Decomposes axis bundles into cubist structures for fully irreducible outer automorphisms.
problem Understanding the geometry and structure of axis bundles for fully irreducible outer automorphisms.
method Develops a 'cubist' decomposition into branched cubes with special combinatorics.
result Locates a canonical finite collection of periodic fold lines in each axis bundle.
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 show how to derive hyperbolicity of the free factor complex of FN from the Handel-Mosher proof of hyperbolicity of the free splitting complex of FN, thus obtaining an alternative proof of a theorem of Bestvina-Feighn. We also show that under the natural map τ from the free splitting complex to free factor co…
It is well-known that a point T∈cvN in the (unprojectivized) Culler-Vogtmann Outer space cvN is uniquely determined by its \emph{translation length function} ∣∣.∣∣T:FN→R. A subset S of a free group FN is called \emph{spectrally rigid} if, whenever T,T′∈cvN are such that $||g||_T=||g||_…
When two free factors A and B of a free group F_n are in "general position" we define the projection of B to the splitting complex (alternatively, the complex of free factors) of A. We show that the projections satisfy properties analogous to subsurface projections introduced by Masur and Minsky. We use the subfactor p…
We generalise the Karrass-Pietrowski-Solitar and the Nielsen realisation theorems from the setting of free groups to that of free products. As a result, we obtain a fixed point theorem for finite groups of outer automorphisms acting on the relative free splitting complex of Handel--Mosher and on the outer space of a fr…
New theorem on subgroup dynamics of Out(F_N).
problem Understanding subgroups of Out(F_N).
method Analogous to Ivanov's and Handel-Mosher's theorems.
result Subgroups either contain atoroidal elements or fix conjugacy classes.
We let φ be an ageometric fully irreducible outer automorphism so that its Handel-Mosher axis bundle consists of a single unique axis. We show that the centralizer Cen(⟨φ⟩) of the cyclic subgroup generated by φ equals the stabilizer Stab(Λφ+) of the attracting lamina…
We provide an example in each rank of an ageometric fully irreducible outer automorphism whose ideal Whitehead graph has a cut vertex. Consequently, we show that there exist examples in each rank of Handel-Mosher axis bundles that are not just a single axis, as well as of "nongeneric" behavior in the sense of the "trai…
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 …
Hierarchically hyperbolic spaces provide a common framework for studying mapping class groups of finite type surfaces, Teichmüller space, right-angled Artin groups, and many other cubical groups. Given such a space X, we build a bordificationcompatible with the hierarchically hyperbolic structure. If $\mathc…
The study examines aperiodicity properties of automorphism groups of free products of groups.
problem Investigating aperiodicity properties of automorphism groups of free products of groups.
method Analyzing the subgroup of Out(G) preserving conjugacy classes and proving aperiodicity properties.
result The group IA(G, G, 3) is torsion-free and has notable aperiodicity properties.
New condition prevents hyperbolic spaces from matching curve complexes.
problem Identifying when hyperbolic spaces cannot match curve complexes.
method Analyzing specific hyperbolic complexes and identifying a condition.
result Identified a condition preventing quasi-isometry between hyperbolic spaces and curve complexes.
Study on complex line fields on almost-complex manifolds, proving existence conditions.
problem Existence of linearly independent complex line fields on almost-complex manifolds.
method Prove necessary and sufficient conditions for the existence of one, two, or three fields over certain manifolds.
result Necessary and sufficient condition for the existence of complex line fields over certain manifolds.
Study complex deformations of compact complex surfaces in Calabi-Yau four-folds.
problem Explaining why complex and Cayley deformations of a compact complex surface are the same.
method Study complex deformations of compact complex submanifolds of Calabi-Yau manifolds.
result Prove that the moduli space of complex deformations of any compact complex embedded submanifold of a Calabi-Yau manifold is a smooth manifold.
Homotopy types of curve and arc complexes are studied.
problem Understanding the homotopy types of curve and arc complexes.
method Proving homotopy equivalence and contractibility of complexes.
result Fine curve complex is homotopy equivalent to curve complex, fine arc complex is contractible.
This research explores complex-valued neural networks and their implementation.
problem The challenges of implementing complex-valued neural networks and their potential for non-complex data.
method Detailed theory and implementation of CVNN, including Wirtinger calculus, complex backpropagation, and modules like complex layers and activation functions. Python implementation using cvnn toolbox.
result Demonstrates the potential of CVNN for non-complex data through simulations.
