Finite capacity proved for a specific manifold.
problem Calculating the capacity of a specific manifold's cotangent bundle.
method Analyzing the unit disk bundle of the cotangent bundle.
result The Hofer-Zehnder capacity is finite for a rationally inessential manifold.
We give the first examples of rationally inessential but macroscopically large manifolds. Our manifolds are counterexamples to the Dranishnikov rationality conjecture. For some of them we prove that they do not admit a metric of positive scalar curvature, thus satisfy the Gromov positive scalar curvature conjecture. Fu…
Proves manifolds with positive scalar curvature have inessential covers.
problem Positive scalar curvature manifolds with abelian fundamental groups.
method Proves existence of finite covers with deformed classifying maps.
result Existence of strongly inessential covers for most manifolds.
We show that for a rationally inessential orientable closed n-manifold M whose fundamental group π is a duality group the macroscopic dimension of its universal cover is strictly less than n:$$ \dim_{MC}\Wi M<n.$$ As a corollary we obtain the following 0.1 Theorem. The inequality $ \dim_{MC}\Wi M<n$ holds for t…
Study essentiality and simplicial volume of manifolds fibered over spheres.
problem When manifolds fibered over spheres are essential or have positive simplicial volume.
method Analyzing mapping tori and fiber bundles over spheres, using results on macroscopic dimension and characteristic classes.
result Mapping tori of odd-dimensional manifolds with non-zero simplicial volume are essential, while fiber bundles over spheres of dimension d > 1 have zero simplicial volume.
Maps obstructing positive scalar curvature and inessentialness.
problem Obstructing metrics with positive scalar curvature and inessential submanifolds.
method Construction of wrong way maps in uniformly finite homology and homology of groups.
result Obstructions to positive scalar curvature metrics and inessential submanifolds.
Study explores solvable Lie groups' actions on closed Lorentzian manifolds.
problem Understanding conformal actions of solvable Lie groups on closed Lorentzian manifolds.
method Analyzes the identity component of the conformal group and uses algebraic hypotheses.
result Establishes conditions for conformal flatness and local embeddings.
Study on 4-manifolds with positive scalar curvature.
problem Characterizing 4-manifolds with positive scalar curvature.
method Geography problem approach focusing on four-dimensional case.
result Survey of mathematical work and strengthening of Carr's result.
Homotopy method removes isolated zeroes in continuous maps.
problem Removing isolated zeroes in continuous maps with local obstructions.
method Constructing a continuous homotopy to a nonvanishing map.
result Existence of a continuous homotopy to a nonvanishing map for certain dimensions.
We show that after generic filling along a torus boundary component of a 3-manifold, no two closed, 2-sided, essential surfaces become isotopic, and no closed, 2-sided, essential surface becomes inessential. That is, the set of essential surfaces (considered up to isotopy) survives unchanged in all suitably generic Deh…
Researchers compute equivariant bordism of certain manifolds.
problem Computing the equivariant bordism of free oriented (Z/p)n-manifolds. method Investigation of the submodule of the Brown-Peterson homology of B(Z/p)n generated by elements coming from proper subgroups. result The module is canonically isomorphic to a direct sum of suspensions of multiple tensor products of Ω∗SO(BZ/p), and is generated by products of standard lens spaces. Using the theory of Weyl structures, we give a natural generalization of the notion of essential conformal structures and conformal Killing fields to arbitrary parabolic geometries. We show that a parabolic structure is inessential whenever the automorphism group acts properly on the base space. As a corollary of the g…
Develops intrinsic curved cosets for Cartan geometries.
problem Defines curved cosets for arbitrary Cartan geometries.
method Defines intrinsic holonomy group and curved cosets.
result Curved cosets retain characteristics of homogeneous counterparts and behave well under automorphisms.
Paper simplifies concentration inequalities for easier probabilistic analysis.
problem Complexity in probabilistic analysis of random variables.
method Compact notations for concentration inequalities.
result Simplified expressions for typical sizes and tails of random variables.
Classifies 3-manifolds bounding rational homology balls.
problem Identifying 3-manifolds that bound rational homology balls.
method Used constraints from Donaldson's diagonalization theorem and Heegaard Floer correction terms.
result Complete classification of spherical 3-manifolds bounding rational homology balls.
New criteria for non-isometric group actions in metric spaces.
problem Understanding group actions on non-isometric spaces.
method Generalizing results from isometric to continuous group actions.
result Criterion for cocompact cyclic groups to be inessential.
