Paper constructs geometric transitions between hyperbolic and Anti-de Sitter geometries.
problem Transitioning between hyperbolic and Anti-de Sitter geometries.
method Construction via Half-pipe geometry on ΣimesS1 with cone singularities. result Deformation of convex core structure as bending laminations collapse.
Constructs geometric decompositions for thick hyperbolic 3-manifolds with bounded rank.
problem Understanding the structure of thick hyperbolic 3-manifolds with bounded rank.
method Geometric decomposition of the convex core of M.
result Upper bounds on Heegaard genus and radius of embedded balls in terms of rank and injectivity radius.
We introduce the notion of the visual core of a hyperbolic 3-manifold N and explore its basic properties. The visual core can be thought of as a harmonic analysis analogue of the convex core. We investigate circumstances in which the visual core of a cover N' of N embeds under the covering map from N' to N. We apply th…
Totally geodesic submanifolds in convex cores are properly immersed and have finite volume.
problem Characterizing totally geodesic submanifolds in geometrically finite manifolds.
method Analysis of totally geodesic submanifolds in the convex core of geometrically finite rank-one locally symmetric manifolds.
result Every maximal totally geodesic submanifold of dimension at least two in the convex core is properly immersed and has finite volume, and only finitely many such submanifolds can occur.
Classifies geodesic planes outside convex core of geometrically finite 3-manifolds.
problem Classifying geodesic planes in geometrically finite 3-manifolds.
method Constructive proof involving exotic rays and roofs.
result Existence of exotic roofs depends on the existence of exotic rays and bending lamination properties.
The paper bounds eigenvalues of hyperbolic manifolds with infinite volume.
problem Bounding eigenvalues of geometrically finite hyperbolic manifolds of infinite volume.
method Provided a lower bound on the kth eigenvalue of the Laplace-Beltrami operator by the kth eigenvalue of a neighborhood of the thick part of the convex core.
result Recovered a theorem bounding the bottom eigenvalue from below by a specific formula involving the volume of the 1-neighborhood of the convex core.
The Hessian of the renormalized volume of geometrically finite hyperbolic 3-manifolds without rank-1 cusps, computed at the hyperbolic metric g with totally geodesic boundary of the convex core, is shown to be a strictly positive bilinear form on the tangent space to Teichmüller space. The metric g is known fro…
In 3-dimensional hyperbolic geometry, the classical Schlafli formula expresses the variation of the volume of a hyperbolic polyhedron in terms of the length of its edges and of the variation of its dihedral angles. We prove a similar formula for the variation of the volume of the convex core of a geometrically finite h…
The abstract proves spherical surface decompositions with conical singularities.
problem Decomposing surfaces with spherical metrics and conical singularities.
method Geometric triangulations and irreducible components of standard shapes.
result Spherical polygons, including half-spherical concave polygons, can be arbitrarily complicated.
We establish a sharp geometric constant for the upper bound on the resonance counting function for surfaces with hyperbolic ends. An arbitrary metric is allowed within some compact core, and the ends may be of hyperbolic planar, funnel, or cusp type. The constant in the upper bound depends only on the volume of the cor…
Geodesic planes in 3-manifolds are either closed or dense.
problem Characterize geodesic planes in geometrically finite acylindrical 3-manifolds.
method Analyzes geodesic planes in the interior of convex cores of 3-manifolds.
result Geodesic planes are either closed or dense, with countably many closed ones.
Estimates covariance matrices for matrix-variate data via core covariance geometry.
problem Estimating covariance matrices for matrix-variate data with partial isotropy.
method Fixed-rank core covariance geometry, partial-isotropy rank-r core shrinkage estimator.
result The geometry of the space of rank-r cores is a smooth manifold.
Introduces Levi core for CR manifolds, linking it to global invariants.
problem Understanding global invariants of CR manifolds.
method Introduces Levi core, relates to Diederich-Fornæss index and D'Angelo class.
result Levi core is trivial under certain conditions, nontrivial otherwise.
We consider hyperbolic structures on the compression body C with genus 2 positive boundary and genus 1 negative boundary. Note that C deformation retracts to the union of the torus boundary and a single arc with its endpoints on the torus. We call this arc the core tunnel of C. We conjecture that, in any geometrically …
We extend the concept of renormalized volume for geometrically finite hyperbolic 3-manifolds, and show that is continuous for geometrically convergent sequences of hyperbolic structures over an acylindrical 3-manifold M with geometrically finite limit. This allows us to show that the renormalized volume attains its…
Combines topological and geometric approaches to data analysis.
problem Understanding when and how geometric objects intersect.
method Connects topological and geometric concepts of curvature.
result Reconceptualizes curvature and links it to hyperconvexity.
