Study on renormalized volume for hyperbolic 3-manifolds, including rank-1 cusps.
problem Defining and studying renormalized volume for geometrically finite hyperbolic 3-manifolds.
method Defined renormalized volume, proved variation formula, and showed convergence for degenerating metrics.
result Renormalized volume converges to the limiting metric for certain families of convex co-compact hyperbolic metrics.
Study shows renormalized volume is locally convex for certain hyperbolic 3-manifolds.
problem Understanding critical points of renormalized volume in hyperbolic 3-manifolds.
method Introduced a modified definition of renormalized volume that is additive under gluing, and studied local properties.
result Renormalized volume is locally convex around critical points in acylindrical geometrically finite hyperbolic 3-manifolds with rank-1 cusps.
We establish the proportionality principle between the Riemannian volume and locally finite simplicial volume for Q-rank 1 locally symmetric spaces covered by products of hyperbolic spaces, giving the first examples for manifolds whose cusp groups are not necessarily amenable. Also, we give a simple direct proof of the…
We define \emph{piecewise rank 1} manifolds, which are aspherical manifolds that generally do not admit a nonpositively curved metric but can be decomposed into pieces that are diffeomorphic to finite volume, irreducible, locally symmetric, nonpositively curved manifolds with π1-injective cusps. We prove smooth (sel…
Study rigidity of geodesic planes in geometrically finite manifolds with cusps.
problem Rigidity of geodesic planes in geometrically finite manifolds with rank 1 cusps.
method Analysis of horocycles recurrence and density criteria for immersions.
result Closed immersions give rise to surfaces with finitely generated fundamental groups.
New method fractures hyperbolic manifolds using cone singularities.
problem Deforming hyperbolic manifolds with cone singularities.
method Direct manipulation of a fundamental polyhedron to change cone angles.
result Upper unknotting tunnels of highly twisted links can be drilled out.
Study on Čech cohomology of Morse boundaries in hyperbolic manifolds.
problem Computing Čech cohomology groups of Morse boundaries.
method Analyzing cusped hyperbolic n-manifolds and relatively hyperbolic groups.
result Reduced Čech cohomology vanishes in dimensions ≤ n-3 and does not vanish in dimension n-2.
The renormalized volume of hyperbolic 3-manifolds is strictly convex.
problem Understanding the convexity of the renormalized volume in hyperbolic 3-manifolds.
method Analyzing the Hessian of the renormalized volume and its positivity.
result Strict convexity of the functional volume of the convex core at its minimum point.
Certain classes of 3-manifolds, following Thurston, give rise to a 'skinning map', a self-map of the Teichmüller space of the boundary. This paper examines the skinning map of a 3-manifold M, a genus-2 handlebody with two rank-1 cusps. We exploit an orientation-reversing isometry of M to conclude that the skinning map …
Proves conditions for Fourier transforms in rank 1 symmetric spaces.
problem Understanding Fourier transform bounds in symmetric spaces.
method Proves sufficient and necessary conditions using Lipschitz and Fourier type integral conditions.
result Establishes bounds for Fourier transforms in rank 1 symmetric spaces with specific moduli of continuity.
Study shows dynamics of rank 1 orbifolds in flat surfaces.
problem Characterize dynamics of rank 1 affine invariant orbifolds.
method Analyzes M-isoperiodic foliations and their ergodic properties.
result Leaves of the isoperiodic foliation are either all closed or all dense.
Sharp isoperimetric inequalities for Neumann eigenvalues in symmetric spaces.
problem Finding bounds for eigenvalues of Neumann Laplacian on domains in symmetric spaces.
method Proving sharp inequalities for eigenvalues in compact and noncompact rank-1 symmetric spaces.
result Generalization of previous results for hyperbolic space and symmetric spaces.
Paper proves existence of minimal surfaces in hyperbolic 3-manifolds.
problem Existence of minimal surfaces in non-compact 3-manifolds.
method Analyzes specific cases of hyperbolic 3-manifolds with bounded geometry and rank-1 cusps.
result Proves existence of minimal surfaces in various hyperbolic 3-manifolds.
Study Morse-Novikov cohomology for 1-forms on rank 1 manifolds.
problem Analyzing cohomology of closed one-forms on manifolds.
method Explicit computation and discussion of locally conformally symplectic manifolds.
result Explicit computation for Inoue surface S^0.
Right inverse found for Cartan differential in rank-1 symmetric spaces.
problem Finding a right inverse for the Cartan differential in symmetric spaces.
method Integral operator approach to the Cartan differential on exact forms.
result Extension of Gauss linking integral to rank-1 symmetric spaces.
