Introduces homotopy momentum sections on multisymplectic manifolds.
problem No specific problem stated; focuses on introducing a new concept.
method Introduces a new concept of homotopy momentum sections on multisymplectic manifolds.
result Shows that a gauged nonlinear sigma model with Wess-Zumino term has homotopy momentum section structure.
The paper generalizes spectral section concepts to non-compact spaces.
problem Generalizing spectral sections to non-compact base spaces.
method Generalization to arbitrary base spaces, applications to cobordism theorems, investigation of Riesz continuity.
result If a family of operators has a spectral section, it is Riesz continuous.
Generalizes momentum sections to higher-dimensional gauged sigma models.
problem Understanding momentum sections in Hamiltonian mechanics and sigma models.
method Introduces a generalization of momentum sections on pre-multisymplectic manifolds.
result Shows a connection between constrained Hamiltonian systems and gauged sigma models.
Study Higgs sections and flat sections for nonlinear harmonic bundles.
problem Equivalence of Higgs and flat sections for nonlinear harmonic bundles.
method Analyze harmonic vector bundles, generalize to sub-fibrations and morphisms.
result Vanishing of a degree obstruction for general nonlinear harmonic bundles.
Proves section conjecture for curves and surface bundles over various fields.
problem Proving Grothendieck's section conjecture for curves and surface bundles.
method Formulated and proved the section conjecture for stable graphs, used Galois cohomology classes to obstruct sections.
result Proved section conjecture for curves and surface bundles over p-adic and number fields.
New global section found for geodesic flows on convex hypersurfaces.
problem Finding global sections for geodesic flows on convex hypersurfaces.
method Constructing a global hypersurface of section with an isometric involution.
result Generalized Birkhoff annulus to higher dimensions.
The paper proves properties of manifolds with negative holomorphic sectional curvature.
problem Negative holomorphic sectional curvature properties of manifolds.
method Analyzing irreducible subvarieties and extending results to quasi-negative curvature.
result Quasi-projective manifolds with negative holomorphic sectional curvature are of log general type.
New maps show some surfaces can't be sections of 4D spheres.
problem Finding sections for certain 4D sphere maps.
method Exhibited singular fibrations with high genus fibers.
result Some regular fibers cannot be sections.
A singular riemannian foliation F on a complete riemannian manifold M is said to admit sections if each regular point of M is contained in a complete totally geodesic immersed submanifold (a section) that meets every leaf of F orthogonally and whose dimension is the codimension of the regular leaves of F. We prove that…
In this paper, we investigate the null (light-like) sectional curvatures of Lorentzian warped product manifolds. We derive the formulas for the null sectional curvature of many well-known warped product space-time models such as multiply generalized Robertson-Walker space-times, generalized Kasner spacetimes and standa…
The paper proves extension theorems for holomorphic sections from divisors.
problem Extension of holomorphic sections from reduced unions of strata of divisors.
method Proves an Ohsawa--Takegoshi type extension theorem.
result Qualitative results on extension from snc divisors and generic global generation of vector bundles.
In this paper, we obtain classification of four-dimensional Einstein manifolds with positive Ricci curvature and pinched sectional curvature. In particular, the first result concerns with an upper bound of sectional curvature, improving a theorem of E. Costa. The second is a generalization of D. Yang's result assuming …
New method bounds sectional curvatures and proves Spectral Theorem.
problem Computing and bounding sectional curvatures for algebraic curvature tensors.
method Algebraic and geometric methods to construct hypersurfaces with prescribed sectional curvatures.
result A relatively short proof of the Spectral Theorem for self-adjoint operators.
The abstract proves that certain Reeb vector fields on 3-manifolds have Birkhoff sections.
problem Existence of Birkhoff sections for Reeb vector fields on 3-manifolds.
method Showed existence of Birkhoff sections for Reeb vector fields satisfying Kupka-Smale condition.
result Reeb vector fields on closed 3-manifolds with Kupka-Smale condition admit Birkhoff sections.
Formula for sectional curvatures on matrix groups.
problem Calculating curvatures on matrix groups.
method Simple formula derivation for sectional curvatures.
result Valid formula for general linear and reductive Lie groups.
TQA improves prediction intervals for time series data by adjusting quantiles for both cross-sectional and longitudinal coverage.
problem Constructing reliable prediction intervals for cross-sectional time series data.
method Temporal Quantile Adjustment (TQA) method that adjusts the quantile in Conformal Prediction to account for both cross-sectional and longitudinal coverage.
result TQA improves longitudinal coverage while preserving cross-sectional coverage, as validated through extensive experimentation.
