Study blow-ups in generalized complex geometry using holomorphic ideals.
problem Blow-ups in generalized complex geometry.
method Introduce holomorphic ideal to define blow-ups in smooth manifolds. Identify suitable submanifolds and provide conditions for blow-ups.
result Necessary and sufficient conditions for generalized Poisson submanifolds to carry a canonical holomorphic ideal and for blow-ups to be generalized complex.
Paper extends cohomology classes and holomorphic sections on subvarieties.
problem Tackles extension of cohomology classes and holomorphic sections on subvarieties.
method Uses quotient sheaves of multiplier ideal sheaves of quasi-plurisubharmonic functions.
result Provides positive answers to questions and generalizes existing L2 extension theorems. We extend Nadel's results on some conditions for the multiplier ideal sheaves to satisfy which are described in terms of an obstruction defined by the first author. Applying our extension we can determine the multiplier ideal sheaves on toric del Pezzo surfaces which do not admit Kähler-Einstein metrics. We also show t…
We generalise work of Young-Eun Choi to the setting of ideal triangulations with vertex links of arbitrary genus, showing that the set of all (possibly incomplete) hyperbolic cone-manifold structures realised by positively oriented hyperbolic ideal tetrahedra on a given topological ideal triangulation and with prescrib…
Minimal discs count as knot invariants in hyperbolic 4-space.
problem Counting minimal discs with a given boundary in hyperbolic 4-space.
method Using J-holomorphic curves in twistor space to count minimal discs, considering singularities at infinity.
result The number of minimal discs with a given boundary is a knot invariant.
Wintgen ideal submanifolds in space forms are those ones attaining equality pointwise in the so-called DDVV inequality which relates the scalar curvature, the mean curvature and the scalar normal curvature. Using the framework of Moebius geometry, we show that in the codimension two case, the mean curvature sphere of t…
The paper proves new injectivity theorems for complex geometry.
problem Injectivity of higher direct image sheaves under Kähler morphisms.
method Transcendental methods, focusing on Kähler morphisms and singular hermitian metrics.
result Established new injectivity theorems for canonical bundles and their twists.
The paper solves conjectures related to strong openness and optimal L2 extension.
problem Strong openness of multiplier ideal sheaves and optimal L2 extension. method Solution of conjectures related to Demailly's strong openness and related conjectures.
result Optimal L2 extension implies Berndtsson's positivity of vector bundles. The note confirms a conjecture for specific Lie groups.
problem The conjecture about constant holomorphic sectional curvature in non-Kähler geometry.
method Compact quotients of Lie groups with specific properties.
result The conjecture is confirmed for almost abelian Lie algebras and those with certain abelian ideals.
Study pseudo-holomorphic disks in real analytic hypersurfaces using exterior differential systems.
problem Existence of pseudo-holomorphic disks in non-integrable real analytic hypersurfaces.
method Theory of exterior differential systems.
result Non-existence of certain equivalent structures in the non-integrable case.
The paper constructs harmonic maps from punctured surfaces to the hyperbolic plane.
problem Harmonic maps from punctured surfaces to hyperbolic planes.
method Constructing polynomial growth harmonic maps from punctured Riemann surfaces to regular polygons in hyperbolic plane.
result Established uniqueness of harmonic maps within a class of maps differing by exponentially decaying variations.
The paper proves vanishing theorems for vector bundles with singular metrics.
problem Analyzing vector bundles with singular Hermitian metrics.
method Using a sheaf of locally square integrable holomorphic sections and a new Nadel-Nakano type vanishing theorem.
result Generalizations of vanishing theorems for vector bundles with singular metrics.
Study of 3d-3d correspondence involving q-Weyl algebra and 3d-index.
problem Understanding the action of a q-Weyl algebra on the 3d-index of knots. method Investigation of the q-Weyl algebra's module action on the 3d-index, conjecturing structural properties. result Bilinear factorization, pair of linear q-difference equations, and rational function matrix for the 3d-index determination. The paper studies vector bundles over surfaces, focusing on singularity formation.
problem Understanding singularity formation in rank two holomorphic vector bundles over surfaces.
method Defining fertile families bearing bubbles and using elementary modifications to prove their existence.
result Existence of fertile families bearing bubbles for certain types of vector bundles.
We propose a new description of 3d N=2 theories which do not admit conventional Lagrangians. Given a quiver Q and a mutation sequence m on it, we define a 3d N=2 theory T[(Q,m)] in such a way that the Sb3 partition function of the theory coincides with the cluster partition f…
We consider the construction of the basic bundle gerbe on SU(n) introduced by Meinrenken and show that it extends to a range of groups with unitary actions on a Hilbert space including U(n), diagonal tori and the Banach Lie group of unitary operators differing from the identity by an element of a Schatten ideal. In all…
Paper proves non-existence of certain balanced metrics on six-manifolds.
problem Existence of balanced metrics on six-manifolds with specific properties.
method Analysis of cohomogeneity one actions and properties of balanced structures.
result Non-existence of balanced non-Kähler SU(3)-structures on six-manifolds.
