Graphs with stronger curvature grow faster.
problem Understanding volume growth on graphs with various curvatures.
method Examined inner-outer and Ricci-Ollivier curvatures to relate them to volume growth.
result Graphs with stronger inner-outer curvature growth have faster volume growth.
Paper proposes a structured approach to improve active learning performance.
problem Improving active learning performance in non-linear feature spaces.
method Approximates version space to a structured hypersphere, divides AL sampling into outer and inner volumes, and uses sequential optimization to globally optimize the kernel space.
result Proves guarantees for AL performance in version space and proposes a new algorithm called Volume-based AL (VAL).
Motivated by Bonahon's result for hyperbolic surfaces, we construct an analogue of the Patterson-Sullivan-Bowen-Margulis map from the Culler-Vogtmann outer space CV(Fk) into the space of projectivized geodesic currents on a free group. We prove that this map is a topological embedding. We also prove that for every $…
Researchers extend monotonicity formulas for harmonic functions in RCD(0,N) spaces.
problem Generalizing monotonicity formulas for harmonic functions in mRCD(0,N) spaces. method New estimates for harmonic functions and a functional version of the outer volume cone theorem.
result Proven rigidity and almost rigidity statements for harmonic functions in mRCD(0,N) spaces. Formula for transgressions on polyhedral manifolds, linking face volumes and outer angles.
problem Computing topological invariants of polyhedral manifolds.
method Defining transgressions for Pfaffian of metric connections and applying to polyhedral manifolds.
result Derivation of an identity linking face volumes and outer angles of spherical and hyperbolic polyhedra.
A remarkable result of McShane states that for a punctured torus with a complete finite volume hyperbolic metric we have \[ \sum_γ \frac{1}{e^{\ell(γ)}+1}={1/2} \] where γ varies over the homotopy classes of essential simple closed curves and ℓ(γ) is the length of the geodesic representative of γ. We prove tha…
Outer billiards maps on foliated surfaces with specific vector fields.
problem Characterizing vector fields that induce outer billiards maps on foliated surfaces.
method Analyzing necessary and sufficient conditions for foliation of the exterior of a hypersurface.
result Explicit periodic and unbounded orbits in a specific outer billiard map.
Computes the outer mass of small metric spheres in time-symmetric slices.
problem Computing the outer mass of small metric spheres in time-symmetric slices.
method Analyzes the deviation from a standard sphere, estimates mass of a static vacuum extension, and uses geodesic ball shrinking to a point as an application.
result Outer mass to first order in the data's deviation from the standard sphere, with an upper bound obtained by estimating the mass of a static vacuum extension.
Outer space and Teichmüller space fail well-rounded retract analogy.
problem Failure of well-rounded retract in Outer space and Teichmüller space.
method Analysed flat tori and metric graphs to show failure.
result Analogue of well-rounded retract does not contain equivariant spine.
We provide an effective algorithm for determining whether an element of the outer automorphism group of a free group is fully irreducible. Our method produces a finite list which can be checked for periodic proper free factors.
We define cusp-decomposable manifolds and prove smooth rigidity within this class of manifolds. These manifolds generally do not admit a nonpositively curved metric but can be decomposed into pieces that are diffeomorphic to finite volume, locally symmetric, negatively curved manifolds with cusps. We prove that the gro…
Convex hypersurfaces evolve to spheres under a specific flow.
problem Volume preserving nonhomogeneous mean curvature flow of convex hypersurfaces.
method Monotonicity of isoperimetric ratio, inner and outer radius control, maximum principle arguments.
result Closed convex hypersurfaces converge to round spheres.
Domain Fusion uses GANs to augment data for low-volume target datasets.
problem High costs in data development for deep learning applications.
method Multi-domain learning GANs to generate new samples.
result Domain Fusion achieves better classification accuracy with less data.
We show that the aspherical manifolds produced via the relative strict hyperbolization of polyhedra enjoy many group-theoretic and topological properties of open finite volume negatively pinched manifolds, including relative hyperbolicity, nonvanishing of simplicial volume, co-Hopf property, finiteness of outer automor…
Study shows smooth convergence of round surfaces in flat space-time models.
problem Volume preserving mean curvature flow of round surfaces in asymptotically flat spaces.
method Volume preserving mean curvature flow in asymptotically flat 3-manifolds.
result The flow converges smoothly to a stable CMC surface.