Paper introduces fat CW complexes including all closed manifolds.
problem No specific problem stated, focuses on introducing new CW complexes.
method Introduces a new smooth version of CW complexes called fat CW complexes.
result Fat CW complexes include all closed manifolds and have desirable properties.
The paper discusses q-deformations of the Aomoto complex.
problem Deformation of cochain complexes associated with hyperplane arrangements.
method Replaces entries of coboundary maps with q-analogues and analyzes the resulting structures. result The q-deformation can be a cochain complex under certain conditions and yields local system cohomology groups. Study calculates global sections on complex curves.
problem Global sections of chiral de Rham complexes on complex curves.
method Calculation on closed complex curves with genus g ≥ 2.
result Space of global sections determined.
The paper studies lifts of complex structures on a manifold.
problem Understanding higher-order lifts of extended almost complex structures.
method Proved theorems on Nijenhuis tensor and introduced a new tensor field.
result Basic results on almost analytic complex vectors are investigated.
In this paper, we first provide an updated survey of the geometry of complex Cartan spaces. New characterizations for some particular classes of complex Cartan spaces are pointed out, e.g. Landsberg-Cartan, strongly Berwald-Cartan and others. We introduce the Cartan-Randers spaces which offer examples of Berwald-Cartan…
The paper defines and constructs almost complex blow-ups on 4D almost complex manifolds.
problem Existence and uniqueness of almost complex blow-ups on almost complex manifolds.
method Definition and construction of almost complex blow-ups, proving their existence and uniqueness.
result Existence and uniqueness of almost complex blow-ups on 4D almost complex manifolds.
Study L2 Hilbert complexes on complex manifolds.
problem Analyse L2 Hilbert complexes on complex manifolds. method Define and study L2 Aeppli-Bott-Chern Hilbert complex; examine properties on various manifolds; use self-adjoint extensions of differential operators. result Kernels of operators on compact Hermitian manifolds are isomorphic to Aeppli or Bott-Chern cohomology.
Odd m-fold connected sums of complex projective spaces admit almost complex structures.
problem Existence of almost complex structures on connected sums of complex projective spaces.
method Analyzing the m-fold connected sum m#CP2n for m odd or even. result Only odd m-fold connected sums of complex projective spaces admit almost complex structures.
Research shows arc complex is not quasi-isometric to sphere complex.
problem Comparing quasi-isometry of arc complex and sphere complex.
method Simple proof of quasi-isometric rigidity of arc complex.
result Arc complex is not quasi-isometric to sphere complex.
Study Hodge-de Rham numbers for almost complex 4-manifolds, extending properties from complex surfaces.
problem Understanding Hodge-de Rham numbers for almost complex 4-manifolds.
method Introduced and studied Hodge-de Rham numbers, extending properties from complex surfaces.
result All Hodge-de Rham numbers for compact almost complex 4-manifolds are determined by the cohomology, except for one (the irregularity).
Proposes a Complex Transformer for complex-valued sequence modeling.
problem Lack of deep learning models for complex-valued data.
method Develops a Complex Transformer using transformer backbone with specialized attention and encoder-decoder networks.
result Achieves state-of-the-art performance on complex-valued datasets.
New proofs for growth series of Coxeter groups using complex structures.
problem Proving new formulae for growth series of Coxeter groups.
method Using the structure of Coxeter complexes, Davis complexes, or Tits non-complexes.
result Several classical formulae for growth series are proved in a new way.
Unified complexity measure for learning theory improves risk bounds.
problem Improving risk bounds in learning theory for various estimators.
method Introduces a new complexity measure interpolating between Rademacher, KL-divergence, and NML complexities.
result Bounded excess risk in terms of the new complexity measure.
Method constructs complex symplectic Lie algebras from simpler ones.
problem Classifying complex symplectic Lie algebras of various dimensions.
method Complex symplectic oxidation method
result Classification of eight-dimensional nilpotent complex symplectic Lie algebras.
A Sasaki-like almost contact complex Riemannian manifold is defined as an almost contact complex Riemannian manifold which complex cone is a holomorphic complex Riemannian manifold. Explicit compact and non-compact examples are given. A canonical construction producing a Sasaki-like almost contact complex Riemannian ma…
New rational parallelisms found on complex manifolds that are not flat.
problem Finding non-flat rational parallelisms on complex manifolds.
method Examined rational parallelisms on compact complex manifolds, discovering non-flat examples.
result Discovered rational parallelisms on compact complex manifolds that are not flat.
