The paper develops a theory of Bott-Chern cohomology and proves a residue theorem.
problem Localization of Bott-Chern classes and Hermitian residues.
method Developed a unified theory of Cech-Bott-Chern cohomology, defined cup product and integration, and introduced local and global duality homomorphisms.
result Proved a residue theorem for vector bundles with compatible Hermitian connections.
New method uses free probability to improve deep ResNet initialization speed.
problem Improving initialization speed of deep ResNets.
method Introducing a novel analytical tool in free probability for non-Hermitian random matrices.
result Proposed initialization scheme learns at a speed of orders of magnitudes faster.
Let P be a non-negative self-adjoint Laplace type operator acting on sections of a hermitian vector bundle over a closed Riemannian manifold. In this paper we review the close relations between various P-related coefficients such as the mollified spectral counting coefficients, the heat trace coefficients, the reso…
Constructs currents representing Baum-Bott residues for foliations.
problem Calculating Baum-Bott residues for complex foliations.
method Explicit construction of currents with support on singular components.
result Currents represent Baum-Bott residues and are independent under certain conditions.
Given a compatible vector field on a compact connected almost-complex manifold, we show in this article that the multiplicities of eigenvalues among the zero point set of this vector field have intimate relations. We highlight a special case of our result and reinterpret it as a vanishing-type result in the framework o…
We compute explicit transgression forms for the Euler and Pontrjagin classes of a Riemannian manifold M of dimension 4 under a conformal change of the metric, or a change to a Riemannian connection with torsion. These formulae describe the singular set of some connections with singularities on compact manifolds as a …
Holomorphic maps are a special case of Hermitian pluriharmonic maps between almost Hermitian manifolds.
problem Characterizing maps between almost Hermitian manifolds.
method Introducing Hermitian pluriharmonic maps and proving their properties.
result Holomorphic or anti-holomorphic maps are Hermitian pluriharmonic.
Link quandles are shown to be residually finite.
problem Residual finiteness of link quandles.
method Proving residual finiteness of free products of residually finite quandles with residually finite associated groups.
result Link quandles are residually finite.
The paper proves Liouville theorems for holomorphic maps on pseudo-Hermitian manifolds.
problem Proving Liouville theorems for holomorphic maps on pseudo-Hermitian manifolds.
method Analyzing maps between different classes of pseudo-Hermitian manifolds, using curvature assumptions.
result Holomorphic maps are constant under certain curvature conditions.
Construct a Hermitian metric on non-Hermitian Yang--Mills moduli spaces near the Hermitian locus.
problem Moduli space of non-Hermitian Yang--Mills connections over a compact Kähler manifold
method Using normalized harmonic metrics
result Near the Hermitian locus, the unobstructed locus carries an almost hypercomplex structure compatible with the associated Riemannian metric.
Free and knot quandles are residually finite.
problem Residual finiteness of quandles.
method Definition and investigation of residual finiteness; proof for free and knot quandles; discussion on automorphism groups.
result Free and knot quandles are residually finite and Hopfian.
Paper proves solutions for deformed Hermitian-Yang-Mills equation on almost Hermitian manifolds.
problem Existence of solutions for deformed Hermitian-Yang-Mills equation on almost Hermitian manifolds.
method Derive a priori estimates under the existence of an admissible C-subsolution; prove existence of solutions under the condition of existence of a supersolution. result Proves existence of solutions for the deformed Hermitian-Yang-Mills equation.
Abstract: Non-residually finite hyperbolic groups imply non-residually finite rigid hyperbolic groups.
problem Existence of non-residually finite hyperbolic groups
method Direct implication
result Existence of non-residually finite rigid hyperbolic groups
We study geometric realization questions of curvature in the affine, Riemannian, almost Hermitian, almost para Hermitian, almost hyper Hermitian, almost hyper para Hermitian, Hermitian, and para Hermitian settings. We also express questions in Ivanov-Petrova geometry, Osserman geometry, and curvature homogeneity in ter…
The paper establishes Schwarz type lemmas for holomorphic maps between pseudo-Hermitian and Hermitian manifolds.
problem Analyzing holomorphic maps between pseudo-Hermitian and Hermitian manifolds.
method Using Bochner formulas and comparison theorems.
result Established Schwarz type lemmas for holomorphic maps.
The paper solves a problem related to Higgs bundles and Hermitian metrics.
problem Existence of Hermitian metrics with prescribed Hermitian-Yang-Mills tensors for Higgs bundles.
method Solving the prescribed Hermitian-Yang-Mills tensor problem for Higgs bundles over compact complex manifolds.
result For any Hermitian positive definite tensor, there exists a unique smooth Hermitian metric on the Higgs bundle.
