Schottky groups constructed from flag manifolds' partial cyclic orders.
problem Constructing Schottky groups from geometric structures.
method Using 3-hyperconvexity and partial cyclic orders on flag manifolds, constructing Schottky groups. result Schottky groups correspond to positive representations in Fock and Goncharov's sense.
This paper solves Yang-Baxter cohomology for cyclic biquandles.
problem Yang-Baxter cohomology of cyclic biquandles.
method Completely determined free parts and computed torsion subgroups of homology groups.
result Upper bounds for torsion orders in higher dimensional homology groups.
This paper extends cyclic branched coverings theory to surfaces with quotient singularities.
problem Develop a theory for cyclic branched coverings of surfaces with quotient singularities.
method Extend Esnault-Viehweg's theory to surfaces with quotient singularities, partially resolve ramification loci, and provide global and local conditions.
result Prove the existence of non-homeomorphic embeddings of cuspidal curves of degree 12 in weighted projective plane.
Exact systole values found for hyperbolic surfaces with specific cyclic symmetries.
problem Finding exact systole lengths for hyperbolic surfaces with maximal cyclic symmetries.
method Analyzing hyperbolic surfaces of different genera with specified cyclic symmetries and calculating exact systole lengths.
result Exact formulas for systole lengths of hyperbolic surfaces with maximal cyclic symmetries.
Let X be a CAT(0) space, and G a discrete cyclic group of isometries of X. We investigate the domain of discontinuity for the action of G on the boundary ∂X.
We describe a construction of Schottky type subgroups of automorphism groups of partially cyclically ordered sets. We apply this construction to the Shilov boundary of a Hermitian symmetric space and show that in this setting Schottky subgroups correspond to maximal representations of fundamental groups of surfaces wit…
Classifies hypersurfaces with specific curvature properties in 4D space.
problem Classifying hypersurfaces with three distinct principal curvatures in 4D space.
method Used classification results for hypersurfaces in R4, S3imesR, and H3imesR to derive new classifications. result Alternative classification of cyclic conformally flat hypersurfaces in R4. Study proves non-left-orderability of 3-manifolds derived from specific knots.
problem Proving non-left-orderability of knot groups in 3-manifolds.
method Analyzing fundamental groups of cyclic branched covers of pretzel knots.
result Non-left-orderability of fundamental groups of n-fold cyclic branched covers of P(3,−3,−2k−1) for all integers k and n≥1. We study the question of when cyclic branched covers of knots admit taut foliations, have left-orderable fundamental group, and are not L-spaces.
We study those Lagrangian surfaces in complex Euclidean space which are foliated by circles or by straight lines. The former, which we call cyclic, come in three types, each one being described by means of, respectively, a planar curve, a Legendrian curve of the 3-sphere or a Legendrian curve of the anti de Sitter 3-sp…
Study on left orderability of specific knot covers.
problem Determining left orderability of knot covers.
method Computing nonabelian SL2(C) character varieties and analyzing real points.
result Left orderability of fundamental groups of cyclic branched covers.
In a recent paper Y. Hu has given a sufficient condition for the fundamental group of the r-th cyclic branched covering of S^3 along a prime knot to be left-orderable in terms of representations of the knot group. Applying her criterion to a large class of two-bridge knots, we determine a range of the integer r>1 for w…
Solves logarithmic ∂-equation on Kähler manifolds with smooth divisors.
problem Closedness of logarithmic forms and injectivity theorems.
method Cyclic covering trick to solve ∂-equation.
result Unobstructed deformations for smooth divisors.
The paper explores circular orderability in 3-manifold groups, related to the L-space conjecture.
problem Circular orderability of 3-manifold groups and its relation to the L-space conjecture.
method Investigation of finite cyclic covers and Dehn surgeries to establish circular orderability.
result Circularly orderable fundamental groups of compact, connected, P^2-irreducible 3-manifolds are characterized.
New findings on L-spaces and left-orderability for specific knot covers.
problem Understanding L-spaces and left-orderability in knot theory.
method Analyzing 3-fold and 5-fold cyclic branched covers of two-bridge knots.
result Fundamental groups of certain knot covers are not left-orderable.
The paper explores left-orderable groups in branched cyclic covers with taut foliations.
problem Left-orderability of fundamental groups in branched cyclic covers.
method Study of left-orderability in cyclic branched covers of links with co-oriented taut foliations.
result Proves left-orderability for specific types of branched covers and Dehn surgeries.