Study shows conditions for rational ellipticity of manifolds with symmetries.
problem Conditions for rational ellipticity of manifolds with symmetries.
method Analyzes conditions on compact simply connected manifolds with G-actions. result Proves rational ellipticity of M/G if M satisfies certain conditions. Classifies rationally elliptic manifolds and proves geometric formality properties.
problem Classifying and understanding geometric formality in rationally elliptic manifolds.
method Cohomology structure analysis and geometric formality criteria.
result Geometrically formal six-dimensional manifolds with specific Betti numbers are identified.
Standard embeddings of Schubert varieties in specific manifolds characterized.
problem Characterizing standard embeddings of Schubert varieties in rational homogeneous manifolds.
method Characterization through varieties of minimal rational tangents.
result Characterization of standard embeddings of smooth Schubert varieties.
New proof for curved 3-cohom manifold rational ellipticity.
problem Rational ellipticity of curved manifolds with specific cohomogeneity.
method Proved rationally elliptic for cohomogeneity-three manifolds with positive curvature and no boundary quotient.
result Closed, simply connected, positively curved, cohomogeneity-three manifolds without boundary are rationally elliptic.
New rational parallelisms found on complex manifolds that are not flat.
problem Finding non-flat rational parallelisms on complex manifolds.
method Examined rational parallelisms on compact complex manifolds, discovering non-flat examples.
result Discovered rational parallelisms on compact complex manifolds that are not flat.
Rational ellipticity proven for G-manifolds with specific quotient properties.
problem Conditions for rational ellipticity in G-manifolds. method Proving rational ellipticity based on quotient properties and manifold submetries.
result Compact, simply connected G-manifolds with certain quotient properties are rationally elliptic. New manifold models solve duality group problems.
problem Finding rational manifolds with specific properties.
method Using rational manifold models to study duality groups.
result Closed manifolds with rationally acyclic covers found.
The paper constructs new rational homology 3-spheres bounding rational homology 4-balls.
problem Constructing rational homology 3-spheres that bound rational homology 4-balls.
method Exploring plumbed 3-manifolds and using rational homology circles.
result Infinite families of rational homology 3-spheres that bound rational homology 4-balls.
We give a characterization of closed, simply connected, rationally elliptic 6-manifolds in terms of their rational cohomology rings and a partial classification of their real cohomology rings. We classify rational, real and complex homotopy types of closed, simply connected, rationally elliptic 7-manifolds. We give par…
Classifies torus bundles bounding 4-manifolds with rational homology.
problem Classifying torus bundles over the circle that bound 4-manifolds with rational homology.
method Completely classified torus bundles over the circle that bound 4-manifolds with rational homology.
result Completely classified torus bundles over the circle that bound 4-manifolds with rational homology.
Upper bound found for torus actions on certain manifolds.
problem Bounding torus actions on rationally elliptic manifolds.
method Classifying manifolds with maximal-rank torus actions up to equivariant rational homotopy type.
result An upper bound on torus rank for these manifolds.
Paper introduces a new invariant for knots in contact 3-manifolds.
problem Bounding rational Thurston-Bennequin invariants for knots.
method Introduces a rational τ invariant for rationally null-homologous knots.
result Provides an upper bound for the sum of rational Thurston-Bennequin invariant and rational rotation number.
Given two maps f1 and f2 from the sphere Sm to an n-manifold N, when are they loose, i.e. when can they be deformed away from one another? We study the geometry of their (generic) coincidence locus and its Nielsen decomposition. On the one hand the resulting bordism class of coincidence data and the corresponding Niels…
The study explores geometric formality in specific dimensions of rationally elliptic manifolds.
problem Investigating geometric formality for rationally elliptic manifolds of dimensions 6 and 7.
method Analyzing real cohomology and homotopy properties to determine geometric formality.
result Geometrically formal six-dimensional biquotients with b2=3 have symmetric space cohomology. New 4-manifold accounts for rationally slice knots.
problem Characterize rationally slice knots.
method Show independence of construction of rational homology ball VK. result Single 4-manifold accounts for all known rationally slice knots.
The study of symplectic fillings for rational cuspidal curves.
problem Understanding symplectic fillings of contact manifolds associated with rational cuspidal curves.
method Exploration through Stein handlebodies and rational blow-downs.
result Examples of contact manifolds that are links of normal surface singularities, and those that do not admit symplectic fillings.
Study embeds ruled surfaces into symplectic manifolds, finds Stein fillability results.
problem Embedding rational ruled surfaces into symplectic manifolds.
method Analyzes symplectic hyperplane sections of rational ruled surfaces.
result Obtains Stein fillability results for rational ruled surfaces.
