Method counts zeros of Betti map for elliptic surface sections.
problem Counting zeros of Betti map for elliptic surface sections.
method Differential-geometric approach using Kähler metrics.
result Explicit linear estimates of Betti map multiplicities.
This paper gives a detailed construction of Seiberg-Witten-Floer homology for a closed oriented 3-manifold with a non-torsion $\spinc$ structure. Gluing formulae for certain 4-dimensional manifolds splitting along an embedded 3-manifold are obtained.
We extend the techniques in a previous paper to calculate the Heegaard Floer homology groups for fibered 3-manifolds M whose monodromy is a power of a Dehn twist about a genus-1 separating circle on a surface of genus g > 1. We only consider non-torsion Spin^c-structures on M.
We exhibit the first examples of links which are homologically thin but not quasi-alternating. To show that they are not quasi-alternating, we argue that none of their branched double-covers bounds a negative definite 4-manifold with non-torsion H_1. Using this method, we also complete the determination of the quasi-al…
Let M be a closed, connected, orientable 3-manifold. The purpose of this paper is to study the Seiberg-Witten Floer homology of M given that S^1 X M admits a symplectic form. In particular, we prove that M fibers over the circle if M has first Betti number 1 and S^1 X M admits a symplectic form with non-torsion canonic…
We determine the Seiberg-Witten-Floer homology groups of the three-manifold which is the product of a surface of genus g≥1 times the circle, together with its ring structure, for spin-c structures which are non-trivial on the three-manifold. We give applications to computing Seiberg-Witten invariants of four-man…
The study shows infinitely many Reeb orbits on star-shaped hypersurfaces with growth rate like prime numbers.
problem Growth rate of Reeb orbits on star-shaped hypersurfaces.
method Analyzing fiberwise star-shaped hypersurfaces in cotangent bundles with topological conditions.
result The number of Reeb orbits with period at most T grows at least like T/log(T).
The twisted torsion of a 3-manifold is well-known to be zero whenever the corresponding twisted Alexander module is non-torsion. Under mild extra assumptions we introduce a new twisted torsion invariant which is always non-zero. We show how this torsion invariant relates to the twisted intersection form of a bounding 4…
We show the non-vanishing of cohomology groups of sufficiently small congruence lattices in SL(1,D), where D is a quaternion division algebras defined over a number field E contained inside a solvable extension of a totally real number field. As a corollary, we obtain new examples of compact, arithmetic, hyperbol…
The study explores S1-invariant Laplacian flow on 6-manifolds.
problem Finding S1-invariant solutions to the Laplacian flow. method Derive and analyze S1-invariant evolution equations. result Discovery of first inhomogeneous shrinking solitons.
Given a finite metric CW complex X and an element α∈πn(X), what are the properties of a geometrically optimal representative of α? We study the optimal volume of kα as a function of k. Asymptotically, this function, whose inverse, for reasons of tradition, we call the volume distortion, turns out to be an…
Paper introduces simplified formulas for Milnor's triple linking number.
problem Computational difficulty in calculating Jones polynomial for topological polymers.
method Developed Gauss diagram formulas for Milnor's Vassiliev invariants.
result Introduced non-torsion valued Milnor's triple linking number.
We calculate the Heegaard Floer homologies$HF^+(M,s) for mapping tori M associated to certain surface diffeomorphisms, where s is any Spin^c structure on M whose first Chern class is non-torsion. Let gamma and delta be a pair of geometrically dual nonseparating curves on a genus g Riemann surface Sigma_g, and let sigma…
The study classifies and disproves gradient properties of certain solitons on specific Lie groups.
problem Characterizing and proving non-graduation of solitons on specific Lie groups.
method Proving structure theorems and analyzing specific examples of solitons.
result Examples of solitons that cannot be made gradient, including specific Lie groups.
Given a self-diffeomorphism h of a closed, orientable surface S and an embedding f of S into a three-manifold M, we construct a mutant manifold N by cutting M along f(S) and regluing by h. We will consider whether there are any gluings such that for any embedding, the manifold and its mutant have isomorphic Heegaard Fl…
Study of solitons in Laplacian flow on 7-manifolds.
problem Understanding finite-time singularities of Laplacian flow on G_2-structures.
method Systematic study of cohomogeneity-one solitons with Sp(2) and SU(3) symmetry groups.
result Existence and classification of smoothly-closing solitons, including complete and incomplete solutions.
The Chern sectional curvature of a Hermitian manifold is derived and related to Kähler metrics.
problem Understanding the relationship between Chern and Riemann sectional curvatures on Hermitian manifolds.
method Derivation of Chern sectional curvature expressions and subsequent results on Ricci and scalar curvatures.
result A Hermitian metric is Kähler if and only if its Riemann sectional curvature equals its Chern sectional curvature.
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.
Defines basic sections of LA-groupoids for simpler modeling.
problem Modeling sections of stacky Lie algebroids.
method Introduces basic sections with injective core-anchor map.
result Basic sections are Morita invariant and equivalent to multiplicative sections.