A new framework learns cyclic causal graphs from incomplete data.
problem Learning causal models in systems with feedback loops and missing data.
method MissNODAGS framework, alternating imputation and likelihood maximization.
result Improved performance compared to imputation followed by causal learning.
Classifies cyclic actions on bordered surfaces up to topological conjugation.
problem Classify cyclic actions on bordered surfaces up to topological conjugation.
method Topological conjugation and algebraic genus analysis.
result Classifies cyclic actions on bordered surfaces up to topological conjugation.
We provide an alternative proof of a sufficient condition for the fundamental group of the nth cyclic branched cover of S3 along a prime knot K to be left-orderable, which is originally due to Boyer-Gordon-Watson. As an application of this sufficient condition, we show that for any (p,q) two-bridge knot, wi…
In this paper we use continuous family of multisections of the moduli space of pseudo holomorphic discs to partially improve, in the case of real coefficient, the construction of Lagrangian Floer cohomology of which the author developed jointly with Oh-Ohta-Ono. Namely we associate cyclically symmetric filtered A infin…
Detect slopes in toroidal 3-manifolds to prove properties of fundamental groups.
problem Prove properties of fundamental groups of toroidal 3-manifolds.
method Slope detection using left-orders, foliations, and Heegaard Floer homology.
result Toroidal integer homology spheres have left-orderable fundamental groups.
Study on Hausdorff dimension of lamination endpoints for fully irreducible automorphisms.
problem Hausdorff dimension of lamination endpoints for fully irreducible automorphisms of free groups.
method Analysis of attracting laminations and ending laminations, using properties of hyperbolic surfaces and free-by-cyclic groups.
result For fully irreducible automorphisms, the set of endpoints of the ending lamination has Hausdorff dimension 0.
MissNODAG learns cyclic causal graphs from incomplete data.
problem Causal discovery in systems with feedback loops and missing data.
method Differentiable framework integrating additive noise model and expectation-maximization.
result MissNODAG uncovers cyclic structures and missingness mechanisms from partially observed data.
We prove a Godbillon-Vey index formula for longitudinal Dirac operators on a foliated bundle with boundary; in particular, we define a Godbillon-Vey eta invariant on the boundary-foliation; this is a secondary invariant for longitudinal Dirac operators on type-III foliations. Moreover, employing the Godbillon-Vey index…
Simple mechanism resolves economic puzzles.
problem High cyclically adjusted P/E ratio, wage vs. capital gains, equity premium persistence.
method Simple mechanism noted.
result Simple mechanism partially resolves economic puzzles.
We use controlled topology applied to the action of the infinite dihedral group on a partially compactified plane and deduce two consequences for algebraic K-theory. The first is that the family in the K-theoretic Farrell-Jones conjecture can be reduced to only those virtually cyclic groups which admit a surjection wit…
Researchers solved a complex problem for a specific type of 4-manifolds.
problem Realizing mapping classes of finite order on del Pezzo surfaces.
method Synthesized results from reflection group theory and 4-manifold topology.
result Provided both positive and negative examples of realizability.
The article proves properties of Seifert links and their cyclic branched covers.
problem Properties of Seifert links and their cyclic branched covers.
method Left-orderability, co-oriented taut foliations, and L-space properties. result Proves the ADE link conjecture for Seifert links. Smooth actions of cyclic groups on 3-manifolds are conjugate to smooth ones.
problem Understanding smoothness in group actions on 3-manifolds.
method Finite cyclic group actions by (1+ε)-bilipschitz homeomorphisms on closed 3-manifolds. result Finite cyclic group actions by (1+ε)-bilipschitz homeomorphisms on closed 3-manifolds are conjugate to smooth actions. We prove that given a Hitchin representation in a real split rank 2 group G0, there exists a unique equivariant minimal surface in the corresponding symmetric space. As a corollary, we obtain a parametrization of the Hitchin components by a Hermitian bundle over Teichmüller space. The proof goes through intr…
Let M be an irreducible, compact, connected, orientable 3-manifold whose boundary is a torus. We show that if M is hyperbolic, then it admits at most six finite/cyclic fillings of maximal distance 5. Further, the distance of a finite/cyclic filling to a cyclic filling is at most 2. If M has a non-boundary-paralle…
Resolving Schwartz's quadratic meander number conjecture
problem Meander number of cyclic permutations
method Constructing families of cyclic permutations
result Meander number is bounded above and below quadratically in n
New groups prevent certain geometric actions on spaces.
problem Preventing certain geometric actions on spaces.
method Analyzing cyclic orders on boundaries of trees.
result Groups prevent actions on PD(n) spaces.