We show that the interior of the convex core of a quasifuchsian punctured-torus group admits an ideal decomposition (usually an infinite triangulation) which is canonical in two different senses: in a combinatorial sense via the pleating invariants, and in a geometric sense via an Epstein-Penner convex hull constructio…
We generalize subset currents on hyperbolic groups to surfaces.
problem Generalizing subset currents to surfaces.
method Developed the theory of subset currents on π_1(Σ), proving they are a measure-theoretic completion of conjugacy classes of subgroups.
result The space of subset currents on Σ is a measure-theoretic completion of conjugacy classes of non-trivial subgroups, each geometrically corresponding to a convex core.
Study on mapping class groups of non-orientable surfaces, proving some conjectures and refuting others.
problem Analogies between Fuchsian groups and mapping class groups of non-orientable surfaces.
method Analyzing limit sets, foliations, and geometric properties.
result Established parts of a conjecture about the limit set and provided evidence for and against the analogy.
This paper explores how pairs of multicurves can be realized as cylinders on translation surfaces.
problem Understanding when pairs of multicurves can be realized as cylinders on translation surfaces.
method Surface topology and flat grafting deformation.
result Pairs of multicurves can be realized as cylinders on some translation surface.
By means of an analogy with Classical Mechanics and Geometrical Optics, we are able to reduce Lagrangians to a kinetic term only. This form enables us to examine the extended solution set of field theories by finding the geodesics of this kinetic term's metric. This new geometrical standpoint sheds light on some founda…
The paper proves unique determination of Dehn fillings in hyperbolic 3-manifolds.
problem Unique determination of Dehn fillings in hyperbolic 3-manifolds with non-symmetric cusps.
method Analyzes core geodesics and their holonomies in Dehn fillings of hyperbolic 3-manifolds.
result Dehn fillings with sufficiently large coefficients are uniquely determined by the product of core geodesics' holonomies.
Profinite rigidity proven for many hyperbolic manifolds.
problem Profinite rigidity of hyperbolic manifolds.
method Geometric topology and bubble-drilling construction.
result Profinite rigidity of many cusped hyperbolic manifolds.
Study how geometric properties of anti-de Sitter structures degenerate along specific paths.
problem Degeneration of geometric properties in anti-de Sitter structures.
method Parameterization of deformation space by Teichmüller space, study of geometric quantities along quadratic differential rays.
result Geometric properties like Hausdorff dimension, core width, and Hölder exponent degenerate along specific paths.
We introduce a coarse combinatorial description of the Weil-Petersson distance d_WP(X,Y) between two finite area hyperbolic Riemann surfaces X and Y. The combinatorics reveal a connection between Riemann surfaces and hyperbolic 3-manifolds conjectured by Thurston: the volume of the convex core of the quasi-Fuchsian man…
Suppose that N is a geometrically finite orientable hyperbolic 3-manifold. Let P(N,C) be the space of all geometrically finite hyperbolic structures on N whose convex core is bent along a set C of simple closed curves. We prove that the map which associates to each structure in P(N,C) the lengths of the curves in the b…
New geometric system from Hessian operators offers solutions to geometric problems.
problem Solving geometric problems using Hessian operators.
method Introducing a new differential-geometric system based on m-Hessian operators. result Deduced an a priori C1-estimate for solutions to the Dirichlet problem for m-Hessian equations. We study the spectral theory of asymptotically hyperbolic manifolds with ends of warped product type. Our main result is an upper bound on the resonance counting function with a geometric constant expressed in terms of the respective Weyl constants for the core of the manifold and the base manifold defining the ends.
Paper studies embedding conditions for homogeneous quandles.
problem Embedding problem of homogeneous quandles.
method Necessary and sufficient condition for quandle homomorphisms to be embeddings.
result Generalization of embedding theorem for generalized Alexander quandles.
This review explores Ricci soliton inequalities in Riemannian geometry.
problem Understanding geometric and analytic characteristics of Riemannian manifolds.
method Comprehensive study of Ricci soliton inequalities, summarizing historical evolution and current developments.
result Complex interactions between curvature conditions and geometric inequalities.
It is unknown whether an unknotting tunnel is always isotopic to a geodesic in a finite volume hyperbolic 3-manifold. In this paper, we address the generalization of this problem to hyperbolic 3-manifolds admitting tunnel systems. We show that there exist finite volume hyperbolic 3-manifolds with a single cusp, with a …
The paper improves alignment methods for deep neural networks using geometric and spectral analysis.
problem Improving alignment methods for deep neural networks.
method Geometric and spectral analysis of residual Jacobian chains.
result Deterministic and margin-verified results on the transport of dominant singular subspaces across layers.