We study the conditions under which a Kählerian structure (G,J) of general natural lift type on the cotangent bundle T∗M of a Riemannian manifold (M,g) has constant holomorphic sectional curvature. We obtain that a certain parameter involved in the condition for (T∗M,G,J) to be a Kählerian manifold, is expres…
Introduces comomentum sections and proves they are Poisson maps.
problem Generalizing Poisson maps to Hamiltonian Lie algebroids.
method Introduces comomentum sections and proves they are Lie algebroid morphisms and Poisson maps.
result Comomentum sections are Poisson maps between proper Poisson manifolds.
This thesis predicts the distribution of smoothed zeros of random sections on line bundles.
problem Predicting the distribution of smoothed zeros of random sections on line bundles.
method Developing smoothing operators on discrete surfaces and computing the expected sum of indices on each face.
result Predictions on the distribution of smoothed section's signed zeros with multiplicity.
In this paper, we give a new generalization of positive sectional curvature called positive weighted sectional curvature. It depends on a choice of Riemannian metric and a smooth vector field. We give several simple examples of Riemannian metrics which do not have positive sectional curvature but support a vector field…
The study extends cobordism theory to complex sections, defining and calculating cobordism groups.
problem Understanding when almost complex manifolds can have complex sections.
method Defined complex section cobordism, determined groups, and introduced an obstruction.
result The obstruction vanishes for certain multiplicative generators in the complex cobordism ring.
The study shows that symplectic Lefschetz fibrations can have infinitely many sections.
problem The finiteness of sections in Lefschetz fibrations.
method General criterion and examples for symplectic Lefschetz fibrations with infinitely many sections.
result Symplectic Lefschetz fibrations can have infinitely many homologically distinct sections.
Classifies sections of Riemannian bundles on Lie groups.
problem Classifying sections of Riemannian bundles on Lie groups.
method Developed variational theory of higher-power energy for mappings and sections.
result Complete classification of left-invariant vector fields on 3D Lie groups.
We show that there exists a non-trivial simplified broken Lefschetz fibration which has infinitely many homotopy classes of sections. We also construct a non-trivial simplified broken Lefschetz fibration which has a section with non-negative square. It is known that no Lefschetz fibration satisfies either of the above …
Study examines Lie algebroids with homological sections, generalizing Q-manifolds and Lie superalgebras.
problem Exploring Lie algebroids with homological sections.
method Derived bracket formalism to define an odd Loday-Leibniz bracket on sections.
result Sections of inner Q-algebroids come equipped with an odd Loday-Leibniz bracket.
Defines new metric space sections with Ahlfors-David regularity.
problem Defining and analyzing new types of sections in metric spaces.
method Introducing intrinsically quasi-symmetric sections and proving their Ahlfors-David regularity.
result Proves Ahlfors-David regularity for intrinsically quasi-symmetric sections.
We study critical points of holomorphic sections of $\ocal(m)$ on $\CP^n$. For quadrics, we give a complete discription of their critical points. When n=1, we prove a spherical Gauss-Lucas theorem. For general situation, we prove that a general section has all its critical points isolated and non-degenerate.
We generalize a construction of Hitchin to prove that, given any compact Kähler manifold M with positive holomorphic sectional curvature and any holomorphic vector bundle E over M, the projectivized vector bundle P(E) admits a Kähler metric with positive holomorphic sectional curvature.
The central problem of strip theory is the calculation of potential flowaround 2D sections. One particular method of solutions to this problem is conformal mapping of the body section to the unit circle over which a solution of potential flow is available. Here, a new multiparameter conformal mapping method is presente…
The goal of this paper is not to introduce a single algorithm or method, but to make theoretical steps towards fully understanding the training dynamics of generative adversarial networks. In order to substantiate our theoretical analysis, we perform targeted experiments to verify our assumptions, illustrate our claims…
The study finds sufficient conditions for Reeb flows to have genus zero global surfaces of section.
problem Finding conditions for Reeb flows to have genus zero global surfaces of section.
method Analyzes linking assumptions on periodic orbits and ambient contact geometry.
result Reveals sufficient conditions for Reeb flows to have genus zero global surfaces of section.
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. The paper classifies translation surfaces with constant curvature in a specific connection.
problem Classifying translation surfaces with constant curvature in a semi-symmetric non-metric connection.
method Completely classified translation surfaces of constant sectional curvature in a semi-symmetric non-metric connection.
result Translation surfaces of constant curvature are generalized cylinders, similar to the Levi-Civita connection but with additional non-constant curvature cases.
Two proofs of Melrose-Piazza theorem on spectral sections.
problem Analytic index of families of Fredholm operators.
method Two independent proofs of the theorem, generalizing and clarifying the analytic index definition.
result Generalization and clarification of the Melrose-Piazza theorem on spectral sections.
Develops a geometric framework for spaces of distributional sections and calculates curvature of conical metrics.
problem Defining spaces of distributional sections and their geometric operations.
method Geometric framework for generalized sections, incorporating tensor calculus.
result Calculates the curvature of conical metrics used to describe cosmic strings.
