Paper proves contractible fake surfaces up to complexity 6 are deformable.
problem Stable Andrews-Curtis conjecture and contractible fake surfaces.
method Induction scheme proving contractibility up to complexity 6.
result Contractible fake surfaces up to complexity 6 are 3-deformable.
Study resolves conjectures on hypercritical deformed Hermitian-Yang-Mills equation.
problem Resolving conjectures on hypercritical deformed Hermitian-Yang-Mills equation.
method Study compact Kähler manifolds and resolves conjectures of Collins-Yau.
result Resolves two conjectures of Collins-Yau.
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.
New 5-manifold without 'spine' challenges deformation conjecture.
problem Existence of exotic smooth structures on 4-manifolds.
method Constructed a compact PL 5-manifold homotopy equivalent to a wedge of 11 spheres.
result The constructed manifold is 'spineless', not a regular neighborhood of any 2-complex.
The paper explores volume product and slicing conjectures using convex body deformations.
problem Volume product and slicing conjectures in convex geometry.
method Study of variational aspects of volume product functional under projective deformations.
result Provides a proof of a theorem by Klartag and identifies critical convex bodies.
New spherical curve deformations solve a conjecture.
problem Solving the Östlund Conjecture for spherical curves.
method Introducing a new type of deformation (β) and proving equivalence under specific deformations.
result Equivalence of spherical curves under specific deformations.
Smooth deformation of Moishezon manifolds preserves their Moishezon property.
problem Preserving Moishezon property under smooth deformation.
method Smooth deformation over a unit disk in C.
result Deformation limit of Moishezon manifolds is Moishezon.
Study shows that deformed Liouville metrics on tori remain Liouville.
problem Tackles the conjecture that only Liouville metrics are integrable on tori.
method Examines deformations of non-flat Liouville metrics and proves they remain Liouville.
result For a broad class of deformations, the deformed metric remains Liouville.
Deforms quasigeodesic flows to pseudo-Anosov ones.
problem Deforming quasigeodesic flows to pseudo-Anosov flows.
method Proves Calegari's conjecture by deforming flows.
result Proves every quasigeodesic flow can be deformed to pseudo-Anosov.
In this paper we state an analog of Calabi's conjecture proved by Yau. The difference with the classical case is that we propose deformation of the complex structure, whereas the complex Monge--Ampère equation describes deformation of the Kähler (symplectic) structure.
The generalized volume conjecture relates asymptotic behavior of the colored Jones polynomials to objects naturally defined on an algebraic curve, the zero locus of the A-polynomial A(x,y). Another "family version" of the volume conjecture depends on a quantization parameter, usually denoted q or ℏ; this quan…
Study disproves conjecture about Hermitian-Yang-Mills solutions.
problem Disproving conjecture about Hermitian-Yang-Mills solutions.
method Analyzes real (1,1)-classes on compact Kähler manifolds.
result Proves conjecture is false by showing proper subset.
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.
The Kobayashi pseudometric on a complex manifold is the maximal pseudometric such that any holomorphic map from the Poincaré disk to the manifold is distance-decreasing. Kobayashi has conjectured that this pseudometric vanishes on Calabi-Yau manifolds. Using ergodicity of complex structures, we prove this conjecture fo…
The paper proves a version of the SYZ conjecture for hyperkahler manifolds.
problem Proving the SYZ conjecture for hyperkahler manifolds with specific conditions.
method Introduced a Teichmuller space to parametrize pairs of hyperkahler manifolds up to isotopy, used a version of the global Torelli theorem to deduce the deformation invariance of semiampleness.
result Proved the SYZ conjecture for hyperkahler manifolds with specific conditions.
Paper proves solvability condition for complex equation on special submanifolds.
problem Solvability condition for supercritical deformed Hermitian-Yang-Mills equation.
method Used integrals on subvarieties to provide necessary and sufficient condition.
result Confirms mirror version of Thomas-Yau conjecture about special Lagrangian submanifolds.
The paper confirms the solvability of a complex equation for a 4D manifold.
problem Solvability of a deformed Hermitian--Yang--Mills equation on a 4D Kähler manifold.
method Used eigenvalues and topological constants to prove the existence of a C-subsolution.
result The existence of a C-subsolution implies the solvability of the deformed Hermitian--Yang--Mills equation when the complex dimension is 4 and θ is close to π.