The first part of this paper discusses general procedures for finding numerical approximations to distinguished Kahler metrics, such as Calabi-Yau metrics, on complex projective manifolds. These procedures are closely related to ideas from Geometric Invariant Theory, and to the asymptotics of high powers of positive li…
We show how supersymmetry conditions for flux compactifications of supergravity and string theory can be described in terms of a flat subalgebra of the Kahler-Atiyah algebra of the compactification space, a description which has wide-ranging applications. As a motivating example, we consider the most general M-theory c…
This paper gives a quantitative version of Thurston's hyperbolic Dehn surgery theorem. Applications include the first universal bounds on the number of non-hyperbolic Dehn fillings on a cusped hyperbolic 3-manifold, and estimates on the changes in volume and core geodesic length during hyperbolic Dehn filling. The proo…
Classifies horocycle flow closures in hyperbolic 3-manifolds.
problem Classifying horocycle flow closures in hyperbolic 3-manifolds.
method Classifies orbit closures of the 1-dimensional horocycle flow on the frame bundle of M.
result The closure of a horocycle in M is a properly immersed submanifold.
Theoretical study explains grokking in neural networks.
problem Understanding the abrupt transition from fitting to generalizing in neural networks.
method Characterized a shell-core topological configuration of the solution space induced by Adam's optimization dynamics.
result Derived grokking scaling laws for learning rate, batch size, and regularization coefficient.
As an example of the transitions between some of the eight geometries of Thurston, investigated before, we study the geometries supported by the cone-manifolds obtained by surgery on the trefoil knot with singular set the core of the surgery. The geometric structures are explicitly constructed. The most interesting phe…
Secure linear regression at speed of plaintext methods.
problem Secure multiparty linear regression and feature selection.
method Distributed algorithms combining geometric ideas.
result Efficient and secure genome-wide association studies.
This paper establishes an equivalence between transitive double Lie algebroids and core diagrams.
problem Understanding and characterizing transitive double Lie algebroids.
method Using core diagrams and equivalence of transitive core diagrams with transitive double Lie groupoids.
result Transitive double Lie algebroids are completely determined by their core diagrams.
New adapted renormalized volume for hyperbolic 3-manifolds with compressible boundary.
problem Analyzing convex co-compact hyperbolic 3-manifolds with compressible boundaries.
method Defining and analyzing a new version of the renormalized volume.
result The adapted renormalized volume is bounded and has properties analogous to the classical renormalized volume.
Study on geometric variational problems for existence, regularity, and uniqueness of solutions.
problem Geometric variational problems, focusing on existence, regularity, and uniqueness of solutions.
method Formulated in Federer and Fleming's theory of currents, discussed the existence theory, and presented core ideas of the (interior) regularity theory for area-minimizing currents and optimal transport paths. Two original results on generic uniqueness of solutions were presented.
result Generic uniqueness of solutions for both Plateau's problem and optimal branched transport problem.
We prove hyperbolic 3-manifolds are geometrically inflexible: a unit quasiconformal deformation of a Kleinian group extends to an equivariant bi-Lipschitz diffeomorphism between quotients whose pointwise bi-Lipschitz constant decays exponentially in the distance form the boundary of the convex core for points in the th…
Recovering core nodes in hypergraphs from fringe interactions.
problem Recovering core nodes from fringe interactions in hypergraphs.
method Modeling core recovery as a hitting set problem in hypergraphs, developing a practical algorithm.
result Demonstrated the effectiveness of the algorithm on real-world datasets.
Minimal surfaces in hyperbolic 4-space degenerate to core of product trees.
problem Degeneration of minimal surfaces in hyperbolic 4-space.
method Study of induced metrics and geometric interpretation of minimal surfaces.
result Limits of minimal surfaces are mixed structures and cores of product trees.
Study reveals multiple core-periphery structures in interbank markets, transforming during financial crises.
problem Understanding the complex structure and transformation of interbank markets during financial crises.
method Novel core-periphery detection method on eMID interbank market data.
result Interbank markets exhibit multiple core-periphery pairs and transition to bipartite structures over short time scales.
New peripheral structure for core groups detects noninvertible knots.
problem Detecting noninvertible knots and links.
method Introduced a new peripheral structure for core groups.
result The new structure detects noninvertibility of some knots and links.
ALℓ0CORE tensor decomposition reduces computational cost for sparse count data.
problem Efficiently decompose sparse count data matrices.
method Probabilistic Tucker decomposition with ℓ0-norm constraint. result ALℓ0CORE achieves similar results to full Tucker decomposition at a fraction of the cost.