New method determines arrangement combinatorics from Milnor fiber boundary.
problem Determining arrangement combinatorics from Milnor fiber boundary.
method Explicit method using plumbing graph in normal form.
result Milnor fiber boundary determines arrangement combinatorics.
Paper generalizes a theorem for real analytic singularities.
problem No specific problem stated; focuses on generalization.
method Generalization of a theorem for complex singularities.
result Generalized Join theorem for real analytic singularities.
Paper proves a conjecture about a Heegaard Floer invariant for certain rational homology spheres.
problem Proving a conjecture about the Heegaard Floer d-invariant for negative-definite plumbed rational homology spheres.
method Using Zemke's isomorphism between lattice and Heegaard Floer homology, the paper proves Némethi's conjecture.
result The conjecture about the Heegaard Floer d-invariant for negative-definite plumbed rational homology spheres is proven.
Computes Pin(2)-equivariant monopole Floer homology for certain plumbed 3-manifolds.
problem Computing Pin(2)-equivariant monopole Floer homology for specific 3-manifolds.
method Uses Heegaard Floer/monopole Floer lattice complex and ranks of monopole Floer homology groups.
result Pin(2)-equivariant monopole Floer homology can be determined by ranks of monopole Floer homology groups.
Proves lattice homology equals Heegaard Floer homology for certain 3-manifolds.
problem Equivalence of lattice homology and Heegaard Floer homology for specific 3-manifolds.
method Proves conjecture using plumbing of disk bundles over S^2.
result Lattice homology equals Heegaard Floer homology for 3-manifolds with specified properties.
Ozsváth and Szabó gave a combinatorial description for the Heegaard Floer homology of boundaries of certain negative-definite plumbings. Némethi constructed a remarkable algorithm for executing these computations for almost-rational plumbings, and his work gives a formula computing the invariants for the Brieskorn homo…
We study the counting function of topological Poincaré series associated with rational homology sphere plumbed 3-manifold with connected negative definite tree, interpreting as an alternating sum of coefficient functions associated with some Taylor expansions. It is motivated by a theorem of Szenes and Vergne which exp…
We define and prove properties of link lattice complexes for plumbed links.
problem Understanding the homology and formality of plumbed L-space links.
method Define and analyze link lattice complexes, proving homotopy equivalence and formality properties.
result Link lattice complexes are homotopy equivalent to link Floer complexes for plumbed links.
Study of first homology group of Milnor fiber boundary for generic hyperplane arrangements in C^3.
problem Computing the first homology group of the Milnor fiber boundary for generic hyperplane arrangements.
method Analyzing the Milnor fiber boundary for hyperplane arrangements in C^3.
result Affirmative answer to the conjecture of Suciu and example of arrangements with non-trivial torsion.
Study on contact structures of surface singularity links using Heegaard Floer homology.
problem Understanding contact structures on singularity links using Heegaard Floer homology.
method Interplay between Heegaard Floer homology and Némethi's lattice cohomology.
result Heegaard Floer invariant cannot lie in the image of the U-action for canonical contact structures on singularity links.
Lattice cohomology, defined by Némethi in (arXiv:0709.0841), is an invariant of negative definite plumbed 3-manifolds which conjecturally computes the Heegaard Floer homology HF^+. We prove a surgery exact triangle for the lattice cohomology analogous to the one for HF^+. This is a step towards comparing these two inva…
Survey connects singularity invariants to link pairings.
problem Understanding connections between singularities and link pairings.
method Use of Hermitian Variation Structures.
result Unified understanding of Picard--Lefschetz invariants and Blanchfield forms.
Study calculates Heegaard Floer homology for Seifert fibered 3-manifolds.
problem Computing Heegaard Floer homology for Seifert fibered 3-manifolds.
method Combinatorial models inspired by lattice homology.
result Connected Floer homology determines local equivalence class of ι-complex. This paper reverses a construction by merging boundary critical points into an interior one.
problem Pushing interior critical points to the boundary and splitting them into two boundary points.
method Specific assumptions allow merging two boundary critical points into one interior critical point.
result Merging two boundary critical points into a single interior critical point.
