The study shows that several properties are not profinite invariants.
problem Determining which properties are profinite invariants.
method Combining Rips constructions and iterated group-theoretic Dehn filling on hyperbolic virtually special groups.
result Several properties (stable commutator length, quasimorphisms, property NL, property FW∞, property FA, and non-abelian free subgroups) are not profinite invariants. In this paper, we explore several Fatou-type properties of risk measures. The paper continues to reveal that the strong Fatou property, which was introduced in [17], seems to be most suitable to ensure nice dual representations of risk measures. Our main result asserts that every quasiconvex law-invariant functional on…
The paper explores basic properties of knot skein invariants.
problem Understanding skein invariants of knots.
method Discussion of basic properties and known examples of skein invariants.
result Discussion of basic properties and known examples of skein invariants.
Defines linking number and Milnor invariants, their properties.
problem None explicitly stated; definitions and properties are discussed.
method Definitions and properties of linking number and Milnor invariants are discussed.
result Definitions and properties of linking number and Milnor invariants are established.
Boundary properties of hyperbolic groups are invariant under a maximization procedure.
problem Proving boundary properties of hierarchically hyperbolic groups are invariant.
method Proving boundary invariance under a maximization procedure.
result Boundary properties of hierarchically hyperbolic groups are invariant under maximization.
The study examines groups with a specific automorphism property using BNS-invariant.
problem When does a group have the finitely generated fixed subgroup property?
method Using BNS-invariant to analyze groups of the form GimesZm. result Provides partial answers and examples for groups with the property.
The paper characterizes risk measures with the Fatou property in function spaces.
problem Investigating the Fatou property of law-invariant risk measures in function spaces.
method Characterization of the Fatou property using the AOCEA property and dual representations.
result Risk measures with the Fatou property exist under the AOCEA property in most classical model spaces.
Study cohomological and metric properties of non-Kähler complex manifolds.
problem Understanding cohomological invariants and metrics of non-Kähler complex manifolds.
method Partial account of problems through cohomological and metric properties.
result Partial insights into cohomological and metric properties of non-Kähler manifolds.
Groups with certain properties have invariant subalgebra rigidity.
problem Invariant subalgebra rigidity in groups with specific properties.
method Analyzing normal subgroups and invariant subalgebras in groups.
result Torsion-free acylindrically hyperbolic groups and hyperbolic groups have the relative ISR property.
New calculus for pseudodifferential operators on manifolds with cylindrical ends.
problem Analyzing layer potentials on manifolds with cylindrical ends.
method Introducing and studying two classes of pseudodifferential operators.
result Spectrally invariant property of the 'essentially translation invariant calculus'.
Study annular concordance invariants and refine transverse link invariants.
problem Understanding annular concordance and transverse links.
method Defined and studied annular concordance invariant, introduced modified chain complex, and refined transverse invariant.
result Refined transverse invariant and studied its relationship with transverse and braid monodromy properties.
Formula derived for a magnetic line invariant.
problem Solving MHD problems with two-scale mean fields.
method Combinatorial definition of M3 invariant for three-component links. result Proven formula for M invariant verified for simple cases. New method uses cycle consistency to enforce invariance in latent space.
problem Learning meaningful and independent factors of variation in datasets.
method Two separate latent subspaces, cycle consistency constraints, deep information bottleneck.
result Identifies more meaningful factors leading to sparser and interpretable models.
Introduces bounded scale measure and generalizes property A.
problem Defining property A for large scale spaces with bounded geometry.
method Introduces bounded scale measure, shows its coarse invariance, and generalizes property A.
result Definition of property A for large scale spaces with bounded scale measure is a coarse invariant.
Clarifies properties of Ozsvath-Szabo contact invariant.
problem Properties and functoriality of Ozsvath-Szabo contact invariant.
method Explains and proves functoriality properties of the invariant.
result Strong functoriality under Stein cobordisms.
The paper studies properties of triangle and shearing invariants in PSL(n,R) and connects them to a slice of Hitchin components.
problem Understanding invariants of PSL(n,R)-Fuchsian representations and their relationship to Hitchin components.
method Examined triangle and shearing invariants, used Bonahon-Dreyer parameterization.
result The Fuchsian locus of Hitchin components corresponds to a slice.
