Study shows compact Sasakian manifolds are locally Heisenberg up to deformation.
problem Characterizing compact Sasakian manifolds.
method Analyzing basic Chern classes and using left invariant Sasakian structures.
result Compact Sasakian manifolds are locally isomorphic to the real Heisenberg group.
Proves conjecture on deformation invariance of big fundamental groups.
problem Stability of big fundamental groups under small deformations.
method Deformation regularity of equivariant pluriharmonic maps and techniques from Shafarevich conjectures.
result Deformation openness of big fundamental groups for varieties with big complex local systems.
Formula proves invariant matches for smooth and orbifold test configurations.
problem Proving equivalence of Donaldson-Futaki invariant and Futaki invariant.
method Equivariant localization formula.
result Invariant matches for smooth and orbifold test configurations.
New principle for supersymmetric localization on Lie groups.
problem Computing supertrace of non-supersymmetric observables.
method Invariant supersymmetric deformations and fermionic zero modes.
result Path integral localizes to periodic orbits.
We study deformations of associative submanifolds Y3⊂M7 of a G2 manifold M7. We show that the deformation space can be perturbed to be smooth, and it can be made compact and zero dimensional by constraining it with an additional equation. This allows us to associate local invariants to associative subm…
In 1979, M. Kashiwara and M. Vergne formulated a conjecture on a Lie group G which implies that the Duflo isomorphism of Z(g) and S(g)^g extends to a natural module isomorphism between the spaces of germs of invariant distributions on G and g=Lie(G), respectively. They also proved their conjecture for G solvable. Using…
We study D-homothetic deformations of almost α-Kenmotsu structures. We characterize almost contact metric manifolds which are CR-integrable almost α-Kenmotsu manifolds, through the existence of a canonical linear connection, invariant under D-homothetic deformations. If the canonical connect…
The study identifies two sources of invariants in 2--nondegenerate CR geometries.
problem Characterizing fundamental invariants of 2--nondegenerate CR geometries.
method Analyzes the harmonic curvature and the difference in complex structures.
result Nontrivial examples of CR geometries can be obtained as deformations of models.
A scattering transform defines a signal representation which is invariant to translations and Lipschitz continuous relatively to deformations. It is implemented with a non-linear convolution network that iterates over wavelet and modulus operators. Lipschitz continuity locally linearizes deformations. Complex classes o…
Researchers prove spaces of positive scalar curvature metrics are contractible with symmetry.
problem Contractibility of spaces of invariant positive scalar curvature metrics.
method Combining equivariant Morse theory with conformal deformations and local flexibility properties.
result Spaces of invariant positive scalar curvature metrics are contractible.
The paper studies deformations and confluences of singularities in meromorphic connections and quadratic differentials.
problem Understanding singularities and deformations in meromorphic connections and quadratic differentials.
method Local formal invariants and jets of meromorphic quadratic differentials, universal isomonodromic deformation, unfolded Stokes phenomenon, horizontal and vertical foliations.
result Establishes a correspondence between local formal invariants and jets of meromorphic quadratic differentials, describing parameter spaces and moduli spaces.
In this paper we propose novel Deformable Part Networks (DPNs) to learn {\em pose-invariant} representations for 2D object recognition. In contrast to the state-of-the-art pose-aware networks such as CapsNet \cite{sabour2017dynamic} and STN \cite{jaderberg2015spatial}, DPNs can be naturally {\em interpreted} as an effi…
We consider a 2-complex in a particular form, called the Quinn model of a 2-complex. It can be sliced in graphs, where a change from one graph to another can be organized by a sequence of local transitions, which are described in a list of F. Quinn [Q1]. The decomposition of that 2-complex into graphs has to be transla…
By use of a natural map introduced recently by the first and third authors from the space of pure-type complex differential forms on a complex manifold to the corresponding one on the small differentiable deformation of this manifold, we will give a power series proof for Kodaira-Spencer's local stability theorem of Kä…
Local study of foliation deformation cohomology.
problem Understanding deformations of singular foliations.
method Introducing and studying local deformation cohomology.
result Local deformation cohomology for singular foliations and related structures.
The paper explores non-metrizability of projective deformations of Finsler sprays.
problem Investigating non-metrizability of projective deformations of Finsler sprays.
method Analyzing projective deformation of Finsler sprays by holonomy invariant functions and proving non-metrizability for most cases.
result For most values of λ and holonomy invariant nontrivial functions P, the projective deformation is not Finsler metrizable.
