Smooth manifolds have equivalent diffeomorphism groups if and only if they are diffeomorphic.
problem Understanding when diffeomorphism groups of smooth manifolds are elementarily equivalent.
method Analyzing the equivalence of Cr and Cs diffeomorphism groups of smooth manifolds. result Equivalent diffeomorphism groups imply diffeomorphic manifolds, strengthening previous results.
We prove:(1) the existence, for every integer n > 3, of a noncompact smooth n-dimensional topological manifold whose diffeomorphism group contains an isomorphic copy of every finitely presented group; (2) a finiteness theorem on finite simple subgroups of diffeomorphism groups of compact smooth topological manifolds.
We construct Anosov diffeomorphisms on manifolds that are homeomorphic to infranilmanifolds yet have exotic smooth structures. These manifolds are obtained from standard infranilmanifolds by connected summing with certain exotic spheres. Our construction produces Anosov diffeomorphisms of high codimension on infranilma…
Smooth solutions found for hydrodynamic equations.
problem Geodesic equations on diffeomorphism groups.
method Proving smoothness of solutions and constructing exponential maps.
result Smooth solutions match boundary conditions.
Theta graph diffeomorphism shows nontrivial mapping class of 4-sphere.
problem Identifying nontrivial elements in the smooth mapping class group of 4-sphere.
method Diagrammatic calculus for smooth mapping class group of 4-sphere, Watanabe's clasper surgery construction.
result Theta graph diffeomorphism is isotopic to a nontrivial element of (1,2)-subgroup.
Real moment-angle manifolds of combinatorially equivalent simple polytopes are equivariantly diffeomorphic.
problem Uniqueness of smooth structures on real moment-angle manifolds.
method Arguments from calculus applied to results from complex moment-angle manifolds.
result Real moment-angle manifolds of combinatorially equivalent simple polytopes are equivariantly diffeomorphic.
Nilmanifolds are shown to be diffeomorphic to trivial bundles over tori.
problem Characterizing diffeomorphisms between nilmanifolds and smooth quasi-projective varieties.
method Analyzing cohomology and diffeomorphism properties of quasi-projective varieties and nilmanifolds.
result Nilmanifolds are diffeomorphic to trivial bundles over tori under certain conditions.
Two smooth 4-manifolds are shown to be diffeomorphic.
problem Identifying diffeomorphic 4-manifolds arising from quotients of S2imesS2. method Free actions of Z/4 on S2imesS2. result Two quotients of S2imesS2 are diffeomorphic. New Virasoro-like structures for circle diffeomorphisms with breaks.
problem Constructing Virasoro-like extensions for non-smooth diffeomorphisms.
method Explicit construction of central extensions of Lie groupoids and algebroids.
result Explicit nontrivial central extensions of Lie groupoids and algebroids for broken diffeomorphisms.
One can define what it means for a compact manifold with corners to be a "contractible manifold with contractible faces." Two combinatorially equivalent, contractible manifolds with contractible faces are diffeomorphic if and only if their 4-dimensional faces are diffeomorphic. It follows that two simple convex polytop…
The paper examines convergence of currents and forms under smooth diffeomorphisms.
problem Analyzing convergence of currents and forms under C0-limits of diffeomorphisms. method Geometric analysis, measure theory, homotopy theory.
result Pushforwards of rectifiable currents converge in the flat norm.
Study controllability of diffeomorphisms of simple polytopes.
problem Controllability of diffeomorphisms of simple polytopes.
method Lie group structure and controllability results for diffeomorphisms of simple polytopes.
result Identity component of the diffeomorphism group is generated by the exponential image.
New cohomology theory reveals Q/Z in group homology.
problem Understanding torsion in group homology of diffeomorphism groups.
method Introduced configured group cohomology, yielding explicit R/Z-valued 3-cocycles. result Found a subgroup isomorphic to $\Q/\Z$ in the third group homology of certain diffeomorphism groups.
In this paper we prove that no complex surface of general type is diffeomorphic to a rational surface, thereby completing the smooth classification of rational surfaces and the proof of the Van de Ven conjecture on the smooth invariance of Kodaira dimension.
The compact-open topology is minimal on diffeomorphism and homeomorphism groups of most smooth manifolds.
problem The minimality of compact-open topology on diffeomorphism and homeomorphism groups.
method Analyzing the compact-open topology on diffeomorphism and homeomorphism groups of smooth manifolds.
result The compact-open topology is minimal on diffeomorphism and homeomorphism groups of most smooth manifolds.