A1GM method improves efficiency in reconstructing missing data using KL divergence.
problem Efficiently reconstructing missing data in matrices.
method Fast non-gradient-based rank-1 NMF using KL divergence.
result A1GM outperforms gradient methods in efficiency with competitive reconstruction errors.
The paper describes hyperkähler geometry of cotangent bundles using rank-1 projections.
problem Understanding hyperkähler geometry of cotangent bundles via algebraic methods.
method Algebraic description via the scheme of rank-1 projections, isometric embeddings, and generalizations.
result Explicit isometric embeddings and generalizations of hyperkähler geometry.
Volume comparison theorem for rank 1 symmetric spaces proved.
problem Volume comparison for symmetric spaces of non-compact type.
method Normalized Ricci--DeTurck flow to analyze volume functional and derive monotonicity properties.
result Volume comparison theorem established for rank 1 symmetric spaces of non-compact type.
Paper presents a rank-1 approximation method for natural policy gradients in deep RL.
problem Computing natural gradients requires inverting the Fisher Information Matrix, which is computationally expensive.
method Develops a rank-1 approximation to the inverse Fisher Information Matrix for efficient natural policy optimization.
result The rank-1 approximation converges faster and has similar sample complexity to stochastic policy gradient methods.
Research examines rank 1 abelian subgroups in 2-knot groups.
problem Identifying rank 1 abelian normal subgroups in 2-knot groups.
method Analyzes properties of 2-knot groups and their subgroups.
result Either 2-knot groups have no minimal Seifert hypersurface or they are topologically equivalent to a specific example.
Defines foliation criterion for dense isoperiodic leaves in rank 1 affine orbifolds.
problem Dynamics of isoperiodic leaves in rank 1 affine invariant suborbifolds.
method Defines foliation FM and establishes density criterion.
result Establishes criterion for density of isoperiodic leaves.
We study the space of L^2 harmonic forms on complete manifolds with metrics of fibred boundary or fibred cusp type. These metrics generalize the geometric structures at infinity of several different well-known classes of metrics, including asymptotically locally Euclidean manifolds, the (known types of) gravitational i…
Rank-1 BNNs improve efficiency and scalability of Bayesian neural nets.
problem Underfitting and lack of scalability in Bayesian neural networks.
method Propose a rank-1 parameterization of BNNs and use mixture approximate posteriors.
result Rank-1 BNNs achieve state-of-the-art performance across various datasets.
Study on diagonal and separating coordinates for symmetric spaces of rank 1.
problem Existence and nonexistence of diagonal and separating coordinates for symmetric spaces of rank 1.
method Generalization of results by Gauduchon and Moroianu, 2020, and analysis of constant sectional curvature and orthogonal separation of variables.
result Diagonal coordinates exist if and only if the symmetric space has constant sectional curvature.
Study of Martin boundary for rank 1 manifolds with nonpositive curvature.
problem Understanding the Martin boundary for specific types of manifolds.
method Modification of Ancona's argument to analyze the geometric and Martin boundaries.
result A residual set in the geometric boundary corresponds naturally to a subset of the Martin boundary.
We use generalised cross--ratios to prove the Ptolemaean inequality and the Theorem of Ptolemaeus in the setting of the boundary of symmetric Riemannian spaces of rank 1 and of negative curvature.
For a given lattice, we establish an equivalence involving a closed zone of the corresponding Voronoi polytope, a lamina hyperplane of the corresponding Delaunay partition and a quadratic form of rank 1 being an extreme ray of the corresponding L-type domain.
STARK learns structured dictionaries for tensor data.
problem Representing multidimensional data with structured dictionaries.
method Solves a convex relaxation of a nonconvex rank-1 tensor recovery problem.
result Empirical results show promising performance for tensors of any order.
New invariant cusp crossing density shows cusp densities of hyperbolic knots and links are dense.
problem Densities of cusp invariants for hyperbolic knots and links.
method Defined cusp crossing density as ratio of cusp volume to crossing number; showed densities are dense.
result Cusp crossing density for links is bounded above by 3.1263... and is dense in [0, 2.120...].
The article proves a unique invariant measure for geodesic flows on certain rank 1 manifolds.
problem Existence and uniqueness of invariant measure for geodesic flows.
method Using Patterson-Sullivan measure and Busemann density.
result Geodesic flow on compact rank 1 manifolds has a unique invariant measure of maximal entropy.
We study rank 1 flat bundles over solvmanifolds whose cohomologies are non-trivial. By using Hodge theoretical properties for all topologically trivial rank 1 flat bundles, we represent the structure theorem of Kähler solvmanifolds as extensions of Hasegawa's result and Benson-Gordon's result for nilmanifolds.