Classifies slant surfaces in almost para-Hermitian manifolds.
problem Classifying slant surfaces in almost para-Hermitian manifolds.
method Defined slant submanifolds and classified pointwise slant surfaces.
result Classified slant surfaces in four-dimensional almost para-Hermitian manifolds.
Develops Hermitian TQFTs from quantum groups, defining new topological phases.
problem Defining Hermitian non-semisimple TQFTs.
method Categorical context and representation theory of quantum groups.
result New pseudo-Hermitian topological phases from quantum group representations.
On Hermitian manifolds, the second Ricci curvature tensors of various metric connections are closely related to the geometry of Hermitian manifolds. By refining the Bochner formulas for any Hermitian complex vector bundle (Riemannain real vector bundle) with an arbitrary metric connection over a compact Hermitian manif…
Paper extends matrix inequality to Hermitian matrices.
problem Extending inequalities to Hermitian matrices.
method Using Frobenius norm of commutators for real and skew matrices, extending to Hermitian and skew-Hermitian.
result DDVV-type inequalities now apply to Hermitian matrices.
Study deforms Hermitian metrics with positive curvature.
problem Deforming Hermitian metrics with positive curvature.
method Adapted conformal perturbation method to Hermitian setting.
result Hermitian metrics with quasi-positive curvature can be deformed to positive curvature.
Every non-trivial knot group is fully residually perfect.
problem Understanding the residual properties of knot groups.
method Analyzing the residual properties of knot groups using group theory.
result Every non-trivial knot group is fully residually perfect.
Residual algorithms improve reinforcement learning performance.
problem Distribution mismatch in model-based planning.
method Bidirectional target network technique for residual algorithms.
result Residual reinforcement learning significantly outperforms vanilla methods.
New flow preserves almost Hermitian metrics for manifold study.
problem Curvature flow for almost Hermitian manifolds.
method Introducing a new curvature flow matching Ricci flow and preserving almost Hermitian condition.
result Ricci flow can be used to study almost Hermitian manifolds.
Study on a nonlinear elliptic equation on compact Hermitian manifolds.
problem Solving a nonlinear elliptic equation on compact Hermitian manifolds.
method Investigation of the deformed Hermitian-Yang-Mills equation.
result Analysis of eigenvalues and their relation to the equation.
Paper solves tensor problem for holomorphic vector bundles.
problem Existence of Hermitian metrics with prescribed Hermitian-Yang-Mills tensors.
method New comparison theorem for Hermitian-Yang-Mills tensors.
result Existence of unique smooth Hermitian metric for any positive-definite tensor.
Study various submanifolds in quaternionic skew-Hermitian spaces.
problem Characterize submanifolds in almost quaternionic skew-Hermitian manifolds.
method Construct explicit examples of submanifolds in semisimple quaternionic skew-Hermitian symmetric spaces.
result Explicit examples of submanifolds for each type considered.
The paper classifies Hermitian manifolds with specific connection properties.
problem Classifying Hermitian manifolds with specific connection properties.
method Algebraic consideration of holonomy systems, structure theorems, and classification theorems.
result The universal cover of such Hermitian manifolds is the product of a complex Lie group and Hermitian symmetric spaces.
Paper generalizes sub-slope definition and solves complex equations on compact manifolds.
problem Solving complex equations on compact almost Hermitian manifolds.
method Generalized sub-slope definition and proved existence of solutions for a class of equations.
result Solved complex Hessian quotient and deformed Hermitian-Yang-Mills equations.
The study defines and analyzes nearly Kähler and Kähler-Codazzi manifolds.
problem Defining and analyzing nearly Kähler and Kähler-Codazzi manifolds.
method Definition and comparison of nearly Kähler and Kähler-Codazzi manifolds in various geometries.
result Nearly Kähler type manifolds exist only in Hermitian and para-Hermitian contexts, while Kähler-Codazzi type manifolds reduce to Kähler type manifolds in all four geometries.
Hermitian symmetric manifolds are Hermitian manifolds which are homogeneous and such that every point has a symmetry preserving the Hermitian structure. The aim of these notes is to present an introduction to this important class of manifolds, trying to survey the several different perspectives from which Hermitian sym…
Introduce a flow for prescribed Hermitian-Yang-Mills tensors
problem Solve the prescribed Hermitian-Yang-Mills tensor equation
method Long-time convergence of the flow
result Uniform C0-estimate of the flow Residual flows are shown to approximate MMD well.
problem Lack of theoretical understanding of normalizing flows' expressiveness.
method Proved residual flows are universal approximators in MMD.
result Residual flows can approximate MMD with a bounded number of blocks.