New theorems on Hodge numbers and Kähler structures derived from complex differential forms.
problem Deformation invariance and local stability of Hodge numbers and Kähler structures.
method Using the exponential operator and power series method, the approach focuses on d-closed extensions and foliated cases. result Local stabilities of transversely p-Kähler structures and new theorems on Hodge numbers. Study on deformations of Lie groupoid morphisms and their properties.
problem Understanding the deformation theory of Lie groupoid morphisms.
method Established deformation theory, cohomology, and properties of morphisms.
result Invariance and stability properties of morphisms, Morita invariance of cohomology, and simultaneous deformations.
Analytic K-semistability connects curvature to metric existence.
problem Establishing constant scalar curvature Kähler metrics.
method Small polarized deformations and Futaki invariant computation.
result K-polystability implies existence of cscK metrics locally.
Constructs universal local deformations for curves and differential forms.
problem Local deformations of curves and differential forms under preservation of periods.
method Develops Kuranishi families for pairs of curves and meromorphic 1-forms, focusing on hyperelliptic cases.
result First paper in a series developing a deformation theory for spectral curve data of integrable systems.
New homology invariant for links in surfaces, a deformation of APS.
problem Defining a new homology invariant for links in surfaces.
method A 1-parameter family of homology invariants, motivated by instanton Floer homology.
result The new invariant recovers APS homology and has a stronger detection property.
In this thesis, we study the deformation problem of coisotropic submanifolds in Jacobi manifolds. In particular we attach two algebraic invariants to any coisotropic submanifold S in a Jacobi manifold, namely the L∞[1]-algebra and the BFV-complex of S. Our construction generalizes and unifies analogous cons…
Holomorphic families lead to Moishezon manifolds and bimeromorphic embeddings.
problem Characterizing and embedding Moishezon manifolds in projective space.
method Holomorphic families, local deformation invariance, strongly Gauduchon metrics, Monge-Ampère equations.
result Moishezon manifolds in a family are still Moishezon and admit a bimeromorphic embedding.
A new method evolves point clouds using B-splines for smooth surfaces.
problem Evolution of smooth surfaces from discrete point clouds.
method Adaptive Lagrangian B-spline framework for geometric evolution.
result Efficient and accurate reproduction of surface evolution phenomena.
We prove that any holomorphic locally homogeneous geometric structure on a complex torus, modelled on a complex homogeneous surface, is translation invariant. We conjecture that this result is true is any dimension. In higher dimension we prove it here for nilpotent models. We also prove that in any dimension the trans…
Paper uses algebraic methods to prove an equivariant Poincaré-Hopf theorem.
problem Equivariant index theory on manifolds.
method Localization algebras and Witten deformation techniques in K-homology.
result Established an equivariant version of the Poincaré-Hopf theorem.
New invariant derived from skein algebra representations at roots of unity.
problem Understanding skein algebras of open surfaces and their invariants.
method Extended skein algebras, constructed isomorphisms, embedding and isomorphism constructions.
result Invariant associated to each skein algebra representation class.
Study invariants of elliptic curves in LCS manifolds, leading to new phenomena in Riemann-Finsler geometry.
problem Defining and studying invariants of elliptic curves in locally conformally symplectic manifolds.
method Using J-holomorphic curves and Gromov-Witten theory to define and study invariants. result Found new phenomena in Riemann-Finsler geometry and an analogue of the Weinstein conjecture.
Let X be a compact complex Calabi-Yau 4-fold. Under certain assumptions, we define Donaldson-Thomas type deformation invariants (DT4 invariants) by studying moduli spaces of solutions to the Donaldson-Thomas equations on X. We also study sheaves counting problems on local Calabi-Yau 4-folds. We relate DT4…
Study shows stability of locally conformally balanced condition under modifications but not under small deformations.
problem Stability of locally conformally balanced condition under small deformations and modifications.
method Proved stability under proper modifications and instability under small deformations using examples and Hilbert-Chow map.
result Stability of locally conformally balanced condition under proper modifications and instability under small deformations.
CR 3-sphere rigidity proven through curvature invariant.
problem Proving rigidity of CR 3-sphere under deformations.
method Analyzing curvature invariant and linearized equation.
result CR 3-sphere does not admit nontrivial obstruction flat deformations.
Study local third Chern class for point singularities on threefolds.
problem Understanding gauge theory singularity contributions on threefolds.
method Local algebraic data and deformation invariance, K-theoretic interpretation.
result Local third Chern class can be computed from family data and is deformation invariant.