We introduce a new invariant, the real (logarithmic)-Kodaira dimension, that allows to distinguish smooth real algebraic surfaces up to birational diffeomorphism. As an application, we construct infinite families of smooth rational real algebraic surfaces with trivial homology groups, whose real loci are diffeomorphic …
We show that on a closed smooth manifold M equipped with k fiber bundle structures whose vertical distributions span the tangent bundle, every smooth diffeomorphism f of M sufficiently close to the identity can be written as a product f=f1...fk, where fi preserves the ith-fiber. The factors …
Smooth isotopy on cube saves energy with extra dimensions.
problem Constructing an isotopy with infinite kinetic energy.
method Explicit construction of isotopy on [0,1]n. result Existence of isotopy with infinite kinetic energy.
Researchers relax the CVF's smoothness requirement to create more flexible flow models.
problem Challenges in constructing flexible density models due to the CVF's smoothness requirement.
method Introduce L-diffeomorphisms as generalized transformations that may violate smoothness on zero Lebesgue-measure sets. result The relaxation allows for the use of non-smooth activation functions like ReLU in residual flows.
Every smooth homotopy 4-sphere is diffeomorphic to the 4-sphere.
We prove that a compactly supported homeomorphism of a smooth manifold of dimension greater or equal to 5 can be approximated uniformly by compactly supported diffeomorphisms if and only if it is isotopic to a diffeomorphism. If the given homeomorphism is in addition volume preserving, then it can be approximated unifo…
Criteria for lifting manifold diffeomorphisms to vector bundle automorphisms.
problem Lifting diffeomorphisms to vector bundles.
method Criteria for lifting diffeomorphisms to linear automorphisms of vector bundles.
result Criteria for lifting diffeomorphisms to linear automorphisms of vector bundles.
In this paper we discuss the relationship between groups of diffeomorphisms of spheres and balls. We survey results of a topological nature and then address the relationship as abstract (discrete) groups. We prove that the identity component Diff_0(S^{2n-1}) of the group of smooth diffeomorphisms of S^{2n+1} admits no …
Study on controllability and groups of manifolds with boundaries.
problem Controllability of vector fields on manifolds with boundaries.
method Establish controllability results for diffeomorphism groups of manifolds with smooth boundaries.
result Diffeomorphism groups of manifolds with smooth boundaries form fibre bundles and are generated by the exponential map.
For any smooth compact manifold W of dimension at least two we prove that the classifying spaces of its group of diffeomorphisms which fix a set of k points or k embedded disks (up to permutation) satisfy homology stability. The same is true for so-called symmetric diffeomorphisms of W connected sum with k co…
Paper proves every stable 4-sphere has a unique diffeomorphism class.
problem Identifying stable 4-spheres and their diffeomorphisms.
method Using Wall's result and properties of surface-knot spaces.
result Every stable 4-sphere has a unique orientation-preserving diffeomorphism class.
Unified framework for smooth structures on coadjoint orbits.
problem Smooth structures on coadjoint orbits of diffeomorphism groups.
method Decorated and augmented nonlinear Grassmannians, functors, smooth structure.
result Uniform description of coadjoint orbits' smooth structures.
It is observed that on many 4-manifolds there is a unique smooth structure underlying a globally hyperbolic Lorentz metric. For instance, every contractible smooth 4-manifold admitting a globally hyperbolic Lorentz metric is diffeomorphic to the standard R4. Similarly, a smooth 4-manifold homeomorphic to the produc…
New corks found that are not strong and exotic.
problem Finding non-strong corks and exotic examples.
method Constructing new corks and analyzing their properties.
result First non-strong corks and new exotic examples.
We consider the group of smooth diffeomorphisms of the circle. We show that any recurrent f (in the sense that {fn}n∈Z is not discrete) is in fact a distortion element (in the sense that its iterates can be written as short compositions involving finitely many smooth diffeomorphisms). Thus rotations are d…
We classify up to diffeomorphism all smooth manifolds homeomorphic to the complex projective m-space CPm for m=5,6,7 and 8. As an application, for m=7 and 8, we compute the smooth tangential structure set of CPm and obtain a bound on the number of smooth homotopy complex projec…
The paper studies exotic diffeomorphisms of 4-manifolds with b_+ = 2.
problem Examining exotic diffeomorphisms in 4-manifolds with a specific topological invariant.
method Utilizes Seiberg-Witten invariants for 1-parameter families of 4-manifolds and a gluing formula for connected sums.
result Proves that the mapping class group of certain 4-manifolds is not finitely generated and surjects to Z^∞.