We study spaces obtained from a complete finite volume complex hyperbolic n-manifold M by removing a compact totally geodesic complex (n-1)-submanifold. The main result is that the fundamental group of M-S is relatively hyperbolic, relative to fundamental groups of the ends of M-S, and M-S admits a complete finite volu…
Outer automorphism groups of Coxeter groups are trivial except for small ranks.
problem Triviality of outer automorphism groups of Coxeter groups.
method Using Guirardel-Levitt outer space for free products, proving triviality and cyclic order for specific ranks.
result Outer automorphism groups are trivial except for small ranks.
New Steiner formula for Lp affine surface area in Minkowski theory.
problem Developing a new Steiner formula for Lp affine surface area. method Proving a new Steiner formula for the Lp affine surface area of a Minkowski outer parallel body. result New curvature measures with properties not previously seen in literature.
Study on symmetric automorphisms of RAAGs, proving finiteness properties and contractibility.
problem Finiteness properties and contractibility of symmetric automorphisms of RAAGs.
method Definition of symmetric automorphism group, construction of symmetric Outer space, proof of contractibility.
result Finiteness properties and contractibility results for symmetric automorphisms of RAAGs.
New insights into optimizing Local SGD's outer optimizer for faster convergence.
problem Understanding the impact of outer optimizer and its hyperparameters in Local SGD.
method Analyzing convergence guarantees with new outer learning rates and momentum.
result Tuning the outer learning rate can improve convergence and handle inner learning rate ill-tuning.
Random free group outer automorphisms are geometric and have nongeometric attracting trees.
problem Understanding the structure of random outer automorphisms of free groups.
method Analyzing the Whitehead graph and ideal Whitehead graph of random outer automorphisms.
result The attracting tree of a random outer automorphism is a nongeometric R-tree with all branch points trivalent.
Criterion for subgroup separability in outer automorphism groups.
problem Subgroup separability in outer automorphism groups.
method Criterion for separability of subgroups.
result Strengthening and generalizing a previous result on mapping class groups.
Estimates dimensions of maximal simplices for rational and irrational trees in Outer space.
problem Understanding the structure of trees in Outer space.
method Associate simplices to R-trees and estimate their dimensions. result Estimates the dimensions of maximal simplices for both rational and irrational trees.
Outer automorphisms act loxodromically on a complex.
problem Understanding the dynamics of outer automorphisms on relative free factor complexes.
method Proving loxodromic action and north-south dynamics.
result Fully irreducible outer automorphisms act loxodromically on the relative free factor complex.
We give estimates on the length of paths defined in the sphere model of outer space using a surgery process, and show that they make definite progress in some sense when they remain in some thick part of outer space. To do so, we relate the Lipschitz metric on outer space to a notion of intersection numbers.
For n>3 we study spaces obtained from finite volume complete real hyperbolic n-manifolds by removing a compact totally geodesic submanifold of codimension two. We prove that their fundamental groups are relative hyperbolic, co-Hopf, biautomatic, residually hyperbolic, not Kähler, not isomorphic to lattices in virtually…
Random trees found in Outer space boundary.
problem Understanding trees in the boundary of Outer space.
method Proving properties of harmonic measure on random walks.
result Typical tree is trivalent and nongeometric.
Study improves curvature estimate for stable marginally outer trapped hypersurfaces with a free boundary.
problem Curvature estimate for stable marginally outer trapped hypersurfaces with a free boundary.
method Iteration argument based on uniform area bound.
result Improved curvature estimate for stable marginally outer trapped hypersurfaces.
The paper shows symmetries of a geometric space for Coxeter groups.
problem Understanding symmetries in the Outer space of a Coxeter group.
method Analyzing the geometric rigidity of the universal Coxeter group of rank n.
result For n ≥ 4, the symmetries of the spine of the outer space are only the outer automorphisms.
Study of non-homogeneous mean curvature flow in hyperbolic space converging to a geodesic sphere.
problem Volume/area preserving curvature flow of convex hypersurfaces in hyperbolic space.
method Generic positive, increasing mean curvature velocity; preserving convexity by horospheres; maximum principle; exponential convergence analysis.
result Exponential convergence to a geodesic sphere.
The paper extends Lipschitz metric isometries between Outer Spaces of virtually free groups.
problem Extending isometry properties of Lipschitz metric to virtually free groups.
method Analyzing finite-index subgroups and their covers, identifying folding paths, and using deformation retraction.
result Existence of candidates for Lipschitz distance and deformation retraction of spine.
