Paper explores cohomology classes on non-compact almost Kähler manifolds.
problem Understanding cohomology classes induced by symplectic forms.
method Provides criteria for non-trivial classes in Lp cohomology. result Symplectic forms induce non-trivial classes in Lp cohomology. We prove a relation between the ∂ˉM cohomology of a minimal orbit M of a real form G0 of a complex semisimple Lie group G in a flag manifold G/Q and the Dolbeault cohomology of the Matsuki dual open orbit X of the complexification K of a maximal compact subgroup K0 of G0, under the assum…
Given a compact stratified pseudomanifold with a Thom-Mather stratification and a class of riemannian metrics over its regular part, we study the relationships between the L2 de Rham and Hodge cohomology and the intersection cohomology of X associated to some perversities. More precisely, to a kind of metric whi…
Generalizes Dolbeault cohomology computation to Levi-flat CR structures on compact Lie groups.
problem Computing Dolbeault cohomology for Levi-flat CR structures on compact Lie groups.
method Algebraic classification of left-invariant CR structures combined with Pittie's result on compact Lie groups.
result Generalization of Dolbeault cohomology computation to Levi-flat CR structures.
Proves conjectures about maximal antipodal sets in symmetric and generalised symmetric spaces.
problem Cohomological descriptions of maximal antipodal sets in symmetric spaces.
method Equivariant cohomology theory.
result Proves several long-standing conjectures by Chen--Nagano and extends them to generalised symmetric spaces.
Let Mˉ be a manifold with boundary Y which is the total space of a fibre bundle, and is defined by the vanishing of a boundary defining function, x. We prove L2 Hodge and signature theorems for M endowed with a metric of the form dx2+x2ch+k, where k is the lift to Y of the metric on the b…
Let X be any subanalytic compact pseudomanifold. We show a De Rham theorem for L∞ forms. We prove that the cohomology of L∞ forms is isomorphic to intersection cohomology in the maximal perversity.
Study L2-cohomology in unbounded geometry manifolds.
problem Invariance of L2-cohomology under quasi-isometries on unbounded ends. method Uniform homotopy equivalence, quasi-isometry on unbounded ends, mapping cone for L2-cohomology. result Invariance of L2-cohomology groups under quasi-isometry on unbounded ends. Researchers compute Hochschild cohomology of Grassmannians.
problem Computing Hochschild cohomology of Grassmannians.
method Explicit description of Gerstenhaber algebra structure, vanishing of higher cohomology.
result Decomposition of Hochschild cohomology concentrated in global sections for certain Grassmannians.
Study on almost complex structures with maximal Nijenhuis tensor rank and cohomological properties.
problem Maximally non-integrable almost complex structures and their cohomological properties.
method h-principle and topological invariants characterization.
result Existence of almost complex structures with maximal Nijenhuis tensor rank on parallelizable and certain manifolds.
Develops o-minimal de Rham cohomology for smooth manifolds.
problem Cohomology of smooth manifolds in o-minimal settings.
method Defines o-minimal de Rham cohomology for smooth manifolds in an o-minimal expansion of the real field.
result Establishes properties of o-minimal cohomology groups and invariance under diffeomorphisms.
Maximal measurable cocycles of complex hyperbolic lattices are cohomologous to representations.
problem Characterizing maximal measurable cocycles of complex hyperbolic lattices.
method Utilizing Zimmer's Superrigidity Theorem and proving the existence of a boundary map.
result Maximal measurable cocycles are cohomologous to representations of PU(p,1) into SU(m,n).
Constructive approach to Lie algebra gradings, computing maximal and enumerating all gradings.
problem Computing and enumerating gradings of Lie algebras.
method Constructive approach to torsion-free gradings, computation of maximal grading, enumeration of all gradings.
result Computation of a maximal grading and enumeration of all torsion-free gradings.
Defines Killing spinors and bosonic backgrounds in 5D supergravity.
problem Characterizing backgrounds in 5D supergravity.
method Calculates Spencer cohomology, defines Killing spinors, and imposes constraints on spinor connection curvature.
result Recover field equations of 5D supergravity and find new field equations for sp(1)-valued one-form. This paper defines maximal measurable cocycles for surface groups into Hermitian Lie groups and studies their algebraic hulls.
problem Understanding maximal measurable cocycles for surface groups into Hermitian Lie groups.
method Introducing the notion of maximal measurable cocycles, defining Toledo invariant, and studying the algebraic hulls.
result The algebraic hull of a maximal cocycle is reductive and its centralizer is compact.
We study an integration theory in circle equivariant cohomology in order to prove a theorem relating the cohomology ring of a hyperkahler quotient to the cohomology ring of the quotient by a maximal abelian subgroup, analogous to a theorem of Martin for symplectic quotients. We discuss applications of this theorem to q…
We describe the cohomology of a specific type of foliation on complex manifolds.
problem Computing the basic cohomology of canonical holomorphic foliations on complex moment-angle manifolds.
method Using an Eilenberg-Moore spectral sequence and the formality of the Cartan model for the torus action.
result The basic cohomology ring of the canonical holomorphic foliation on a moment-angle manifold is similar to that of a complete simplicial toric variety.
