gluing construction for collapsing warped-QAC Calabi-Yau manifolds.
result Verification of Yang Li's conjecture on warped QAC Calabi-Yau metrics.
Study convexity of geodesics and balls in Outer space.
problem Convexity properties of geodesics and balls in Outer space.
method Introduced balanced folding paths and used them to show weak convexity of out-going balls.
result Weak convexity of out-going balls in Outer space.
New geometries defined for string models, filling gaps in the literature.
problem Developing mathematical structures for string models.
method Defining E-metric-connection geometries and locality structures.
result Unified framework for metric-affine and generalized geometries.
We consider scattering by an abstract compactly supported perturbation in R^n. To include the traditional cases of potential, obstacle and metric scattering without going into their particular nature we adopt the "black box" formalism developed jointly with Sjostrand [23]. It is quite likely that one could extend the r…
Paper defines Fisher co-metric on cotangent bundle and clarifies its relation to variance.
problem Defines Fisher co-metric on cotangent bundle and clarifies its relation to variance.
method Defines Fisher co-metric directly from Fisher metric without going through tangent bundle, using a natural correspondence between cotangent vectors and random variables.
result Clarifies the relation between Fisher co-metric and variance/covariance, trivializing the Cramér-Rao inequality.
Pearson distance fails to be a metric, but can be fixed.
problem The Pearson distance is not a metric, leading to issues in applications.
method Showed that Pearson distance is not a metric and provided alternative metrics.
result Pearson distance can be repaired by using alternative metrics derived from the correlation coefficient.
We study the geometry of type II supergravity compactifications in terms of an oriented vector bundle E, endowed with a bundle metric of split signature and further datum. The geometric structure is associated with a so-called generalised G-structure and characterised by an E-spinor ρ, which we can regard as a …
Research confirms Streets-Tian conjecture for 2-step solvmanifolds.
problem Streets-Tian conjecture for compact complex manifolds.
method Used special non-unitary frames to reveal hidden symmetries.
result Confirms conjecture for all 2-step solvmanifolds.
New metrics improve uncertainty estimation on graph data.
problem Current GNNs focus only on nodewise scores, limiting uncertainty estimation.
method Proposed edgewise metrics for uncertainty estimation on graphs.
result GNN models with structured prediction perform better in uncertainty estimation.
We study Lipschitz models in reinforcement learning to bound prediction and value-function errors.
problem Bounding errors in reinforcement learning models with Lipschitz continuity constraints.
method We provide bounds on multi-step prediction error and value-function estimate using the Wasserstein metric for Lipschitz models.
result Lipschitz models lead to bounded errors in prediction and value-function estimates.
KataGo accelerates Go self-play learning by 50x.
problem Efficiently learning in large state spaces like Go.
method Improved AlphaZero process and architecture.
result 50x reduction in computation time.
Random walks generate quasi-convex subgroups in acylindrically hyperbolic groups.
problem Understanding the structure of subgroups generated by random walks in hyperbolic groups.
method Probabilistic approach using random walks in acylindrically hyperbolic groups.
result Subgroups generated by random walks are quasi-convex and isomorphic to the original subgroup and the random walk subgroup.
Go-Explore improves performance on hard-exploration problems in Atari games.
problem Challenges in reinforcement learning, especially with sparse or deceptive rewards.
method Exploits principles of remembering states, returning to promising states, and solving simulated environments.
result Scores significantly higher than previous state-of-the-art on Montezuma's Revenge and Pitfall.
We prove that the Ricci flow on CP^n blown-up at one point starting with any rotationally symmetric Kahler metric must develop Type I singularities. In particular, if the total volume does not go to zero at the singular time, the parabolic blow-up limit of the Type I Ricci flow along the exceptional divisor is a comple…
Paper examines Go AI robustness against adversarial attacks.
problem Superhuman Go AIs are vulnerable to simple adversarial strategies.
method Three defenses tested: adversarial training, iterated adversarial training, and changing network architecture.
result No defense is robust against newly trained adversaries, and attacks are similar to cyclic attacks.
Analyzes Kähler-Einstein metrics on families of Fano varieties.
problem Establishing Kähler-Einstein metrics on Fano varieties in families.
method Analytic method to show unique Kähler-Einstein metrics on neighboring fibers.
result Uniform a priori estimates and continuous variation of Kähler-Einstein potentials.
Characterizes Finsleroid-Finsler metrics as solutions to a conformal rigidity problem.
problem Existence of regular Landsberg metrics that are not of Berwald type.
method Characterizes Finsleroid-Finsler metrics as solutions to a conformal rigidity problem involving a specific invariant curvature tensor.
result Solutions of class at least C2 on the complement of the zero section are conformal to Finsleroid-Finsler metrics. The paper studies phase transitions in geodesic flows on curved manifolds.
problem Understanding phase transitions in geodesic flows on geometrically finite manifolds.
method Defined a class of potentials and constructed geometrically finite manifolds to exhibit phase transitions.
result Geometric potential exhibits a phase transition on certain manifolds.
The curvature discussed in this paper is a rather far going generalization of the Riemannian sectional curvature. We define it for a wide class of optimal control problems: a unified framework including geometric structures such as Riemannian, sub-Riemannian, Finsler and sub-Finsler structures; a special attention is p…
The paper introduces a volume invariant for Hermitian-symplectic metrics and proves its critical points are Kähler.
problem Investigating volume invariants for Hermitian-symplectic metrics.
method Introducing a functional acting on metrics in Aeppli cohomology classes and proving critical points are Kähler.
result The volume invariant generalises the volume of a Kähler class and vanishing is a necessary condition for the existence of a Kähler metric.
Optimizes identifying the best arm with fixed samples.
problem Finding the arm with the highest mean in a fixed number of samples.
method Characterizes minimax optimal rates and introduces algorithms to achieve them.
result Characterizes and introduces algorithms for optimal best arm identification.
GO-OED maximizes predictive information gain on nonlinear QoIs.
problem Maximizing information gain on nonlinear predictive quantities.
method Nested Monte Carlo estimator, Markov chain Monte Carlo, kernel density estimation, Bayesian optimization.
result GO-OED outperforms conventional OED in nonlinear settings.
Open-source Go AI achieves perfect record against top professionals.
problem Understanding and usability of AlphaZero algorithms.
method Open-source reimplementation of AlphaZero algorithm.
result Achieves perfect (20:0) record against global top professionals.
Metric learning enhances combinatorial coverage metrics' ability to predict classification errors.
problem Dataset dependence of combinatorial coverage metrics in anticipating classification errors.
method Metric learning to improve latent space separation of data classes.
result Metric learning increases SDCCMs' ability to distinguish between correctly and incorrectly classified data.
This note explains Ricci flow method for Kähler-Einstein metrics.
problem Existence problem of Kähler-Einstein metrics.
method Ricci flow method based on Cao.H.D and Yau.S.T's papers, with additional estimates from Jian Song and Weinkove's note.
result Illustrates and details the Ricci flow method for Kähler-Einstein metrics.