We use the 3d-3d correspondence together with the DGG construction of theories Tn[M] labelled by 3-manifolds M to define a non-perturbative state-integral model for SL(n,C) Chern-Simons theory at any level k, based on ideal triangulations. The resulting partition functions generalize a widely studied k=1 state-integ…
We construct from first principles the operator 'A-hat' that annihilates the partition functions (or wavefunctions) of three-dimensional Chern-Simons theory with gauge groups SU(2), SL(2,R), or SL(2,C) on a knot complement M. The operator 'A-hat' is a quantization of the knot complement's classical A-polynomial A(l,m).…
Study on GL(1∣1) Higgs bundles over Riemann surfaces.
problem Investigate the moduli space of GL(1∣1) Higgs bundles. method Explicit description of moduli space, study of Narasimhan-Seshadri theorem, nonabelian Hodge correspondence, Hitchin equations.
result Derive an explicit description of the moduli space and study its properties.
Study of hyperbolic 3-manifolds using gluing equations and torsion.
problem Understanding the geometry and topology of hyperbolic 3-manifolds.
method Linking gluing equations to cohomology of infinitesimal isometries and geometrically reformulating torsion.
result Verification of generalized 1-loop Conjecture for sister manifold of figure-eight knot complement.
Study ideals in the Goldman Lie algebra S by mapping to a simpler algebra.
problem Identify and classify ideals in the Goldman Lie algebra S. method Construct an algebra homomorphism to a simpler structure and classify ideals.
result Found infinite classes of ideals in both Z-module and Q-module cases. The study proves poor ideal three-edge triangulations are minimal for certain 3-manifolds.
problem Finding minimal ideal triangulations for specific 3-manifolds.
method Analyzing properties of poor ideal three-edge triangulations and applying them to construct minimal triangulations.
result Poor ideal three-edge triangulations are proven to be minimal for certain 3-manifolds.
We give a simple method to find ideal points of the character variety of a 3-manifold from an ideal triangulation.
Paper provides new Alexander ideal-based obstruction to 0-concordance of knotted surfaces.
problem Tackles the 0-concordance problem for knotted surfaces in S4. method Uses Alexander ideals to induce a homomorphism and prove non-sliceness.
result Alexander ideal determines 0-concordance classes and non-sliceness.
Short proof for ideal polygons with near optimal orthogeodesic decomposition.
problem Decomposing ideal polygons into orthogeodesics.
method Short proof with orthogeodesic decomposition of length at most 2log(n). result Optimal orthogeodesic decomposition of ideal polygons with length 2log(n). A method uses CG to create efficient channels for ideal observers.
problem Computational intractability of ideal observers for high-dimensional image data.
method Conjugate gradient (CG) method for constructing efficient channels.
result CG-based channels approximate IO and HO performance efficiently.
The paper explores genericity of triples of antipodal chambers in affine buildings.
problem Investigating genericity of triples of antipodal ideal chambers in locally finite affine buildings.
method Defined ideal-genericity and affine-genericity at the ideal boundary and within the affine building, respectively. Established a barycenter map and showed its continuity.
result Affine-genericity implies ideal-genericity, but the latter is more suitable for constructing a barycenter map.
Study introduces dynamical ideals for non-commutative rings and classifies knots and links.
problem Classifying surface knots and links in smooth 4-manifolds.
method Introduced dynamical analog of prime ideals for non-commutative rings and proved a factorization theorem.
result Classified surface knots and links in smooth 4-manifolds.
We investigate the rigidity of hyperbolic cone metrics on 3-manifolds which are isometric gluing of ideal and hyper-ideal tetrahedra in hyperbolic spaces. These metrics will be called ideal and hyper-ideal hyperbolic polyhedral metrics. It is shown that a hyper-ideal hyperbolic polyhedral metric is determined up to i…
Defines timelike ideal boundary for non-positively curved Lorentzian spaces.
problem Understanding the geometry of non-positively curved Lorentzian spaces.
method Introduces timelike ideal boundary as asymptotic classes of geodesic rays, endows with topology and metric, and studies upper curvature bounds.
result Established upper curvature bounds for the resulting metric space.