We prove a continuity property for ending invariants of convergent sequences of Kleinian surface groups. We also analyze the bounded curve sets of such groups and show that their projections to non-annular subsurfaces lie a bounded Hausdorff distance from geodesics joining the projections of the ending invariants.
Sarkar and Wang have given a combinatorial algorithm for computing Heegaard Floer homology and Plamenevskaya has improved their method to compute Ozsvath-Szabo invariant. In this paper, applying the combinatorial method to stabilizations of an open book, we prove basic properties of Ozsvath-Szabo invariant.
Introduced coloring-allowed invariants of planar knotoids with the coloring number.
problem Construction of polynomial invariants of knotoids with signs of crossings.
method Defined coloring-allowed invariants of planar knotoids with the coloring number.
result Discussed the 4-phases functions of coloring-allowed invariants.
We give a 3-page description of the Gassner invariant / representation of braids / pure braids, along with a description and a proof of its unitarity property.
Paper explores closedness properties of convex sets in rearrangement invariant spaces.
problem Closedness properties of law-invariant convex sets in rearrangement invariant spaces.
method Analyzes equivalence of different closedness types in rearrangement invariant spaces.
result Order closedness, σ(X,Xn∼)-closedness and σ(X,L∞)-closedness of a law-invariant convex set are equivalent. Starting from ideas of Furuta, we develop a general formalism for the construction of cohomotopy invariants associated with a certain class of S1-equivariant non-linear maps between Hilbert bundles. Applied to the Seiberg-Witten map, this formalism yields a new class of cohomotopy Seiberg-Witten invariants which hav…
A new Möbius invariant discretization and decomposition of the Möbius energy is proposed.
problem Lack of Möbius invariant discretization and decomposition in existing discrete Möbius energy.
method Proposed a new discretization of Möbius energy that is Möbius invariant and can be decomposed into Möbius invariant components.
result The proposed discretization and decomposition maintain Möbius invariance and converge to the original components in the continuum limit.
Study continuity of phi-invariant for degenerating graphs.
problem Continuity of phi-invariant for degenerating graphs.
method Use Yuan--Zhang's adelic divisors and follow Yuan's globalization of phi-invariants.
result Asymptotic expression of Zhang--Kawazumi's invariants for Riemann surfaces near the boundary of the moduli space.
Study proves all left-invariant contact structures on 3D Lie groups are tight.
problem Characterizing tightness of left-invariant contact structures on 3D Lie groups.
method Riemannian methods and unique factorization property for Lie groups.
result All left-invariant contact structures on 3D Lie groups are tight.
A one-to-one correspondence is drawn between law invariant risk measures and divergences, which we define as functionals of pairs of probability measures on arbitrary standard Borel spaces satisfying a few natural properties. Divergences include many classical information divergence measures, such as relative entropy a…
Study on Riemannian properties of SU_n using bi-invariant metric.
problem Properties of SU_n with a specific bi-invariant metric.
method Analyzes distance, diameter, and geodesics in SU_n.
result Parametrizes minimizing geodesic segments using a complex Grassmannian.
Interpolation can prevent classifiers from having desired invariance properties.
problem Invariance properties in over-parameterized models are often ineffective.
method Theoretical analysis and algorithm design for non-interpolating classifiers.
result Interpolating classifiers cannot satisfy desired invariance properties.
Characterizes graphs with leveled embeddings and introduces new graph invariants.
problem Understanding the properties of leveled embeddings in spatial graphs.
method Characterization of graphs with leveled embeddings, introduction of new invariants.
result Characterization of graphs with low level number and determination of specific invariants for complete graphs and complete bipartite graphs.
Learn invariances in neural networks by optimizing over augmentation parameters.
problem Lack of knowledge about present invariances and their extent in data.
method Parameterize a distribution over augmentations and optimize network parameters and augmentation parameters simultaneously.
result Recover correct set and extent of invariances on various tasks from training data alone.
Counterexample found for Stein property of certain solvable Lie groups.
problem Stein property of simply connected unimodular solvable Lie groups with left-invariant complex structures.
method Constructing a solvable Lie group with specific properties.
result A simply connected solvable Lie group with a left-invariant complex structure whose universal cover is not Stein.