Paper extends multiplicative constants to measurable cocycles theory.
problem Maximal measurable cocycles in bounded cohomology.
method Extending multiplicative constants to measurable cocycles theory.
result Defined and studied Cartan invariant for measurable PU(m,1)-cocycles.
The study finds that certain curved manifolds can be mapped to symmetric spaces.
problem Understanding singular Riemannian foliations in positively curved manifolds.
method Generalizing fixed point homogeneous actions to singular Riemannian foliations.
result Positively curved manifolds with point leaf maximal SRF's are diffeo/homeomorphic to compact rank one symmetric spaces.
Paper calculates Torelli group's cohomology second group.
problem Calculating the second rational cohomology group of the Torelli group.
method Building on Hain's and Kupers-Randal-Williams's work, the paper provides an exposition of prerequisite material and the two key results.
result Calculation of the second rational cohomology group of the Torelli group.
The purpose of this paper is applying minimality of hyperplane arrangements to local system cohomology groups. It is well known that twisted cohomology groups with coefficients in a generic rank one local system vanish except in the top degree, and bounded chambers form a basis of the remaining cohomology group. We det…
The paper extends Dolbeault cohomology to almost complex manifolds and provides new tools for studying their properties.
problem Extending Dolbeault cohomology to almost complex manifolds.
method Developed a spectral sequence and harmonic theory for Dolbeault cohomology.
result Dolbeault cohomology can be used to prohibit the existence of nearly Kähler metrics.
Existence of minimizers and singular solutions for Hodge energy on manifolds.
problem Existence of minimizers and singular solutions for Hodge energy.
method Analyzing the generalized minimal surface energy on compact Riemannian manifolds.
result Existence of unique minimizers for k=1 and singular solutions for k>1. We prove that the group of Hamiltonian automorphisms of a symplectic 4-manifold contains only finitely many conjugacy classes of maximal compact tori with respect to the action of the full symplectomorphism group. We also extend to rational and ruled manifolds a result of Kedra which asserts that, if M is a simply co…
Let (M,g) an open and oriented riemannian manifold. The aim of this paper is to study some properties of the two following sequences of L2 cohomology groups: H2,m→Mi(M,g) defined as the image $\im(H^i_{2,min}(M,g)\rightarrow H^i_{2,max}(M,g))$ and Hˉ2,m→Mi(M,g) defined as $\i…
Standard cohomology of Courant algebroids identified via minimal models.
problem Cohomology of Courant algebroids.
method Minimal model construction and Hodge-to-de Rham spectral sequence.
result Standard cohomology of Courant algebroids identified with function space cohomology.
Study Dolbeault cohomology on complex manifolds with torus action.
problem Describe Dolbeault cohomology of complex manifolds with torus action.
method Describe Dolbeault cohomology algebra of canonical foliation, provide dga model, prove Hodge decomposition.
result Hodge decomposition for basic Dolbeault cohomology proved.
We prove existence and regularity of minimizers for Hölder densities over general surfaces of arbitrary dimension and codimension in \(\R^n \), satisfying a cohomological boundary condition, providing a natural dual to Reifenberg's Plateau problem. We generalize and extend methods of Reifenberg, Besicovitch, and Adams,…
We consider the T-equivariant cohomology of Bott-Samelson desingularisations of Schubert varieties in the flag manifold of a connected semi-simple complex algebraic group of adjoint type with maximal torus T. We construct a combinatorially pure (in the sense of T. Braden and R. Macpherson) sheaf on the Bruhat graph of …
We define an integer graded symplectic Floer cohomology and a Fintushel-Stern type spectral sequence which are new invariants for monotone Lagrangian sub-manifolds and exact isotopes. The Z-graded symplectic Floer cohomology is an integral lifting of the usual Z_Sigma(L)-graded Floer-Oh cohomology. We prove the Kunneth…
Study shows equivalence between cohomology class existence and polynomial properties for 3D manifolds.
problem Characterizing closed 3D manifolds based on cohomology class existence and polynomial properties.
method Analyzes closed one-forms and twisted Alexander polynomials in relation to cohomology classes.
result Equivalence between cohomology class existence and polynomial properties for most 3D manifolds.
The study examines stability of fibres on Hopf surfaces as harmonic maps and minimal surfaces.
problem Stability of fibres on Hopf surfaces as harmonic maps and minimal surfaces.
method Construction of Hermitian metrics on Hopf surface and analysis of fibres as harmonic maps and minimal surfaces.
result Two toric fibres are stable minimal surfaces, while others are unstable.
