Asymptotically CAT(0) metrics and Z-structures for HHGs, proving Farrell-Jones Conjecture.
problem Proving Farrell-Jones Conjecture for HHGs.
method Asymptotically CAT(0) metrics and Z-structures construction.
result Many HHGs satisfy Farrell-Jones Conjecture.
Let $\imath: M\to \RR^{p+2}$ be a smooth embedding from a connected, oriented, closed p-dimesional smooth manifold to $\RR^{p+2}$, then there is a spin structure ♯(ςp+2) on M canonically induced from the embedding. If an orientation-preserving diffeomorphism τ of M extends over as an o…
In this paper, we investigate finitely generated groups of isometries of CAT(0) spaces containing some central hyperbolic isometry, and study CAT(0) groups. We show that every CAT(0) group Γ has the semi-direct product structure $Γ=(\cdots(((Γ'\rtimes\langleδ_{n}\rangle)\rtimes\langleδ_{n-1}\rangle)\rtimes\langleδ_{n…
Study on median algebra structures on Euclidean spaces and manifolds with local CAT(0) cubulation.
problem Understanding median algebra structures on Euclidean spaces and manifolds.
method Showed local CAT(0) cubulation for median structures on ER homology manifolds.
result Median structures on ER homology manifolds have a local CAT(0) cubulation structure.
We announce results on the structure of CAT(0) groups, CAT(0) lattices and of the underlying spaces. Our statements rely notably on a general study of the full isometry groups of proper CAT(0) spaces. Classical statements about Hadamard manifolds are established for singular spaces; new arithmeticity and rigidity state…
We prove that a minimal disc in a CAT(0) space is a local embedding away from a finite set of "branch points". On the way we establish several basic properties of minimal surfaces: monotonicity of area densities, density bounds, limit theorems and the existence of tangent maps. As an application, we prove Fary-Milnor's…
New findings on geometric flows and equidistribution in Hilbert geometry.
problem Characterizing dynamical and counting results in Hilbert geometry.
method Study of dynamical and counting results in rank-one properly convex projective structures with Hilbert metrics.
result Hilbert geodesic flow is strongly mixing and orbits and primitive closed geodesics equidistribute.
New Coxeter groups have unique boundary structures.
problem Understanding boundaries of Coxeter groups.
method Recursive construction and amalgamation of CAT(0) groups.
result Totally disconnected Morse boundaries for new Coxeter groups.
CATNet predicts CAT bond spreads using graph-based deep learning.
problem Complex, relational data in CAT bonds not well captured by traditional models.
method CATNet applies R-GCN to CAT bond primary market as a graph.
result CATNet outperforms Random Forest and XGBoost benchmarks.
Mahan Mitra (Mj) proved Cannon--Thurston maps exist for normal hyperbolic subgroups of a hyperbolic group. We prove that Cannon--Thurston maps do not exist for infinite normal hyperbolic subgroups of non-hyperbolic CAT(0) groups with isolated flats with respect to the visual boundaries. We also show Cannon--Thurston ma…
CAT method learns causal structure of directed trees efficiently.
problem Learning causal structure from directed trees.
method Chu-Liu-Edmonds algorithm for fast and scalable structure learning.
result Consistency in asymptotic regime with vanishing identifiability gap for Gaussian errors.
Homotopy equivalent boundaries of cube complexes are studied.
problem The equivalence of different boundaries of cube complexes.
method Using a partial order on a quotient of the Roller boundary, we obtain the simplicial Roller boundary and show homotopy equivalence among the Tits, simplicial, and simplicial Roller boundaries.
result The Tits, simplicial, and simplicial Roller boundaries are homotopy equivalent.
Paper uses machine learning to uncover nonlinear dynamics in CAT bond pricing.
problem Traditional linear models miss nonlinear relationships in CAT bond pricing.
method Advanced machine learning techniques applied to CAT bond transaction records.
result Machine learning enhances CAT bond pricing accuracy and reveals complex risk interactions.
We study complex surfaces with locally CAT(0) polyhedral Kahler metrics and construct such metrics on CP^2 with various orbifold structures. In particular, in relation to questions of Gromov and Davis-Moussong we construct such metrics on a compact quotient of the two-dimensional unite complex ball. In the course of th…
CaTs use DAGs with transformers to enforce causal constraints, improving neural network robustness.
problem Neural networks lack inherent causal structure respect, leading to reliability issues.
method Introducing Causal Transformers (CaTs) that operate under predefined causal constraints specified by DAGs.
result CaTs improve robustness and interpretability of neural networks under causal constraints.