4-manifolds show every flat 3-manifold as cusp sections.
problem Realizing flat 3-manifolds as cusp sections of hyperbolic 4-manifolds.
method Transitive action on cusps, dense flat metrics realization.
result Existence of many cusp-transitive 4-manifolds.
Survey on discrete subgroups of symmetric spaces with rank 1 behavior.
problem Characterizing discrete subgroups with rank 1 behavior.
method Various characterizations and dynamical/geometric properties.
result Found domains of proper discontinuity and constructed compactifications.
A tensor-based model reduces weight parameters and spatial structure for high-order data classification.
problem High-order data classification with limited training samples and preserved spatial structure.
method Rank-1 FNN model based on modified feedforward neural network with rank-1 canonical decomposition and new learning algorithm.
result The proposed model outperforms state-of-the-art methods, especially in cases with small training samples.
Characterizes holonomies of convex projective cusps.
problem Understanding holonomies in strictly convex projective geometry.
method Complete characterization of holonomies for strictly convex and round cusps, building families of generalized cusps.
result Produces the first example of generalized cusps with non-virtually nilpotent fundamental group.
We consider the heat kernel (and the zeta function) associated with Laplace type operators acting on a general irreducible rank 1 locally symmetric space X. The set of Minakshisundaram- Pleijel coefficients {A_k(X)}_{k=0}^{\infty} in the short-time asymptotic expansion of the heat kernel is calculated explicitly.
A new bandit algorithm finds the max entry of a rank-1 matrix efficiently.
problem Online learning with unobserved values in matrix products.
method Rank1Elim algorithm for stochastic rank-1 bandits.
result Rank1Elim achieves linear regret bound in K+L, 1/Δ, and logn. Study geodesics entering a fixed cusp neighborhood multiple times.
problem Understanding geodesics entering a specific cusp neighborhood multiple times.
method Investigate reciprocal geodesics entering a fixed cusp neighborhood a fixed number of times.
result Characterized the class of reciprocal geodesics entering a fixed cusp neighborhood a fixed number of times.
Proves methods for creating convex projective 3-manifolds with cusps.
problem Creating convex projective 3-manifolds with generalized cusps.
method Properly convex deformations of hyperbolic structures, controlling cusp types.
result First known example of a 1-cusped hyperbolic 3-manifold with a type 2 cusp.
The study classifies Hessian rank 1 hypersurfaces in dimensions 2, 3, and 4.
problem Classifying Hessian rank 1 affinely homogeneous hypersurfaces in specific dimensions.
method Power Series Method of Equivalence, infinitesimal calculations.
result Identified all non-product constant Hessian rank 1 affinely homogeneous hypersurfaces in dimensions 2, 3, and 4.
A new algorithm completes rank-1 tensors with minimal samples and time.
problem Completing rank-1 tensors with minimal samples and time.
method Gauss-Jordan on random linear systems.
result Gauss-Jordan algorithm uses O(d2logd) samples and runs in O(md2) time. The paper classifies different types of cusps on plane curves.
problem Investigating various types of cusps on plane curves.
method Examining criteria for (n,n+1) cusps with differential conditions and relations to evolutes of fronts. result Complete classifications for (4,5)-cusps. The paper characterizes links in 3D from divides with cusps.
problem Characterizing links in 3D from divides with cusps.
method Defines and characterizes divides with cusps and their associated links.
result Every strongly invertible link and 2-periodic link can be described as the link of a divide with cusps.
Extends techniques to show existence of all cusp types in convex projective manifolds.
problem Existence of all cusp types in convex projective manifolds.
method Extension of techniques by Ballas-Marquis.
result Existence of all cusp types in all dimensions except diagonalizable.
Paper proves geodesic ball maximizes second Robin eigenvalue in non-compact symmetric spaces.
problem Maximizing the second Robin eigenvalue in non-compact rank-1 symmetric spaces.
method Quantitative spectral inequality for the second Robin eigenvalue.
result Geodesic ball maximizes the second Robin eigenvalue among domains of the same volume.
The paper explores cusp types in hyperbolic 4-manifolds and their commensurability classes.
problem Understanding cusp types in hyperbolic 4-manifolds and their commensurability.
method Criteria for commensurability classes containing specific cusp types.
result Infinitely many examples of commensurability classes without certain cusp types.
The paper classifies cusp types in hyperbolic knot complements.
problem Classifying cusp types in hyperbolic knot complements.
method Analyzing quotients of hyperbolic knot complements.
result All cusp types arise in the quotients of link complements.
New findings on cusped Borel Anosov representations and their properties.
problem Characterizing and understanding cusped Borel Anosov representations.
method Analyzing representations of lattices in PGL2(R) to PGLd(R). result Cusped Borel Anosov representations with specific properties are Hitchin representations.