A new compactification of outer space is an absolute retract.
problem Compactification of outer space for FN-actions on R-trees. method Defining a new compactification CVN as a blow-up of the classical compactification CVN, assigning orientations to arcs. result The new compactification CVN is an absolute retract. PACMAN provides bounds for classification tasks considering accuracy vs. negative log-loss mismatch.
problem Mismatch between accuracy and negative log-loss in classification tasks.
method Point-wise PAC approach over generalization gap, using likelihood ratio and concentration inequalities.
result PACMAN provides point-wise PAC bounds for the generalization problem.
Study shows Satake compactifications as horofunctions with polyhedral metrics.
problem Understanding horofunction compactifications of symmetric spaces.
method Investigated invariant Finsler metrics and polyhedral metrics for compactification.
result Generalized Satake compactifications realized as horofunction compactifications.
We construct a triangulation of a compactification of the Moduli space of a surface with at least one puncture that is closely related to the Deligne-Mumford compactification. Specifically, there is a surjective map from the compactification we construct to the Deligne-Mumford compactification so that the inverse image…
The paper constructs examples of compactifications for Einstein metrics.
problem Compactifying Einstein metrics with specific properties.
method Constructing projective and c--projective compactifications. result Neutral signature Einstein metrics can be compactified canonically.
Extends harmonic maps compactification to punctured Riemann surfaces.
problem Compactifying Teichmüller spaces for punctured Riemann surfaces.
method Using harmonic maps rays to extend compactification.
result The compactification still coincides with Thurston's compactification.
Paper relates new compactification to classical moduli space.
problem Relating new compactification to classical moduli space.
method Defining a continuous finite-to-one surjection between compactifications.
result Uniform upper bound on cardinalities of fibers.
The horoboundary of Teichmüller space is path connected and has non-dense Busemann points.
problem Characterizing the horoboundary of Teichmüller space.
method Using the relationship between the horofunction and visual compactifications of Teichmüller spaces.
result The horoboundary of Teichmüller space is path connected and has non-dense Busemann points.
Existence of Kähler-Einstein metrics on compactifications of Lie groups.
problem Existence of Kähler-Einstein metrics on Q-Fano compactifications of Lie groups. method Proving existence through compactifications of Lie groups.
result Classification of Q-Fano compactifications of SO4(C) with Kähler-Einstein metrics. Satake has constructed compactifications of symmetric spaces D=G/K which (under a condition called geometric rationality by Casselman) yield compactifications of the corresponding locally symmetric spaces. The different compactifications depend on the choice of a representation of G. One example is the Baily-Borel-Sata…
Characterizes toroidal and semi-toric compactifications as log minimal models and applies to weak K-moduli.
problem Characterizing and applying toroidal and semi-toric compactifications to weak K-moduli.
method Characterizes toroidal and semi-toric compactifications as log minimal models and applies to weak K-moduli.
result Different proof of a theorem of Alexeev-Engel on weak K-moduli compactifications.
The paper explores ball quotient compactifications and their properties.
problem Smooth toroidal compactifications of ball quotients cannot contain properly holomorphically embedded 3-punctured spheres.
method Use totally geodesic punctured spheres to prove ampleness of KX+αD for α∈(41,1). result First examples of bielliptic ball quotient compactifications are produced.
The paper studies compactifications of SL(2,C) character varieties for punctured surfaces.
problem Compactifying character varieties of punctured surfaces.
method Projective compactifications using ideal triangulations and Komyo's method.
result Boundary divisors are toric varieties and the boundary complex is a sphere.
Compactifies semi-simple Lie groups to manifolds with corners.
problem Compactification of semi-simple Lie groups.
method hd-compactification to a manifold with corners, 1-1 correspondence with conjugacy classes of parabolic subgroups.
result Harish-Chandra's Schwartz space identified with conormal functions of rapid-logarithmic decay.
New coordinates for Teichmüller space compactification.
problem Compactification of Teichmüller space.
method Defined new coordinates.
result Thurston compactification is radial compactification of Euclidean space.
New compactification for character varieties with good topological properties.
problem Compactification of character varieties with good topological properties.
method Announced a new compactification with interpretations of ideal points.
result Relates to Weyl chamber length compactification and applies to maximal and Hitchin representations.