Quiver varieties' geometry at infinity studied using Nakajima metric.
problem Understanding the geometry at infinity of quiver varieties.
method Using Melrose's approach to study the geometry at infinity of the Nakajima metric on reduced Hilbert schemes.
result Quiver varieties are quasi-asymptotically conical under generic conditions.
The space of Lie algebra cohomology is usually described by the dimensions of components of certain degree even for the adjoint module as coefficients when the spaces of cochains and cohomology can be endowed with a Lie superalgebra structure. Such a description is rather imprecise: these dimensions may coincide for co…
Study reveals a link between Ruelle-Pollicott resonances and cohomology eigenvalues for Anosov diffeomorphisms.
problem Understanding the speed of mixing in Anosov diffeomorphisms.
method Investigates Ruelle-Pollicott resonances on manifolds of any dimension, connecting them to cohomology eigenvalues of a quasi-compact transfer operator.
result Established a cohomological bound for the speed of mixing of Anosov diffeomorphisms.
We determine the Killing superalgebras underpinning field theories with rigid unextended supersymmetry on Lorentzian four-manifolds by re-interpreting them as filtered deformations of Z-graded subalgebras with maximum odd dimension of the N=1 Poincaré superalgebra in four dimensions. Part of this calcula…
Proves an index theorem for foliations using spectral triples.
problem Proving an Atiyah L2 covering index theorem for foliations. method Symbol calculus for foliations and spectral triples.
result Induces the same map on K-theory for two types of spectral triples.
Let Y=G/L be a flag manifold for a reductive G and K a maximal compact subgroup of G. We define an equivariant differential operator on G/(L cap K) playing the role of an equivariant Dolbeault Laplacian when restricted to the complex manifold G/L, using a distribution transverse to the fibers and satisfying the Hormand…
The paper extends Euler class theory to measurable cocycles.
problem Understanding the structure of measurable cocycles and their cohomology.
method Constructing a parametrized Euler class in bounded cohomology and studying semicohomologous cocycles.
result The parametrized Euler class vanishes if and only if the cocycle can be lifted and admits an equivariant family of points.
Study transverse Dolbeault cohomology for almost complex structures.
problem Understanding cohomology of transverse structures on manifolds.
method Define transverse Dolbeault cohomology, extend transverse complex structure, introduce involutive limit distribution.
result Cohomology spaces of (p,0) for almost complex structures coincide with transverse Dolbeault cohomology.
If a closed smooth n-manifold M admits a finite cover whose Z/2Z-cohomology has the maximal cup-length, then for any riemannian metric g on M, we show that the systole Sys(M,g) and the volume Vol(M,g) of the riemannian manifold (M,g) are related by the following isosystolic inequality: Sys(M,g)^n \leq n! Vol(M,g). The …
The paper proves vanishing cohomology groups for free boundary hypersurfaces.
problem Proving vanishing cohomology groups for free boundary hypersurfaces.
method Using a universal constant and traceless second fundamental form condition.
result The pth cohomology group of a compact free boundary submanifold vanishes. Proposes MEDM to balance entropy minimization and diversity maximization for better domain adaptation.
problem Trivial solutions in entropy minimization for unsupervised domain adaptation.
method Introduces diversity maximization to balance with entropy minimization, controlled by deep embedded validation.
result MEDM outperforms state-of-the-art methods on four domain adaptation datasets.
Study characterizes cohomology and homotopy types for M-theory extensions.
problem Anomaly cancellation in M-theory extensions.
method Characterized integral cohomology and rational homotopy type of combined fibration.
result Subtle cohomology relations match Green-Schwarz mechanism.
New findings on hypersurfaces in Euclidean space that are both maximal and minimal.
problem Characterizing hypersurfaces in Euclidean space that are both maximal and minimal.
method Analyzing the level curves of the hypersurfaces and showing they are minimal hypersurfaces in the lower-dimensional Euclidean space.
result The level curves of these hypersurfaces are minimal hypersurfaces in the lower-dimensional Euclidean space.
We investigate the duality between minimal surfaces in Euclidean space and maximal surfaces in Lorentz-Minkowski space in the family of rotational surfaces. We study if the dual surfaces of two congruent rotational minimal (or maximal) surfaces are congruent. We show that in the duality process by means of a one-parame…
Maximal surfaces in L3 correspond to timelike minimal surfaces.
problem Establishing a correspondence between maximal and timelike minimal surfaces in L3. method Linear transformation between maximal surfaces and timelike minimal surfaces, preserving singularities and Gauss map.
result One-one correspondence and preservation of properties between maximal and timelike minimal surfaces.
The refined analytic torsion on compact Riemannian manifolds with boundary has been discussed by B. Vertman and the authors, but these two constructions are completely different. Vertman used a double of de Rham complex consisting of the minimal and maximal closed extensions of a flat connection and the authors used we…