Most neural networks are trained using first-order optimization methods, which are sensitive to the parameterization of the model. Natural gradient descent is invariant to smooth reparameterizations because it is defined in a coordinate-free way, but tractable approximations are typically defined in terms of coordinate…
Sharp lower bounds for modular invariants and Dehn twist coefficients in genus 2 and 3.
problem Finding sharp lower bounds for modular invariants and Dehn twist coefficients.
method Analyzing the relation between fractional Dehn twists and modular invariants, classifying pseudo-periodic maps, and proving rigidity properties.
result Sharp lower bounds for modular invariants and Dehn twist coefficients in genus 2 and 3.
Study magnetic curvature on Lie groups, extending Milnor's work.
problem Exploring magnetic curvatures on Lie groups.
method Computing magnetic curvatures and analyzing algebraic properties.
result Extending results from Milnor's classic paper on left-invariant metrics.
New insights link algebraic and geometric properties of connections.
problem Understanding numerical integration on manifolds.
method Relating invariant connections to Lie algebra actions.
result Generalized classical results for invariant connections on algebroids.
The paper introduces new structures for colored HOMFLY-PT invariants using skein theory.
problem Proving strong integrality and deriving symmetric properties for HOMFLY-PT invariants.
method Purely using HOMFLY-PT skein theory and applying to LMOV conjecture.
result Strong integrality and symmetric properties for colored HOMFLY-PT invariants.
We classify invariant almost complex structures on homogeneous manifolds of dimension 6 with semi-simple isotropy. Those with non-degenerate Nijenhuis tensor have the automorphism group of dimension either 14 or 9. An invariant almost complex structure with semi-simple isotropy is necessarily either of specified 6 homo…
We employ the language of Cartan's geometry to present a model for studying vector spaces of Killing two-tensors defined in pseudo-Riemannian spaces of constant curvature under the action of the corresponding isometry group. We also discuss geometric properties of joint invariants of Killing two-tensors defined in the …
The paper examines properties of self-affine Sierpiński sponges using metric invariants.
problem Investigating properties of self-affine Sierpiński sponges using metric invariants.
method Examined through maximal power law property and perfectly disconnectedness.
result Characterized self-affine Sierpiński sponges by their metric properties.
We define a Floer-homology invariant for links in S3, and study its properties.
Properties of invariant, anti-invariant and slant isometrically immersed submanifolds of metallic Riemannian manifolds are given with a special view towards the induced Σ-structure. Examples of such metallic manifolds are also given.
Study on deformations of Lie groupoid morphisms and their properties.
problem Understanding the deformation theory of Lie groupoid morphisms.
method Established deformation theory, cohomology, and properties of morphisms.
result Invariance and stability properties of morphisms, Morita invariance of cohomology, and simultaneous deformations.
We consider the oscillator group equipped with a bi-invariant Lorentzian metric, and then some geometrical properties of this group i.e. homogeneous Ricci solitons and harmonicity properties of invariant vector fields are obtained. We also determine all vector fields which are critical points for the energy functional …
Abstract: Studies differential systems on compact Lie groups, extending Greenfield and Wallach's methods.
problem Global properties of left-invariant differential systems on compact Lie groups.
method Abstract: Extends Greenfield and Wallach's methods to systems, obtaining characterizations for regularity, range closeness, and cohomology spaces.
result Abstract: Derives generalizations of results and global versions of Caetano and Cordaro's result.
Study on geometric properties of h-conformal semi-invariant submersions.
problem Exploring geometric characteristics of h-conformal semi-invariant submersions.
method Investigation of quaternionic Kähler manifolds and Riemannian manifolds, focusing on integrability of distributions and foliations.
result Established necessary and sufficient conditions for total geodesic submersions and twisted product manifolds.
Twisted Neumann--Zagier matrices for quantum invariants.
problem Constructing quantum invariants from ideal triangulations.
method Define and compute twisted Neumann--Zagier matrices from combinatorics.
result Twisted 1-loop invariant equals adjoint twisted Alexander polynomial.
Grid homology confirms the Upsilon invariant in knot theory.
problem Verifying the equivalence of Upsilon invariants in knot theory.
method Reconstructed Upsilon invariant using grid homology and proved equivalence.
result Upsilon invariants in knot Floer and grid homology are equivalent.
New 4-manifold invariants from Khovanov-Rozansky link homology.
problem Defining invariants for 4-manifolds.
method Using Khovanov-Rozansky gl(N) link homology, constructing skein modules from 4-categories with duals.
result Proof of the sweep-around property for well-defined link homologies in the 3-sphere.