Formula calculates residues for maps near holomorphic distributions.
problem Calculating residues for maps near holomorphic distributions.
method Residues formula for maps generically transversal to regular holomorphic distributions.
result Established a residues formula for maps near holomorphic distributions.
Let p be a prime. In this paper, we classify the geometric 3-manifolds whose fundamental groups are virtually residually p. Let M=M3 be a virtually fibered 3-manifold. It is well-known that G=π1(M) is residually solvable and even residually finite solvable. We prove that G is always virtually residually p…
Researchers identify critical protein residues using advanced graph theory.
problem Identifying essential residues in proteins for function.
method Learning Random Geometric Graphs (RGG) with Cramer's V correlation and organic thresholding.
result Advanced RGG methods accurately identify critical residues compared to existing techniques.
Study of Hermitian metrics on Lie algebroids over complex spaces.
problem Exploring Hermitian metrics on Lie algebroids over analytic spaces.
method Introduced Hermitian metrics on holomorphic Lie algebroids and examined associated characteristics foliations.
result Extended equivariant de Rham cohomology to Hermitian Lie algebroids.
The paper establishes Schwarz type lemmas for pseudo-Hermitian manifolds.
problem Understanding the geometry and mappings of pseudo-Hermitian manifolds.
method Using sub-Laplacian or Hessian type Bochner formulas and comparison theorems.
result Established Schwarz type results for \emph{CR} maps and transversally holomorphic maps.
The paper examines positivity properties of singular Hermitian metrics.
problem Investigating positivity for singular Hermitian metrics.
method Exploring Griffiths, ω-trace, and RC positivity.
result These positivity notions imply cohomology vanishing and rational connectedness.
Defines Wodzicki residue using groupoids and fibered distributions.
problem Defining and understanding the Wodzicki residue in noncommutative geometry.
method Using groupoid language and filtered manifolds, defining the residue and showing its properties.
result The groupoidal residue is a trace on pseudodifferential operators and matches the usual residue in certain cases.
Paper derives formulas for holomorphic maps between Hermitian manifolds and proves related theorems.
problem Holomorphic maps between Hermitian manifolds and their properties.
method Derives ∂∂-Bochner formulas and proves Schwarz lemma type estimates. result Generalizes Ni's results to Hermitian manifolds, proving rigidity and degeneracy theorems.
New bounds for geometric flows of Hermitian metrics established.
problem Regularity of geometric flows of Hermitian metrics.
method Establishing a C1 a priori bound for smooth curves of Hermitian metrics. result New regularity result for Hermitian curvature flows, including the second Chern-Ricci flow.
Paper explores prescribing Chern scalar curvatures on noncompact Hermitian manifolds.
problem Prescribing Chern scalar curvatures on complete noncompact Hermitian manifolds.
method Generalizes Aviles-McOwen's existence results to higher-dimensional Hermitian manifolds.
result Existence results for Chern scalar curvatures on Hermitian manifolds.
Proves infinitesimal rigidity of Hermitian gravitational instantons.
problem Understanding the moduli space of Hermitian gravitational instantons.
method Proof of infinitesimal rigidity and integrability using boundary conditions and conformal Kähler properties.
result Completes the understanding of Hermitian gravitational instantons, both compact and non-compact.
Study on pseudo-Hermitian quadratic nilpotent Lie algebras with methods and classifications.
problem Characterizing and classifying pseudo-Hermitian quadratic nilpotent Lie algebras.
method Construction methods and double extension by planes.
result Complete classification of nilpotent quadratic Lie algebras and pseudo-Hermitian metrics up to dimension 8.
The paper studies four Hermitian structures on twistor spaces.
problem Understanding the geometry of Hermitian surfaces via twistor spaces.
method Defining four almost Hermitian structures on twistor spaces using canonical connections.
result Relations between twistor space geometry and Hermitian surface geometry.
Holomorphic residue formula for complex supermanifolds.
problem Residue localization on complex supermanifolds.
method Holomorphic residue localization formula for odd vector fields.
result Explicit local residue formula under isolated non-degeneracy hypotheses.
In this work we prove a Baum-Bott type residue theorem for flags of holomorphic foliations. We prove some relations between the residues of the flag and the residues of their correspondent foliations. We define the Nash residue for flags and we give a partial answer to the Baum-Bott type rationality conjecture in this …