Study rational homology spheres with two legs, classifying those bounding rational balls.
problem Classifying Seifert rational homology spheres with two complementary legs.
method Complete classification of Seifert manifolds with 3 exceptional fibers and two complementary legs.
result Classification of Seifert manifolds bounding rational homology balls.
The study classifies manifolds with specific rational cohomology properties.
problem Classifying smooth manifolds with rational cohomology of a sphere.
method Cohomogeneity one actions on smooth manifolds.
result Such manifolds are diffeomorphic to specific known manifolds.
The paper shows how uniform RC-positivity on manifolds implies projectivity and rational connectedness.
problem Understanding the conditions for projectivity and rational connectedness in Kähler manifolds.
method Analyzing the properties of uniformly RC-positive metrics and their relationship to projectivity and rational connectedness.
result Uniformly RC-positive metrics on rationally connected manifolds imply projectivity and rational connectedness.
New rational homology 3-spheres found that can't bound definite 4-manifolds.
problem Finding rational homology 3-spheres that can't bound simply connected definite 4-manifolds.
method Constructing infinite families of irreducible rational homology 3-spheres.
result Infinite families of rational homology 3-spheres that are homology cobordant to given examples but can't bound simply connected definite 4-manifolds.
The paper studies Kähler manifolds with partially semi-positive curvature and rational connectedness.
problem Analyzing compact Kähler manifolds with partially semi-positive curvature and rational connectedness.
method Proving rational connectedness for manifolds with BC-p positive tangent bundles, and applying these results to curvature conditions. result Confirming a conjecture and generalizing results on rational connectedness and curvature conditions.
Study minimal rational curves on complex manifolds with isotropic VMRT.
problem Understanding minimal rational curves tangent to distributions on complex manifolds.
method Partial equivariant compactification of metabelian groups.
result Any isotropic VMRT can be realized as VMRT of minimal rational curves tangent to a distribution.
New rational homology balls found in specific knot 2-handlebodies.
problem Finding rational homology balls in 2-handlebodies.
method Proving existence and properties of rational homology balls through smooth embeddings and rational blow-ups.
result Rationally blown-up 4-manifolds cannot be obtained by ordinary blow-ups under certain conditions.
We investigate rational homology cobordisms of 3-manifolds with non-zero first Betti number. This is motivated by the natural generalization of the slice-ribbon conjecture to multicomponent links. In particular we consider the problem of which rational homology S1×S2's bound rational homology S1×D3'…
Classifies certain types of geometric shapes with specific algebraic properties.
problem Classifying geometric shapes with specific algebraic properties.
method Classifies simply connected, closed cohomogeneity one manifolds with specific rational cohomology properties.
result Classifies manifolds with singly generated or 4-periodic rational cohomology.
Survey on minimal rational curves and their geometric structures.
problem Germ-equivalence problem of minimal rational curves on uniruled projective manifolds.
method Analysis of isotrivial families of projective varieties and G-structures.
result Natural G-structure on Zariski-open subset of uniruled projective manifolds.
New 3-manifolds bound rational 4-balls through specific operations.
problem Finding rational homology 3-spheres that bound rational homology 4-balls.
method Two operations that preserve lattice embedding obstruction to bounding rational homology balls.
result Explicit examples of rational surgeries on torus knots that bound rational homology balls.
The Nielsen Conjecture for Homeomorphisms asserts that any homeomorphism f of a closed manifold is isotopic to a map realizing the Nielsen number of f, which is a lower bound for the number of fixed points among all maps homotopic to f. The main theorem of this paper proves this conjecture for all orientation pre…
Nonorientable surfaces in rational 4-manifolds linked to Pontrjagin square.
problem Representing nonorientable Lagrangian surfaces in rational 4-manifolds.
method Using the Pontrjagin square of classes in H2(X;Z2) to determine nonorientable surfaces. result A nonzero class A in H2(X;Z2) is represented by a nonorientable embedded Lagrangian surface if and only if P(A)≡(L)mod4. Smooth Schubert varieties are rigid in rational homogeneous manifolds.
problem Classifying and understanding rigidity of Schubert varieties.
method Classification and analysis of homology classes.
result Smooth Schubert varieties are rigid unless they are linear.
The paper connects curvature positivity to rational connectedness in complex geometry.
problem Establishing a geometric criterion for rational connectedness.
method Uhlenbeck-Yau's continuity method applied to mean curvature positivity.
result Holomorphic tangent bundle mean curvature positivity is equivalent to rational connectedness of compact Kähler manifolds.