Embeds Higson compactification into adelic solenoids.
problem Embedding Higson compactification into solenoids.
method Induces isomorphism of 1-dimensional cohomology.
result Higson compactification embeds into adelic solenoids.
New memory module improves relational reasoning in neural networks.
problem Standard memory architectures struggle with relational reasoning tasks.
method Introduce a new memory module called Relational Memory Core (RMC) using multi-head dot product attention.
result Achieves state-of-the-art results on various datasets, including WikiText-103, Project Gutenberg, and GigaWord.
We define a compactification of symmetric spaces of noncompact type, seen as spaces of isometry classes of marked lattices, analogous to the Thurston compactification of the Teichmüller space, and we show that it is equivariantly isomorphic to a Satake compactification. We then use it to define a new compactification o…
Researchers create a new compactification of character varieties using geometric and algebraic methods.
problem Compactifying character varieties of finitely generated groups in PSL2(R). method Geometric interpretation of elements of the real spectrum compactification as Γ-actions on R-trees, endowed with an orientation. result Continuous surjection from real spectrum compactification to oriented Gromov equivariant compactification.
Paper studies compactifications of Higgs bundles and self-duality equations.
problem Compactification of Hitchin moduli space and Higgs bundles.
method Analyzes maps between algebraic and analytic compactifications.
result Map between compactifications fails to be continuous at boundary over discriminant locus.
We show that the horofunction compactification of Teichmüller space with the Teichmüller metric is homeomorphic to the Gardiner-Masur compactification.
This review discusses G2-Manifolds and their role in M-Theory compactifications.
problem Understanding the mathematical structure of G2-Manifolds for M-Theory compactifications. method Mathematical and physical considerations of G2-Manifolds and their compactifications. result Progress in constructing and understanding G2-Manifolds. Paper shows geometric properties preserved by compactifications in relation to coarse structures and group actions.
problem Geometric properties preserved by compactifications in relation to coarse structures and group actions.
method Analyzes compactifications of spaces with coarse structures and group actions, proving preservation of geometric properties.
result Geometric properties are preserved by compactifications when coarse structures and group actions are involved.
No natural topological compactification for Fulton-MacPherson.
problem Compactification of configuration spaces on topological manifolds.
method Using recent work on configuration spaces and compactifications.
result Fulton-MacPherson compactification cannot be extended topologically.
The paper classifies and computes limits of equivariant compactifications of groups.
problem Classifying and computing limits of equivariant compactifications of groups.
method Equivariant normal R-test configurations and semistable limits.
result Semistable limits of K-unstable Fano group compactifications are computed.
The paper extends end concepts to arbitrary groups and spaces.
problem Extending end concepts to arbitrary groups and spaces.
method Description of maximal coarse compactification and geometric proof of end properties.
result Generalization of Stallings' theorem and definition of ends for coarse spaces.
The study finds only finitely many Kähler-Einstein compactifications for semisimple groups.
problem Classifying Q-Fano compactifications of semisimple groups with Kähler-Einstein metrics. method Proving finiteness through classification of compactifications.
result There are only finitely many Q-Fano compactifications of semisimple groups with Kähler-Einstein metrics. In this paper we present a topological way of building a compactification of a symmetric space from a compactification of a Weyl Chamber.
To study a noncompact Riemannian manifold, it is often useful to find a compactification. We discuss several common compactifications and survey some recent results.
Paper proves Kähler-Ricci solitons on specific compactifications.
problem Existence of Kähler-Ricci solitons on compact manifolds.
method Continuity method applied to wonderful group compactifications.
result Existence of Kähler-Ricci solitons on certain compactifications.
Consider a finite dimensional (generally reducible) polynomial representation ρof GL_n. A projective compactification of GL_n is the closure of ρ(GL_n) in the space of all operators defined up to a factor (this class of spaces can be characterized as equivariant projective normal compactifications of GL_n). We give an …
The group of direct isometries of the real n-dimensional hyperbolic space is G=SOo(n,1). This isometric action admits many differentiable compactifications into an action on the closed ball. We prove that all such compactifications are topologically conjugate but not necessarily differentiably conjugate. We give the cl…
Schwartz functions smoothly extend to real projective spaces.
problem Identifying functions that extend smoothly to compactifications.
method Showed a similar result for real projective spaces.
result Schwartz functions extend smoothly to real projective spaces.