In this paper we generalize the main result of [13] in two different situations: in the first case for MOTSs of genus greater than one and, in the second case, for MOTSs of high dimension with negative σ-constant. In both cases we obtain a splitting result for the ambient manifold when it contains a stable closed MOT…
We study two sorts of actions on the space of conjugacy classes of irreducible SU2-representations of a knot group. One of them is an involution which comes from the algebraic structure of SU2 and the other is the action by the outer automorphism group of the knot group. In particular, we consider them on an 1-di…
Outer automorphisms of free products are represented by CTs.
problem Representing outer automorphisms of free products by relative train track maps.
method Developed theory of attracting laminations and CTs for free products.
result Outer automorphisms of free products satisfy an index inequality.
We show that any vacuum initial data set containing a marginally outer trapped surface S and satisfying a "no KIDs" condition can be perturbed near S so that S becomes strictly outer trapped in the new vacuum initial data set. This, together with the results in [9], gives a precise sense in which generic initial data c…
Unfolding paths in Outer space accumulate on a simplex, not converge.
problem Understanding accumulation points in Outer space.
method Constructing an unfolding path in Outer space.
result Unfolding paths accumulate on a 1-simplex, not converge.
Subsurface projection has become indispensable in studying the geometry of the mapping class group and the curve complex of a surface. When the subsurface is an annulus, this projection is sometimes called relative twisting. We give two alternate versions of relative twisting for the outer automorphism group of a free …
We study the asymmetry of the Lipschitz metric d on Outer space. We introduce an (asymmetric) Finsler norm that induces d. There is an Out(F_n)-invariant potential Ψon Outer space such that when the Lipschitz norm is corrected by the derivative of Ψ, the resulting norm is quasisymmetric. As an application, we give new …
Outer automorphism group of hyperbolic groups is HHG under certain conditions.
problem Characterizing the outer automorphism group of hyperbolic groups.
method Proving finite-index subgroups are central extensions of orbifold mapping class groups with bounded Euler class.
result Outer automorphism group of a one-ended hyperbolic group is virtually a hierarchically hyperbolic group.
Making use of Murakami's classification of outer involutions in a Lie algebra and following the Morse-theoretic approach to harmonic two-spheres in Lie groups introduced by Burstall and Guest, we obtain a new classification of harmonic two-spheres in outer symmetric spaces and a Weierstrass-type representation for such…
Defines fibered commensurability for outer automorphisms of free groups, proving uniqueness of minimal elements.
problem Understanding symmetry of outer automorphisms of free groups.
method Defines fibered commensurability and proves uniqueness of minimal elements for certain classes of outer automorphisms.
result For certain outer automorphisms, there is a unique minimal element in each fibered commensurability class.
We show that every virtually torsion-free subgroup of the outer automorphism group of a conjugacy separable relatively hyperbolic group is residually finite. As a direct consequence, we obtain that the outer automorphism group of a limit group is residually finite.
Study shows RAAG automorphisms and outer automorphisms are not relatively hyperbolic.
problem Characterizing automorphism and outer automorphism groups of RAAGs.
method Analyzing groups of RAAGs with at least 3 vertices, categorizing based on graph structure.
result Automorphism and outer automorphism groups of RAAGs are not relatively hyperbolic.
The paper develops a method to create non-asymptotic confidence ellipsoids for linear regression without strong noise distribution assumptions.
problem Constructing reliable confidence regions for linear regression with finite sample sizes and general noise distributions.
method The paper introduces the SPS EOA algorithm to create non-asymptotically guaranteed confidence ellipsoids for linear regression problems.
result The sizes of SPS outer ellipsoids are shown to decrease at the optimal rate for linear regression problems.
Outer billiards studied in complex hyperbolic plane, proving smooth and symplectic properties.
problem Outer billiards in complex hyperbolic plane.
method Symplectic form and geodesic foliation analysis.
result Outer billiard map is a diffeomorphism and symplectomorphism.
Random walks on free groups reveal asymmetric expansion factors.
problem Understanding expansion factors in free groups.
method Random walks and BGIP on metric spaces.
result Generic outer automorphisms have different forward and backward expansion factors.
Geometric model for a specific group in Artin groups.
problem Understanding the structure of outer automorphism groups in Artin groups.
method Proving proper cocompact action on a subcomplex of the spine of outer space.
result U(A; G, Ht) acts properly cocompactly on a finite-dimensional subcomplex.
Two-dimensional Yang-Mills theory defects and orbifolds studied.
problem Discrete symmetries and their defects in Yang-Mills theory.
method Path integral over twisted G-bundles, orbifold construction, defect network approach.
result Exact computation of gauge theory partition function with defects.