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.
New proof shows holomorphic sectional curvature fully determines curvature tensor.
problem Determining the curvature tensor from holomorphic sectional curvature.
method Representation-theoretic means to calculate L2-norm of holomorphic sectional curvature. result Holomorphic sectional curvature fully determines the curvature tensor.
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.
Let P(M,G) be a principal fiber bundle and E(M,N,G,P) be an associate fiber bundle. Our interested is to study harmonic sections of the projection πE of E into M. Our first purpose is to give a stochastic characterization of harmonic section from M into E and a geometric characterization of harmonic se…
Classifies surfaces of section for Seifert fibrations.
problem Classifying surfaces of section for Seifert fibrations.
method Discussing branched coverings and relating surfaces of section to algebraic curves.
result Relates surfaces of section to algebraic curves in weighted complex projective planes.
Some optimization problems coming from the Differential Geometry, as for example, the minimal submanifolds problem and the harmonic maps problem are solved here via interior solutions of appropriate multitime optimal control problems. Section 1 underlines some science domains where appear multitime optimal control prob…
The paper proves that Kähler manifolds with positive holomorphic sectional curvature have a limited diameter.
problem Diameter rigidity of Kähler manifolds with positive holomorphic sectional curvature.
method Establishing diameter rigidity for Kähler manifolds with positive holomorphic sectional curvature.
result Kähler manifolds with positive holomorphic sectional curvature have a limited diameter.
New method constructs Birkhoff sections for pseudo-Anosov flows with controlled complexity.
problem Constructing Birkhoff sections for pseudo-Anosov flows with specific properties.
method Uses connection between pseudo-Anosov flows and veering triangulations to explicitly construct sections with controlled complexity.
result Shows that any transitive pseudo-Anosov flow has a Birkhoff section with two boundary components.
The paper explores conditions for sections in Lefschetz fibrations and bundles over 2-complexes.
problem Conditions for sections in Lefschetz fibrations and bundles over 2-complexes.
method Algebraic criteria and topological conditions.
result Complete algebraic criterion for sections in Lefschetz fibrations over the disk.
Study shows normal distribution in divisor counts of random sections on complex manifolds.
problem Distribution of divisors on complex manifolds.
method Central limit theorem for smooth linear statistics of Gaussian sections.
result Asymptotic normality of divisor counts.
We prove the following results: An almost Hermitian manifold of indefinite metric is of pointwise constant holomorphic sectional curvature if the holomorphic sectional curvature is bounded from above and from below. If the antiholomorphic sectional curvature is bounded either from above or from below, then the manifold…
It is proved that if an AK2-manifold of dimension greater or equal to 6 is of pointwise constant antiholomorphic sectional curvature, then it is a 6-dimensional manifold of constant negative sectional curvature or a Kähler manifold of constant holomorphic sectional curvature.
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.
Formula for sectional curvature on 2D Lorentzian manifolds derived.
problem Calculating sectional curvature on 2D Lorentzian manifolds.
method Obtained a formula for sectional curvature.
result Formula for sectional curvature on 2D Lorentzian manifolds.
The φ-sectional curvature of statistical structures on almost contact metric manifolds is always non-positive.
problem Analyzing the φ-sectional curvature of statistical structures on almost contact metric manifolds.
method Investigating the φ-sectional curvature induced by a statistical structure and deriving sufficient conditions.
result The φ-sectional curvature is always non-positive.
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 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.
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.
The study examines when a section exists for graph configuration spaces.
problem When a surjective map of configuration spaces has a section.
method Investigates homotopy type dependence and provides construction techniques.
result Provides a complete answer to when the answer depends only on the graph's homotopy type.
We prove that a product complex manifold cannot admit a complete Kähler metric with sectional curvature K<c<0 and Ricci curvature Ric>d, where c and d are constants. In particular, a product domain in $\C$ cannot cover a compact Kähler manifold with negative sectional curvature. On the other hand, we observe …
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.
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.
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 paper extends a theorem about Kähler manifolds with quasi-negative curvature to almost quasi-negative curvature.
problem Understanding the ampleness of canonical line bundles for Kähler manifolds with specific curvature properties.
method Introducing a new notion of almost quasi-negative holomorphic sectional curvature and extending the theorem to this setting.
result The theorem is extended to compact Kähler manifolds with almost quasi-negative holomorphic sectional curvature, and a gap-type theorem is derived.
Segre varieties' hyperplane sections are unstable under certain conditions.
problem Stability of hyperplane sections of Segre varieties under different conditions.
method Proving instability with respect to any polarization for non-smooth or meqn cases. result Normal hyperplane sections of Segre varieties are K-unstable under specified conditions.
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.
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.
The paper introduces sections in metric spaces with properties related to Ahlfors-David regularity and convexity.
problem Understanding properties of sections in metric spaces.
method Definition and investigation of intrinsically quasi-isometric sections in metric spaces.
result Properties of sections, including Ahlfors-David regularity and convexity, are defined and investigated.
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…