Study on curvature properties and Shafarevich conjecture for complex hyperbolic manifolds.
problem Existence and nonexistence of Kähler metrics with nonpositive curvature on toroidal compactifications.
method Analysis of toroidal compactifications of finite volume complex hyperbolic manifolds, verification of Shafarevich conjecture.
result Verification of Shafarevich conjecture for compactifications of quotients of complex hyperbolic space by non-uniform arithmetic lattices.
Alexander self-dual complexes yield smooth compactifications of M0,n with explicit Chow rings.
problem Understanding the structure of moduli spaces M0,n using Alexander self-dual complexes. method Explicit description of Chow rings, computation of Chern classes and intersection numbers.
result Derivation of a recursion for intersection numbers in ASD compactifications.
The arc metric is an asymmetric metric on the Teichm{ü}ller space T(S) of a surface S with nonempty boundary. In this paper we study the relation between Thurston's compactification and the horofunction compactification of T(S) endowed with the arc metric. We prove that there is a natural homeomorphism between the two …
Study log canonical thresholds on curved metrics on group compactifications.
problem Computing log canonical thresholds for curved metrics on group compactifications.
method Associated convex functions to metrics, using asymptotic behavior to determine thresholds.
result Formula for alpha invariant in terms of polytope associated to group compactification.
The paper solves a geometric P=W conjecture for SL(2,C) using Thurston's compactification.
problem Addressing the geometric P=W conjecture for SL(2,C) in projective compactifications of character varieties of closed surfaces.
method Using Thurston's compactification of Teichmüller space and new results, the paper constructs a projective compactification of the SL(2,C)-character variety of any closed surface of genus g>1.
result The boundary divisors are toric varieties and the dual intersection complex is a sphere.
Any nonpositively curved symmetric space admits a topological compactification, namely the Hadamard compactification. For rank one spaces, this topological compactification can be endowed with a differentiable structure such that the action of the isometry group is differentiable. Moreover, the restriction of the actio…
The paper explores metrics and compactifications of flat tori.
problem Exploring metrics and compactifications of Teichmüller spaces of flat tori.
method Definition and study of Thurston, Teichmüller, and Weil-Petersson metrics; Thurston-type compactification using measured foliations.
result The Thurston metric is a symmetrization of the Thurston metric, and the Weil-Petersson metric is the Riemannian metric of the symmetric space.
The Thurston compactification of Teichmuller spaces has been generalized to many different representation spaces by J. Morgan, P. Shalen, M. Bestvina, F. Paulin, A. Parreau and others. In the simplest case of representations of fundamental groups of closed hyperbolic surfaces in PSL(2,R), we prove that this compactific…
Geometric compactification for complex structures on Lie groups.
problem Compactifying moduli stack of complex structures on Lie groups.
method Describes a geometric compactification using CR structures transverse to a real foliation.
result Extra points represent CR structures transverse to a real foliation.
Explains compactification for heterotic superstrings on Calabi-Gray manifolds.
problem Compactification of heterotic superstrings with torsion.
method Generalized Calabi-Gray geometry.
result Expository overview of compactification methods.
The paper studies horofunction compactifications of symmetric cones under Finsler distances.
problem Understanding horofunction compactifications of symmetric cones under Finsler distances.
method Establishing a correspondence between horofunction compactifications of symmetric cones and normed spaces, using Thompson and Hilbert distances.
result Explicit extensions of the exponential map and characterizations of horofunctions for Thompson and Hilbert distances.
The paper compactifies stability conditions on curves, akin to Teichmüller theory.
problem Compactifying spaces of stability conditions on curves.
method Constructing a Thurston-like compactification.
result Obtained a compactification of stability conditions space, analogous to Teichmüller theory.
Study compactifications of Hermitian-Yang-Mills connections on projective manifolds.
problem Analyzing compactifications of moduli spaces of Hermitian-Yang-Mills connections.
method Defined a gauge theoretic compactification and compared it with sheaf theoretic compactifications.
result Gauge theoretic compactification endowed with complex analytic space structure.
We compare two combinatorial models for the moduli space of two-dimensional cobordisms: Bödigheimer's radial slit configurations and Godin's admissible fat graphs, producing an explicit homotopy equivalence using a "critical graph" map. We also discuss natural compactifications of these two models, the unilevel harmoni…