We prove Chern conjecture, which states that the Euler characteristic vanishes for closed flat affine manifolds. Our key innovation is a deformation argument for the Euler form.
Paper confirms conjecture for projective manifolds in supercritical phase.
problem Stability condition for deformed Hermitian-Yang-Mills equation.
method Establishes stability result not involving uniform constants.
result Confirms conjecture for projective manifolds in supercritical phase.
Study Kähler-Einstein edge metrics on Hirzebruch surfaces, verifying a conjecture and finding a rigid singularity.
problem Verifying a conjecture about Kähler-Einstein edge metrics on Hirzebruch surfaces.
method Using the Calabi ansatz, constructing a family of metrics and studying their angle deformation.
result Verification of a conjecture and finding a rigid singularity.
Wedge product on deRham complex of a Riemannian manifold M can be pulled back to H∗(M) via explicit homotopy, constructed using Green's operator, to give higher product structures. We prove Fukaya's conjecture which suggests that Witten deformation of these higher product structures have semiclassical limits as op…
New integrable deformations for topological hierarchies from Frobenius manifolds.
problem Integrable deformations of topological hierarchies from Frobenius manifolds.
method Construction of integrable deformations with polynomial tau-structures.
result Conjecture of universal object for Riemann--Hopf hierarchy.
Proves stable degeneration preserves symplectic forms and confirms Kaledin's conjecture.
problem Symplectic singularities and their degenerations.
method Combining volume minimization, deformation theory, and rigidity results.
result Kaledin's conjecture confirmed for symplectic singularities.
We show that representations up to homotopy can be differentiated in a functorial way. A van Est type isomorphism theorem is established and used to prove a conjecture of Crainic and Moerdijk on deformations of Lie brackets.
We analyze the indicial roots of the self-dual deformation complex on a cylinder (R×Y3,dt2+gY), where Y3 is a space of constant curvature. An application is the optimal decay rate of solutions on a self-dual manifold with cylindrical ends having cross-section Y3. We also resolve a conjectu…
Study complex structures and curvature equations on compact manifolds.
problem Equations coupling scalar curvature with complex structure deformations.
method Infinite-dimensional Kaehler reduction, flat connections, variational characterization.
result Verification of conjecture in toric manifolds.
This paper defines RII number for knot projections and shows it can be any nonnegative number.
problem Defining and quantifying the minimum number of specific types of deformations for knot projections.
method Using deformations of types 1, 2, and 3, analogs of Reidemeister moves, to simplify knot projections and define RII number.
result RII number can be any nonnegative number, not just zero as previously conjectured.
Solves a problem about deforming symplectic forms on a Klein bottle.
problem Deforming symplectic forms on a nonorientable Lagrangian surface.
method Analyzes a specific Lagrangian Klein bottle in a symplectic 4-manifold.
result Shows how far symplectic forms can be deformed before isotopy classes contain no Lagrangians.
We introduce a novel combinatorial method to study Q∗∗-transformations of group presentations or, equivalently, 3-deformations of CW-complexes of dimension 2. Our procedure is based on a refinement of discrete Morse theory that gives a Whitehead simple homotopy equivalence from a regular CW-complex to the simplifi…
Extends Gromov non-squeezing to locally conformally symplectic structures.
problem Generalizing Gromov non-squeezing to new geometric structures.
method Deformation theory applied to locally conformally symplectic structures.
result Proves a new extension of the Gromov non-squeezing phenomenon.
Study tangle equations linking enzyme actions to knot theory.
problem Proving the Jones Unknot conjecture and understanding tangle solutions.
method Analyzing framed tangle equations and introducing Kauffman bracket ratios.
result Unique rational solutions for tangle equations imply the Jones Unknot conjecture.
The paper proves a stronger Petersen--Wilhelm conjecture for principal bundles.
problem Conditions for positive sectional curvature submersion metrics on principal bundles.
method Cheeger deformations, good triples, Chaves-Derdzinski-Rigas type condition.
result Any principal bundle over a positively curved base admits a metric of positive sectional curvature if the submersion is fat.