Study shows deformations of quaternionic Kähler manifolds are locally inhomogeneous.
problem Understanding deformations of quaternionic Kähler manifolds.
method Proved one-loop deformation of quaternionic Kähler manifolds are locally inhomogeneous.
result Full isometry group of one-loop deformations has cohomogeneity one.
New deformations of lattice cohomology help calculate knot invariants.
problem Calculating knot invariants using lattice cohomology.
method Using holomorphic triangles counting and lattice cohomology.
result Combinatorial formulae for the upsilon invariant are derived.
Extends extension formulas for Hodge numbers on complex manifolds.
problem Deformation invariance of Hodge numbers on complex manifolds.
method Introduces a canonical isomorphism between complex differential forms on a manifold and its infinitesimal deformations, generalizing an extension formula.
result Proves several deformation invariance theorems for Hodge numbers.
Researchers confirm a relation between knot invariants and provide formulas for torus knots.
problem Confirming a relation between knot invariants and providing formulas.
method Explicit formulas and algorithms for certain ADO-invariants of torus knots obtained from the series invariant of knot complements.
result Explicit formulas and algorithms for certain ADO-invariants of torus knots.
Study complex structure deformations on Lie algebras and Dolbeault cohomology.
problem Deformations of complex structures on Lie algebras and their associated Dolbeault cohomology.
method Construct a complete deformation of complex structures similar to the Kuranishi family, showing extension isomorphism validity.
result Analytic open subset of deformations where Dolbeault cohomology can be computed by left invariant tensor fields.
New invariant metrics preserved under deformed Markov embeddings.
problem Preserving invariance in probability measure spaces under deformed embeddings.
method Deforming Markov embeddings while maintaining sufficiency, proving existence and uniqueness of invariant families.
result Existence and uniqueness of invariant families of tensor fields under deformed embeddings.
In this paper, we study deformations of coisotropic submanifolds in a locally conformal symplectic manifold. Firstly, we derive the equation that governs C∞ deformations of coisotropic submanifolds and define the corresponding C∞-moduli space of coisotropic submanifolds modulo the Hamiltonian isotopies.…
Study Lie algebras with complex structures, focusing on degenerations and deformations.
problem Understanding the space of Lie algebras with complex structures and their transformations.
method Identifying invariants that remain consistent under degenerations and applying to four-dimensional case.
result Found invariants that help in understanding the behavior of Lie algebras under complex structures.
Localized deformation of scalar curvature and mean curvature on manifolds.
problem Deforming scalar curvature and mean curvature on compact manifolds with boundary.
method Proving localized surjection of scalar curvature and mean curvature map, handling non-variational linearized problem.
result Localized deformations of scalar curvature and mean curvature on compact manifolds are possible.
New invariant distinguishes non-orientable surfaces.
problem Distinguishing non-orientable surfaces bounded by the same knot.
method Mixed invariant from Lee and Bar-Natan deformations of Khovanov homology.
result Distinguishes exotic non-orientable surfaces.
LOCA learns standardized data coordinates from measurements.
problem Learning invariant data coordinates from non-linearly deformed manifolds.
method LOCA, a LOcal Conformal Autoencoder, learns an isometric embedding.
result LOCA preserves geometric information while learning invariant coordinates.
New construction of Riemannian deformation sequence using differential operators.
problem Linearized deformation theory of Riemannian metrics.
method Explicit linear connection on natural bundle, twisted de Rham sequence, BGG-like construction.
result Sequence computes cohomology of local Killing fields and relates to Cartan geometry deformation theory.
The paper shows measures equidistribute on affine submanifolds with a rate.
problem Understanding equidistribution of measures on affine invariant submanifolds.
method Analyzing unstable foliations and using results from homogeneous dynamics.
result Measures of large dimension equidistribute on affine invariant submanifolds with an effective rate.
Using spinc structure we prove that Kähler-Einstein metrics with nonpositive scalar curvature are stable (in the direction of changes in conformal structures) as the critical points of the total scalar curvature functional. Moreover if all infinitesimal complex deformation of the complex structure are integrable, th…
Non-invariant complex structures on Lie groups are not biholomorphic to invariant ones.
problem Understanding non-invariant deformations of complex structures on Lie groups.
method Computed cohomologies to show non-biholomorphicity.
result Non-invariant complex structures are not biholomorphic to invariant ones.
We develop deformation theory for abelian invariant complex structures on a nilmanifold, and prove that in this case the invariance property is preserved by the Kuranishi process. A purely algebraic condition characterizes the deformations leading again to abelian structures, and we prove that such deformations are uno…