New differential complexes on symplectic manifolds.
problem Developing calculus on symplectic manifolds.
method Coupling a symplectic manifold to a vector bundle with a constrained curvature.
result Construction of new differential complexes.
Study Sp(n)-orbits in complex and Σ-complex subspaces of Hermitian quaternionic vector spaces.
problem Characterize Sp(n)-orbits in Grassmannians of complex and Σ-complex subspaces. method Decompose subspaces into 4-dimensional complex addends and 2-dimensional totally complex subspace. Use properties of isoclinic subspaces and principal angles.
result Determine full set of invariants for Sp(n)-orbits in GrR(2k,4n). Study on invariant almost complex structures on real flag manifolds.
problem Existence of invariant almost complex structures on real flag manifolds.
method Analysis of real flag manifolds associated to split real forms of complex simple Lie algebras.
result Some real flag manifolds do not admit invariant almost complex structures.
Tree complex linked to polyhedral shapes like associahedra and cyclohedra.
problem Understanding the structure of mapping class groups and complex dynamics.
method Characterizing associahedra and cyclohedra using planar tree embeddings and barycentric subdivision.
result Tree complex is a barycentric subdivision of a polyhedral cell complex made of associahedra and cyclohedra.
We show that any compact almost-complex manifold of complex dimension m can be pseudo-holomorphically embedded in R^(6m) equipped with a suitable almost-complex structure.
Geometric model for Hodge filtered complex cobordism constructed.
problem Constructing a geometric model for Hodge filtered complex cobordism.
method Refinement of Pontryagin-Thom construction to create an explicit isomorphism.
result Explicit isomorphism between geometric and abstract models for complex manifolds.
New calculations of topological complexity for symplectic CW-complexes.
problem Calculating topological complexity for symplectic CW-complexes.
method Using atoroidal cohomology classes and CW-complexes, proving topological complexity for symplectic spaces.
result Every atoroidally symplectic CW-complex of dimension 2n has topological complexity 4n.
The study defines and explores properties of complex Sasakian manifolds.
problem Understanding the geometric structure of complex Sasakian manifolds.
method Definition and analysis of properties, curvature relations, and flatness conditions.
result Obtained useful curvature relations and examined flatness conditions for general curvature tensor B.
Proposes a method to learn representations of higher-dimensional simplicial complexes.
problem Lack of methods for representing entire simplicial complexes.
method Geometric message passing schemes for end-to-end learning of simplicial complex representations.
result First method for learning representations of entire simplicial complexes.
This note constructs complex structures on specific isoparametric hypersurfaces.
problem Building complex structures on isoparametric hypersurfaces.
method Constructing almost or complex structures on isoparametric hypersurfaces in unit spheres.
result Complex structures on S1imesS7imesS6 and S1imesS3imesS2 are built. The paper explores complex Poisson structures on smooth functions in complex manifolds.
problem Exploring complex Poisson structures on smooth functions in complex manifolds.
method Considering structures of complex Poisson brackets generated by a (1,1)-form. result Examples of complex Poisson structures are provided in $\C^\ast$.
Proposes CXNs for neural network computations on cell complexes.
problem Performing neural network computations on complex topological spaces.
method Introduces a message passing scheme and a unified encoder-decoder framework for cell complexes.
result Generalizes message passing to cell complexes and provides a cell2vec representation.
We consider options that pay the complexity deficiency of a sequence of up and down ticks of a stock upon exercise. We study the price of European and American versions of this option numerically for automatic complexity, and theoretically for Kolmogorov complexity. We also consider run complexity, which is a restricte…
Characterizes alternating link exteriors using cubed complexes.
problem Understanding the structure of alternating link exteriors.
method Introducing signed BW cubed-complexes and characterizing their homeomorphism to link exteriors.
result Characterization of alternating link exteriors in terms of cubed complexes.
Note on connectedness of primitive disk complex.
problem Whether primitive disk complex is connected for genus > 3 Heegaard splittings.
method Defined and quotiented primitive disk complex to prove connectedness.
result Homotopy primitive disk complex is connected.