In [11], I. M. Gelfand, V. Retakh, and M. Shubin defined the symplectic sectional curvature of a torsion-free connection preserving a symplectic form. The present article defines the corresponding notion of constant symplectic sectional curvature and characterizes this notion in terms of the curvature tensor of the sym…
Study of section conjecture analogues over complex numbers and Kodaira fibrations.
problem Investigate Grothendieck's section conjecture over complex numbers and Kodaira fibrations.
method Topological and Hodge theoretic analogues of the section conjecture over complex numbers, studied in the context of Kodaira fibrations and families of Jacobians.
result Both topological and Hodge-theoretic analogues of the injectivity part of the section conjecture hold for families of curves, but the topological analogue of the surjectivity part does not hold in general.
This paper is a continuation of the paper F. A. Arias and M. Malakhaltsev "A generalization of the Gauss-Bonnet and Hopf-Poincaré theorems", ArXiv:1510.01395 [MathDG] 5 Oct 2015. Let π:E→M be a locally trivial fiber bundle over a two-dimensional manifold M, and Σ⊂M be a discrete subset. A subset $Q \s…
This paper generalizes biharmonic Riemannian submersions to higher dimensions.
problem Classifying biharmonic Riemannian submersions from manifolds with constant sectional curvature.
method Constructing an adapted orthonormal frame to simplify the biharmonic equation and analyzing curvature properties.
result A Riemannian submersion is biharmonic if and only if it is harmonic from an (n+1)-dimensional manifold with constant sectional curvature to an n-dimensional manifold. In this paper, we prove a general maximum principle for the time dependent Lichnerowicz heat equation on symmetric tensors coupled with the Ricci flow on complete Riemannian manifolds. As an application we construct complete manifolds with bounded nonnegative sectional curvature of dimension greater than or equal to fo…
Study of energy functional on Deligne-Hitchin moduli space sections.
problem Understanding energy functionals on sections of Deligne-Hitchin moduli space.
method Generalizes energy of equivariant harmonic maps to holomorphic sections, links to meromorphic connections, and uses Willmore energy analogy.
result Shows functional is essentially Willmore energy for certain sections, distinguishes new components from twistor lines.
The paper estimates variance of random sections on complex manifolds.
problem Estimating variance of random holomorphic sections on compact Kahler manifolds.
method Analyzes a sequence of smooth Hermitian holomorphic line bundles on a compact Kahler manifold X, considering specific probability measures.
result Provides variance estimates for various measures including Gaussian and Fubini-Study measures.
Conditions for flat manifolds as cusp cross-sections in arithmetic hyperbolic manifolds.
problem Determining when a flat manifold can be a cusp cross-section in arithmetic hyperbolic manifolds.
method Analyzing rational representations of holonomy groups and quasi-arithmetic manifolds.
result Conditions for a flat manifold to appear as a cusp cross-section in every commensurability class of arithmetic hyperbolic manifolds.
Every biharmonic Wintgen ideal submanifold in a Riemannian manifold of constant sectional curvature is either minimal or has constant mean curvature.
problem Biharmonic Wintgen ideal submanifolds in Riemannian manifolds of constant sectional curvature
method Show that every biharmonic Wintgen ideal submanifold in a Riemannian manifold of nonpositive constant sectional curvature is minimal and that every biharmonic Wintgen ideal submanifold in a Riemannian manifold of positive constant sectional curvature has constant mean curvature.
result Partial affirmative answers to Chen's conjecture, generalized Chen's conjecture in hyperbolic spaces, and Balmuş-Montaldo-Oniciuc conjecture in spheres within the class of Wintgen ideal submanifolds.
The classical Hadamard three circle theorem is generalized to complete Kähler manifolds. More precisely, we show that the nonnegativity of the holomorphic sectional curvature is a necessary and sufficient condition for the three circle theorem. As corollaries, two sharp monotonicity formulae for holomorphic functions a…
Geometrically, Kostant's Convexity Theorem is extended to submetries with a fat section.
problem Extending Kostant's Convexity Theorem to a broader class of representations.
method Introducing a new concept of 'fat section' and proving the theorem for submetries with this property.
result Kostant's Convexity Theorem is partially extended to submetries with a fat section.
Motivated by a question of Hirzebruch on the possible topological types of cusp cross-sections of Hilbert modular varieties, we give a necessary and sufficient condition for a manifold M to be diffeomorphic to a cusp cross-section of a Hilbert modular variety. Specialized to Hilbert modular surfaces, this proves that e…
The sectional curvature of the volume preserving diffeomorphism group of a Riemannian manifold M can give information about the stability of inviscid, incompressible fluid flows on M. We demonstrate that the submanifold of the volumorphism group of the solid flat torus generated by axisymmetric fluid flows with swi…