In a recent paper, Dimca and Nemethi pose the problem of finding a homogeneous polynomial f such that the homology of the complement of the hypersurface defined by f is torsion-free, but the homology of the Milnor fiber of f has torsion. We prove that this is indeed possible, and show by construction that, for each pri…
We prove that an embedded cobordism between manifolds with boundary can be split into a sequence of right product and left product cobordisms, if the codimension of the embedding is at least two. This is a topological counterpart of the algebraic splitting theorem for embedded cobordisms of the first author, A. Nemethi…
In a recent paper of Akhmedov, Etnyre, Mark and Smith, it was shown that there exist infinitely many contact Seifert fibered 3-manifolds each of which admits infinitely many exotic (homeomorphic but pairwise non-diffeomorphic) simply-connected Stein fillings. Here we extend this result to a larger set of contact Seifer…
Study on contact structures of singularity links and existence of Stein cobordisms.
problem Existence problem of Stein cobordisms between contact structures of singularity links.
method Construction of explicit Stein cobordism and detection of contact Ozsvath-Szabo invariants.
result U-filtration depth obstructs the existence of Stein cobordism from proper almost rational to rational singularity.
We establish three rank inequalities for the reduced flavor of Heegaard Floer homology of Seifert fibered integral homology spheres. Combining these inequalities with the known classifications of non-zero degree maps between Seifert fibered spaces, we prove that a map f from Y' to Y between Seifert homology spheres yie…
Algorithm calculates polynomial Seiberg-Witten invariant from zeta-function of plumbed 3-manifolds.
problem Computing the Seiberg-Witten invariant for plumbed 3-manifolds.
method Némethi's division algorithm for zeta-functions.
result Explicit calculation of polynomial Seiberg-Witten invariant.
It has been recently conjectured by Boyer-Gordon-Watson that a closed, orientable, irreducible 3-manifold M is a Heegaard Floer L-space if and only if π1(M) is not left-orderable. In this article, we study this conjecture from the point of view of lattice cohomology, an invariant introduced by Némethi which is…
Study rational cuspidal curves in projective surfaces with topological and algebraic obstructions.
problem Obstructing possible configurations of singular points on rational cuspidal curves.
method Two criteria: one based on Bezout theorem, the other on Ozsvath-Szabo inequalities.
result Explicit calculations show similar obstructions from both approaches.
Study confirms conjectures for topologically slice knots' concordance group.
problem Primary decomposition conjectures for knot concordance groups.
method Use of amenable L2-signatures, Ozsváth-Szabó d-invariants, and Némethi's Heegaard Floer homology results. result Smooth concordance group of topologically slice knots has large subgroup with true primary decomposition.
We encode the variation structure of a quasihomogeneous polynomial with an isolated singularity as introduced by Nemethi in a set of spectral flows of the signature operator on the Milnor bundle by varying global elliptic boundary conditions in a specific way using the quasihomogeneous circle action on the Brieskorn la…
Legendrian surgery describes canonical contact structures and calculates Gompf's θ-invariant.
problem Understanding canonical contact structures and their properties.
method Legendrian surgery and explicit formulas for Gompf's θ-invariant.
result Explicit description and closed-form formula for Gompf's θ-invariant.
The paper develops a bordered HF− theory using link surgery formula.
problem Computing Heegaard Floer homology of 3-manifolds with torus boundaries.
method Introducing an associative algebra K and interpreting link surgery complexes as type-D modules over K. result Proves a connected sum formula and computes Heegaard Floer homology of various 3-manifolds.
Study Cremona transformations in weighted projective planes to find rational cuspidal curves and Zariski pairs.
problem Finding rational cuspidal curves and Zariski pairs in weighted projective planes.
method Construct families of curves using Cremona transformations, compute fundamental groups, and use blow-up-down decompositions.
result Discover new examples of rational cuspidal curves and Zariski pairs in weighted projective planes.
The paper examines lattice homology invariants of Seifert homology spheres.
problem Understanding homology cobordism invariants for Seifert fibered integral homology 3-spheres.
method Utilizes lattice homology and Heegaard Floer homology to study invariants.
result Reproves and extends the invariance of Seifert homology spheres' d-invariants and maximal monotone subroots. Unbraided wiring diagrams for Stein fillings of lens spaces are described.
problem Constructing Stein fillings of lens spaces with canonical contact structures.
method Algorithm to draw unbraided wiring diagrams equivalent to Lefschetz fibrations.
result Wiring diagrams can be extended to symplectic graphical disks with marked points.
Study on pricing options tied to stock tick complexity.
problem Pricing options based on stock tick complexity.
method Numerical and theoretical analysis of European and American options.
result Numerical and theoretical pricing results for different complexity types.
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.