A taut ideal triangulation of a 3-manifold is a topological ideal triangulation with extra combinatorial structure: a choice of transverse orientation on each ideal 2-simplex, satisfying two simple conditions. The aim of this paper is to demonstrate that taut ideal triangulations are very common, and that their behavio…
The paper solves a problem related to exterior multiplication with singularities.
problem Finding sections of a p-exterior product given sections of a vector bundle.
method Analyzes the sufficiency of a condition involving regular singularities and depth of ideals.
result Shows that the condition is sufficient for existence of a continuous right inverse in smooth, real analytic, and holomorphic categories.
Defines ideal simplicial volume for manifolds with boundary, showing it's bounded by classical volume.
problem Measuring the complexity of manifolds with boundary.
method Introduces ideal simplicial volume, compares it to classical volume, and computes specific examples.
result Ideal simplicial volume is bounded by classical volume and can be strictly smaller for certain manifolds.
The paper studies dynamical properties in semigroups modulo ideals.
problem Analyzing shadowing, expansivity, and stability in semigroups with ideals.
method Investigates shadowing, expansivity, and stability properties in uniform transformation semigroups modulo an ideal.
result Establishes that if a semigroup exhibits shadowing and expansivity modulo an ideal, it is also topologically stable modulo that ideal.
Covering maps help determine ideal embeddings in homogeneous spaces.
problem Determining ideal embeddings in compact homogeneous spaces.
method Study ideal embeddings via covering maps of irreducible compact homogeneous spaces.
result Covering maps show that some homogeneous spaces can have ideal embeddings while others cannot.
The notion of ideal immersions was introduced by the author in 1990s. Roughly speaking, an ideal immersion of a Riemannian manifold into a real space form is a nice isometric immersion which produces the least possible amount of tension from the ambient space at each point. In this paper, we classify all ideal hypersur…
New formula calculates volumes of ideal hyperbolic drums.
problem Computing volumes of ideal hyperbolic drums.
method Proved a volume formula for arbitrary ideal hyperbolic antiprisms (drums).
result Volume formula for ideal hyperbolic drums.
Study of combinatorial Calabi flow on ideal circle patterns.
problem Finding ideal circle patterns with prescribed curvatures.
method Combinatorial Calabi flow in hyperbolic and Euclidean geometry.
result Flow converges exponentially to ideal circle patterns.
The paper studies deformations of Lie ideals in Lie algebras.
problem Understanding deformations of Lie ideals in Lie algebras.
method Develops deformation theory, compares cohomologies, enriches deformation complex.
result Deformation cohomology classes differentiate smooth deformations of ideals.
Paper solves long-standing problem of infinite ideal polyhedra in hyperbolic space.
problem Characterize infinite ideal polyhedra in hyperbolic 3-space.
method Introduced combinatorial Ricci flow for infinite ideal circle patterns.
result Proved characterization of infinite ideal circle patterns under specific conditions.
Wintgen ideal surfaces in E^4 form an important family of surfaces, namely surfaces with circular ellipse of curvature. Obviously, Wintgen ideal surfaces satisfy the pointwise equality K+K_N=H^2. In the present study we consider the Wintgen ideal surfaces in n-dimensional Euclidean space E^4. We have shown that Wintgen…
We discuss two different in general natural approaches to the ideal closure and ideal boundary of Busemann nonpositively curved metric space. It is shown that the identity map of the space admits surjective continuation from its coarse ideal closure to the weak one. We consider some situations when these closures coinc…
Essential loops found in taut ideal triangulations.
problem Finding essential loops in taut ideal triangulations.
method Combinatorialisation of Novikov's technique.
result Vertical or normal loops are essential in the fundamental group.
Combinatorial description of 3-manifolds using ordered triangulations.
problem Understanding closed 3-manifolds through ideal triangulations.
method Combining ordered ideal triangulations and Pachner moves.
result Closed 3-manifolds can be described via ordered triangulations and moves.
Minimal ideal triangulations studied for hyperbolic 3-manifolds.
problem Finding minimal triangulations of hyperbolic 3-manifolds.
method Characterization of low degree edges, layered solid torus subcomplexes, and 1-dimensional cohomology.
result Monodromy ideal triangulations of once-punctured torus bundles are minimal.
Proof of existence for ideal triangulations that normalize fibers in certain 3-manifolds.
problem Existence of ideal triangulations that normalize fibers in specific 3-manifolds.
method Proof and algorithm construction for ideal triangulations.
result Existence of ideal triangulations that normalize fibers in certain 3-manifolds.
The paper generalizes results on biconservative hypersurfaces in Euclidean spaces.
problem Characterizing biconservative hypersurfaces in Euclidean spaces.
method Analyzing hypersurfaces with specific divergence properties and curvature conditions.
result Biconservative hypersurfaces with constant scalar curvature are of constant mean curvature.