The study finds non-pseudoisotopic diffeomorphisms in certain 4-manifolds.
problem Identifying diffeomorphisms that are homotopic but not pseudoisotopic.
method Examples of diffeomorphisms in specific 4-manifolds.
result No non-pseudoisotopic diffeomorphisms in orientable 4-manifolds with free fundamental groups.
New 4-manifolds with exotic diffeomorphisms found.
problem Existence of exotic diffeomorphisms in 4-manifolds.
method Proves existence of infinitely many contractible 4-manifolds with exotic diffeomorphisms.
result Infinitely many contractible 4-manifolds with absolutely exotic diffeomorphisms.
Classifies smooth manifolds homotopy equivalent to sphere products
problem Classifying smooth manifolds homotopy equivalent to sphere products
method Using normal-invariant map and explicit families of manifolds
result Classifies smooth manifolds up to almost diffeomorphism
The paper classifies smooth structures on product manifolds of 3-connected 8-manifolds with spheres.
problem Classifying smooth structures on product manifolds.
method Computational and classification methods for concordance and diffeomorphism.
result Diffeomorphism classification of MimesS1 for specific M and k. Dual pairs constructed for volume preserving diffeomorphisms using symplectic geometry.
problem Understanding the group of volume preserving diffeomorphisms through symplectic geometry.
method Using cotangent bundles of spaces of smooth embeddings, symplectic reduction, and nonlinear Grassmannians of augmented submanifolds.
result Descriptions of coadjoint orbits of the group of volume preserving diffeomorphisms in terms of submanifolds of augmented spaces.
Let X be a smooth, compact, oriented 4-manifold. Building upon work of Li-Liu, Ruberman, Nakamura and Konno, we consider a families version of Seiberg-Witten theory and obtain obstructions to the existence of certain group actions on X by diffeomorphisms. The obstructions show that certain group actions on $H^2(X…
Book on infinite-dimensional Lie groups, covering basics and various classes.
problem Understanding Lie groups in infinite-dimensional spaces.
method Develops smooth manifolds and Lie groups in locally convex spaces, discussing various classes.
result Detailed exploration of infinite-dimensional Lie groups and their properties.
A smooth diffeomorphism is said to be distributionally uniquely ergodic (DUE for short) when it is uniquely ergodic and its unique invariant probability measure is the only invariant distribution (up to multiplication by a constant). Ergodic translations on tori are classical examples of DUE diffeomorphisms. In this ar…
Every homeomorphism of Euclidean space is a commutator of two homeomorphisms.
problem Understanding the structure of homeomorphisms in Euclidean space.
method Proving every orientation-preserving homeomorphism can be written as a commutator of two such homeomorphisms.
result Every orientation-preserving homeomorphism of Euclidean space is a commutator of two homeomorphisms.
Detects exotic embeddings in 4-manifolds with boundary.
problem Detecting exotic diffeomorphisms in 4-manifolds with boundary.
method Defined family versions of Kronheimer-Mrowka invariants and used them to find exotic embeddings.
result First example of exotic 3-spheres in a smooth closed 4-manifold with diffeomorphic complements.
Generators of the 4-sphere's smooth mapping class group via diffeomorphisms of Montesinos twins.
problem Understanding the smooth mapping class group of the 4-sphere.
method Homomorphism from loops of 2-spheres to smooth mapping class group, generators as twists along Montesinos twins.
result Generators of the image of the homomorphism as diffeomorphisms of Montesinos twins.
We construct smooth 4-manifolds homeomorphic but not diffeomorphic to CP^2+6CP^2-bar.
In this article, we study the smooth mapping class group of a surface S relative to a given Cantor set, that is the group of isotopy classes of orientation-preserving smooth diffeomorphisms of S which preserve this Cantor set. When the Cantor set is the standard ternary Cantor set, we prove that the subgroup consisting…
Proves rigidity of 3D partially hyperbolic systems via autonomous dynamics.
problem Rigidity of partially hyperbolic diffeomorphisms in 3D.
method Introducing autonomous dynamical systems to prove rigidity.
result Rigidity of partially hyperbolic diffeomorphisms on 3-manifolds.
FineMorphs models smooth transformations for multivariate regression.
problem Efficiently modeling complex transformations for multivariate regression.
method Optimal control of affine and diffeomorphic transformations using smooth vector fields.
result FineMorphs can reduce dimensionality and adapt to large datasets.
In terms of Turaev's shadows, we provide a sufficient condition for a compact, smooth, acyclic 4-manifold with boundary the 3-sphere to be diffeomorphic to the standard 4-ball. As a consequence, we prove that if a compact, smooth, acyclic 4-manifold with boundary the 3-sphere has shadow-complexity at most 2, then it is…