Getzler-Jones-Petrack introduced A∞ structures on the equivariant complex for manifold M with smooth S1 action, motivated by geometry of loop spaces. Applying Witten's deformation by Morse functions followed by homological perturbation we obtained a new set of A∞ structures. We extend and …
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…
Study deformed Hermitian-Yang-Mills equation on complex projective space blowup.
problem Solving the deformed Hermitian-Yang-Mills equation on complex projective space blowup.
method Expressed the equation as an ODE and solved it using combinatorial methods under an algebraic stability condition.
result Evidence supporting a conjecture on general compact Kahler manifolds.
We use the theory of Berezin-Toeplitz operators of Ma and Marinescu to study the spaces of holomorphic sections of a prequantizing line bundle over compact Kähler manifolds under deformations of the complex structure. We show that the parallel transport in the induced vector bundle over the deformation space behaves li…
We classify simply connected compact Sasaki manifolds of dimension 2n+1 with positive transverse bisectional curvature. In particular, the Kähler cone corresponding to such manifolds must be bi-holomorphic to $\C^{n+1}\backslash \{0\}$. As an application we recover the Mori-Siu-Yau theorem on the Frankel conjecture a…
Consider a geometrically finite Kleinian group G without parabolic or elliptic elements, with its Kleinian manifold $M=(\H^3\cup Ω_G)/G$. Suppose that for each boundary component of M, either a maximal and connected measured lamination in the Masur domain or a marked conformal structure is given. In this setting, w…
Proves solvability of general inverse σ_k equations with constant coefficients.
problem Solvability of general inverse σ_k equations with constant coefficients.
method Proves existence of unique solution if a C-subsolution exists.
result Confirms analytical conjecture for deformed Hermitian--Yang--Mills equation.
New theory connects string theory to swampland distance conjecture.
problem Connecting string theory to swampland distance conjecture.
method Deformations of the heterotic superpotential, treating separately for large fluxes or large distances, integrating out fields to obtain a new field theory.
result New holomorphic theory defined, connects to swampland distance conjecture.
Develops a method to deform metrics on manifolds with non-compact boundaries.
problem Creating metrics with positive scalar curvature on manifolds with boundary.
method General deformation principle for Riemannian metrics on manifolds with non-compact boundaries.
result Non-existence of metrics with positive scalar curvature and mean convex boundary.
Study resolves conjecture linking two algebraic structures on surfaces.
problem Compatibility of skein and cluster algebra structures on surfaces.
method Established compatibility between skein and cluster algebras of surfaces.
result Cluster algebra of positive genus surfaces is not finitely generated.
For any closed surface S of genus g≥2, we show that the deformation space of marked hyperbolic 3-manifolds homotopy equivalent to S, AH(S×I), is not locally connected. This proves a conjecture of Bromberg who recently proved that the space of Kleinian punctured torus groups is not locally connected.…
Let G be a simple complex algebraic group and g its Lie algebra. We show that the g-Witten-Reshetikhin-Turaev quantum invariants determine a deformation-quantization, C_q[X_G(torus)], of the coordinate ring of the G-character variety of the torus. We prove that this deformation is in the direction of the Goldman's brac…
Surveying progress on deformed Hermitian-Yang-Mills equation and its connections to stability.
problem Solvability of the deformed Hermitian-Yang-Mills equation and its relation to geometric stability.
method Utilizing geometric invariant theory (GIT) and Bridgeland stability theory to analyze the equation.
result On the blow-up of \(\mathbb{P}^2\), line bundles admitting a solution of the deformed Hermitian-Yang-Mills equation are Bridgeland stable, but not conversely.
Study of manifolds with flat connections and diagonal metrics leading to vanishing Euler characteristic.
problem Understanding the geometry and topology of affine-orthogonal manifolds.
method Deformation of flat connections into Levi-Civita connections and analysis of Euler characteristic.
result Deformations force the Euler characteristic to vanish, supporting Chern's conjecture.
This is an introduction to the Volume Conjecture and its generalizations for nonexperts. The Volume Conjecture states that a certain limit of the colored Jones polynomial of a knot would give the volume of its complement. If we deform the parameter of the colored Jones polynomial we also conjecture that it would also g…
The study shows boundedness of certain fibered varieties in algebraic geometry.
problem Bounding fibered varieties in algebraic geometry.
method Analyzing Calabi-Yau varieties and their fibrations by abelian or symplectic varieties.
result There are only finitely many deformation classes of certain fibered varieties.