The study broadens the concept of cyclic polytopes to Veronese polytopes.
problem Extending the framework of cyclic polytopes to a broader class of polytopes.
method Described facial structure and combinatorial characterisation of facets via σ-parity alternating sequences.
result Established a bijective correspondence between combinatorial types of Veronese polytopes and partitions of finite sets.
The paper removes a condition for the Nielsen realisation problem using equivariant Poincaré duality.
problem The Nielsen realisation problem for aspherical manifolds.
method Application of equivariant Poincaré duality to cyclic groups of prime order.
result Removal of a technical condition in the Nielsen realisation problem.
Toroidal 3-manifolds have special group structures that can be shown through specific covers.
problem Characterizing the fundamental groups of toroidal 3-manifolds.
method Proving toroidal 3-manifolds have circularly-orderable fundamental groups by showing they admit finite cyclic covers with left-orderable fundamental groups.
result Toroidal 3-manifolds have circularly-orderable fundamental groups, which can be shown through specific covers.
The Trek Separation Theorem (Sullivant et al. 2010) states necessary and sufficient conditions for a linear directed acyclic graphical model to entail for all possible values of its linear coefficients that the rank of various sub-matrices of the covariance matrix is less than or equal to n, for any given n. In this pa…
New classification of nonorientable 4-manifolds with specific fundamental groups.
problem Classifying nonorientable 4-manifolds with cyclic fundamental groups.
method Simple cut-and-paste construction, using results from Hambleton-Kreck-Teichner and Khan.
result Plausible classification of a large set of nonorientable 4-manifolds with cyclic fundamental groups of order 2p.
The virtually cyclic dimension of Out(F_N) is finite and related properties are established.
problem Understanding the virtually cyclic dimension of Out(F_N) and related properties.
method Proving properties of finite index congruence subgroups and using exact sequences.
result The virtually cyclic dimension of Out(F_N) is finite.
Solves Nekhoroshev's problem on invariant tori for Hamiltonian systems with cyclic variables.
problem Finding invariant isotropic tori under Hamiltonian phases flows with involution Hamilton functions.
method Constructs monodromy operator and provides conditions for existence and uniqueness of complex germ without simple spectrum condition.
result Full solution to Nekhoroshev's problem, including Hamiltonian systems with cyclic variables.
The paper solves the Nielsen realization problem for cyclic actions on surfaces.
problem Realizing finite order mapping classes as isometries of hyperbolic structures.
method Inductive procedure to construct hyperbolic structures and combinatorial perspective.
result Explicit solutions to the Nielsen realization problem for cyclic subgroups.
The study shows that certain groups can be uniquely identified by their finite abelian summands.
problem Identifying groups based on their finite abelian summands.
method Analyzing hyperbolic groups as graphs of free groups with cyclic edge groups.
result Free products of free and surface groups are profinitely rigid.
Let Σg(g>1) be a closed surface embedded in S3. If a group G can acts on the pair (S3,Σg), then we call such a group action on Σg extendable over S3. In this paper we show that the maximum order of extendable cyclic group actions is 4g+4 when g is even and 4g−4 when g is odd; the maximum ord…
We give an algorithm to compute stable commutator length in free products of cyclic groups which is polynomial time in the length of the input, the number of factors, and the orders of the finite factors. We also describe some experimental and theoretical applications of this algorithm.
We show that a hyperbolic 3-manifold can be the cyclic branched cover of at most fifteen knots in S3. This is a consequence of a general result about finite groups of orientation preserving diffeomorphisms acting on 3-manifolds. A similar, although weaker, result holds for arbitrary irreducible 3-mani…
We calculate a_4 term in heat kernel expansion for noncommutative tori.
problem Calculating the term a_4 in the heat kernel expansion for noncommutative tori.
method Local expression calculation and functional relations derivation.
result Validated the calculated expressions through functional relations and partial differential system.
The study examines Lee metrics on groups and their properties.
problem Characterizing groups that admit Lee metrics.
method Analyzing conditions for groups to have or not have Lee metrics, studying specific families of groups, and providing tables for groups of order ≤ 31.
result Conditions for groups to have Lee metrics, including specific families and non-cyclic groups.
The paper calculates cohomology of complex manifolds with cyclic group actions.
problem Calculating the integral cohomology of complex manifolds with cyclic group actions.
method Investigates toric blow-ups and uses spectral sequences of equivariant cohomology.
result Provides necessary and sufficient conditions for spectral sequence degeneration.