We prove that every open subset of a euclidean building is a finite dimensional absolute neighborhood retract. This implies in particular that such a set has the homotopy type of a finite dimensional simplicial complex. We also include a proof for the rigidity of homeomorphisms of euclidean buildings. A key step in our…
We classify Dehn functions of Bestvina-Brady groups.
problem Understanding the complexity of Bestvina-Brady groups.
method Explicit criteria on defining graphs to determine Dehn function degree.
result Explicitly classify the Dehn functions of Bestvina-Brady groups.
New method proves heat flow of harmonic maps into CAT(0) spaces.
problem Global existence and uniqueness of suitable weak solutions.
method Elliptic regularization and variational structure.
result Lipschitz continuity of solutions in spatial variables.
Busemann G-spaces with Finsler metrics
problem Admitting a differentiable DC atlas with a continuous Finsler metric
method Comparison geometry
result Generalizes previous work on G-spaces with Riemannian curvature bounds
Study of geometric actions on CAT(0) spaces and their limits.
problem Understanding limits of geometric actions on CAT(0) spaces.
method Analysis of convergence and splitting/collapsing phenomena in CAT(0) lattices and orbispaces.
result Proof of compactness theorem for CAT(0) homology orbifolds.
CAT(0) spaces with small volume growth are homeomorphic to Euclidean space.
problem Characterizing CAT(0) spaces with specific volume growth properties.
method Analyzing asymptotic topological regularity and volume growth.
result CAT(0) spaces with small volume growth are homeomorphic to Euclidean space.
Locally finite complexes with polyhedral CAT(0) metrics are arborescent.
problem Characterizing locally finite complexes with CAT(0) metrics. method Proving arborescence for complexes with polyhedral CAT(0) metrics. result Locally finite complexes with polyhedral CAT(0) metrics are arborescent. This paper proves properties of CAT(κ) surfaces with bounded curvature.
problem Properties of CAT(κ) surfaces with bounded curvature.
method Proof of bounded curvature, approximation by smooth surfaces.
result CAT(κ) surfaces have bounded curvature.
New CAT(0) groups show superexponential subgroup Dehn functions.
problem Understanding subgroups of CAT(0) groups with superexponential Dehn functions.
method Construction of specific 4- and 6-dimensional CAT(0) groups.
result Found subgroups with Dehn functions exp(n)(xm) and exp(n)(xα). CAT(0) study of curve complements and families.
problem Understanding CAT(0) behavior of curve complements and families.
method Investigation of fundamental groups of plane curve complements and universal families.
result Fundamental groups of certain families are not CAT(0).
CaT-GNN improves credit card fraud detection by integrating causal reasoning into GNNs.
problem Credit card fraud detection overlooks causal structure of transactions.
method CaT-GNN combines causal invariant learning and temporal graph neural networks.
result CaT-GNN outperforms existing methods on various datasets.
Suppose a group G is relatively hyperbolic with respect to a collection $\PP$ of its subgroups and also acts properly, cocompactly on a $\CAT(0)$ (or δ--hyperbolic) space X. The relatively hyperbolic structure provides a relative boundary $\partial(G,\PP)$. The $\CAT(0)$ structure provides a different boundary at…
Braid group proof shows 7-strand is CAT(0).
problem Proving the 7-strand braid group is CAT(0).
method Elaborating on Haettel, Kielak, and Schwer's argument.
result The 7-strand braid group is proven to be CAT(0).
In this paper, we show some splitting theorems for CAT(0) spaces on which a product group acts geometrically and we obtain a splitting theorem for compact geodesic spaces of non-positive curvature. A CAT(0) group Γ is said to be {\it rigid}, if Γ determines the boundary up to homeomorphisms of a CAT(0) space on whi…
Classifies 3-manifold groups with equivariant hierarchically hyperbolic structures.
problem Classifying 3-manifold groups with equivariant hierarchically hyperbolic structures.
method Construction of suitable quasimorphisms on Seifert pieces to construct actions on quasi-lines.
result 3-manifold groups admit equivariant hierarchically hyperbolic structures.
Toyoda proved 5-point CAT(0) spaces; this paper offers a new proof.
problem Describing 5-point subspaces in CAT(0) length spaces.
method Provides an alternative proof of Toyoda's theorem.
result Offers a new proof of Toyoda's theorem.
We show that groups satisfying Kazhdan's property (T) have no unbounded actions on finite dimensional CAT(0) cube complexes, and deduce that there is a locally CAT(-1) Riemannian manifold which is not homotopy equivalent to any finite dimensional, locally CAT(0) cube complex.
We give a generalized and self-contained account of Haglund-Paulin's wallspaces and Sageev's construction of the CAT(0) cube complex dual to a wallspace. We examine criteria on a wallspace leading to finiteness properties of its dual cube complex. Our discussion is aimed at readers wishing to apply these methods to pro…
The paper extends geometric results from negatively-curved spaces to strictly convex Hilbert geometry.
problem Extending geometric results from negatively-curved spaces to strictly convex Hilbert geometry.
method Demonstrates dynamical and counting results for geometrically-finite strictly convex projective structures with Hilbert metric.
result Hilbert geodesic flow is strongly mixing and orbits and primitive closed geodesics equidistribute.
This work achieves exponential concentration in heavy-tailed data over CAT(κ) spaces using the Fréchet median.
problem Achieving robust estimation in heavy-tailed data distributions.
method Developing a concentration bound for the Fréchet median in CAT(κ) spaces.
result Exponential concentration of the Fréchet median in CAT(κ) spaces over heavy-tailed data.
Let X be a proper, geodesically complete CAT(0) space under a proper, non-elementary, isometric action by a group Γ with a rank one element. We construct a generalized Bowen-Margulis measure on the space of unit-speed parametrized geodesics of X modulo the Γ-action. Although the construction of Bowen-Margulis m…
Study mapping class groups on CAT(0) cube complexes.
problem Asymptotically rigid mapping class groups of infinitely-punctured surfaces.
method Introduced CAT(0) cube complexes and determined their properties.
result Determined CAT(0) properties of cube complexes.
Affine cactus groups are CAT(0) and hyperbolic.
problem Characterizing geometric properties of affine cactus groups.
method Analyzing CAT(0) and hyperbolic properties through group theory.
result Affine cactus groups of degree three are hyperbolic.
Groups on CAT(0) cube complexes grow exponentially uniformly.
problem Uniform exponential growth of groups acting on CAT(0) cube complexes.
method Study groups acting without global fixed points on CAT(0) square complexes.
result Groups with uniform exponential growth or stabilize Euclidean subcomplexes.
Lecture notes on CAT(0) cube complexes for beginners.
problem Introduction to CAT(0) cube complexes for students.
method Elementary and self-contained lectures.
result Accessible introduction to advanced topics for students.
Geodesics spiral around compact subsets in CAT(0) spaces.
problem Understanding geodesic spiraling in CAT(0) spaces.
method Logarithm law-type result for geodesics in quotients of rank one CAT(0) spaces.
result Proved logarithm law for geodesic spiraling in certain CAT(0) spaces.
CAT(0) spaces quasi-isometric to Euclidean spaces are homeomorphic to R^n.
problem Characterizing CAT(0) spaces quasi-isometric to Euclidean spaces.
method Analyzing properties of CAT(0) spaces and using quasi-isometry.
result CAT(0) spaces quasi-isometric to R^n are not necessarily homeomorphic to it.
We examine volume pinching problems of CAT(1) spaces. We characterize a class of compact geodesically complete CAT(1) spaces of small specific volume. We prove a sphere theorem for compact CAT(1) homology manifolds of small volume. We also formulate a criterion of manifold recognition for homology manifolds on volume g…
We show that the class of CAT(0) spaces is closed under suitable conformal changes. In particular, any CAT(0) space admits a large variety of non-trivial deformations.
New examples show right-angled Artin groups can have connected boundaries.
problem CAT(0) group boundaries not always path connected.
method Provided examples of right-angled Artin groups with connected boundaries.
result Right-angled Artin groups can have boundaries that are path connected.
In this paper, we study CAT(0) spaces with non-locally connected boundary. We give some condition of a CAT(0) space whose boundary is not locally connected.
A simpler proof shows L2-metric completion is CAT(0).
problem Completing Riemannian metrics space.
method Easier proof of existing result by Brian Clarke.
result Completion of Riemannian metrics is CAT(0).
CAT is a new ASR toolkit using CRF and CTC for state-of-the-art speech recognition.
problem Improving automatic speech recognition systems.
method CRF-based discriminative training with CTC-inspired state topology.
result CAT achieves state-of-the-art results with fewer parameters and is competitive with hybrid models.