Study projective klt pairs with nef anti-canonical divisor and their properties.
problem Characterize properties of projective klt pairs with nef anti-log canonical divisors.
method Analyze maximally rationally connected fibration and use numerical dimension analogy.
result Numerical dimension of anti-log canonical divisor on X matches that on a general fiber. 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. The quotients Y=X/conj by the complex conjugation conjX→X for complex rational and Enriques surfaces X defined over R are shown to be diffeomorphic to connected sums of $\barCP2$, whenever Y are simply connected.
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.
Proves existence of Lefschetz fibrations with arbitrary slopes.
problem Finding Lefschetz fibrations with specific slopes.
method Proves existence for rational numbers in (2,8) with large genus.
result Arbitrary slopes (r in (2,8)) are possible for genus-g Lefschetz fibrations.
Proves rational slopes characterize knot 5_2, except for integers.
problem Characterizing slopes for knot 5_2.
method Proves all rational slopes except integers characterize 5_2, classifies Dehn surgeries, studies almost L-spaces.
result Rational slopes characterize knot 5_2 except for positive integers.
The paper proves conditions for Kähler and Riemannian manifolds to be simply connected.
problem Conditions for Kähler and Riemannian manifolds to be simply connected.
method Spectral positivity assumptions for Kähler manifolds and a specific spectral positivity assumption for Riemannian manifolds.
result Compact Kähler manifolds and Riemannian manifolds under the specified spectral positivity assumptions are simply connected.
New dHYM connections found on complex vector bundles.
problem Existence of dHYM connections on higher rank vector bundles.
method Constructing explicit non-trivial examples and providing algebraic conditions.
result First explicit non-trivial dHYM connections on higher rank holomorphic vector bundles.
New surgery exact triangles in Heegaard Floer homology for rational slopes.
problem Constructing new surgery exact triangles in Heegaard Floer homology.
method Combining combinatorial triangle and quadrilateral counting in genus 1 Heegaard diagrams.
result Solving the combinatorial problem for rational slopes, including tricky cases.
We derive an explicit formula for the asymptotic slope of the Aubin-Yau functional along a Bergman geodesic on a surface of complex dimension 2, extending the work of Phong-Sturm on Riemann surfaces. This is equivalent to an explicit calculation of the Donaldson-Futaki invariant of a test configuration. The slope is gi…
Solved a conjecture about rational homology projective planes with quotient singularities.
problem A conjecture about rational homology projective planes with quotient singularities.
method Combining Donaldson's diagonalization theorem with a distinguished spin^c structure on the smooth locus.
result Proved that rational homology projective planes with quotient singularities have at most three singular points.
In this work, it is shown that a simply-connected, rationally-elliptic torus orbifold is equivariantly rationally homotopy equivalent to the quotient of a product of spheres by an almost-free, linear torus action, where this torus has rank equal to the number of odd-dimensional spherical factors in the product. As an a…
Paper generalizes sub-slope definition and solves complex equations on compact manifolds.
problem Solving complex equations on compact almost Hermitian manifolds.
method Generalized sub-slope definition and proved existence of solutions for a class of equations.
result Solved complex Hessian quotient and deformed Hermitian-Yang-Mills equations.
Study slopes on knot manifolds to understand their fundamental groups.
problem Characterize slopes on knot manifolds to determine fundamental group properties.
method Develops new order-detection notions, parallels existing slope detection methods, and uses dynamics of 3-manifold group actions.
result Conjectured structure theorems connecting Heegaard-Floer homology and foliation dynamics to left-orderability.
We give an explicit formula for the Jones polynomial of any rational link in terms of the denominators of the canonical continued fraction of the slope of the given rational link.
The study bounds exceptional surgeries for hyperbolic knots.
problem Identifying the range of slopes for exceptional surgeries.
method Analyzing meridional and non-meridional surgeries, and investigating the relationship between boundary slopes and exceptional surgeries.
result There are boundary slopes b1<b2 such that all non-trivial exceptional surgeries occur in the interval [b1,b2]. The integers in $[\ceil{b_1}, \floor{b_2}]$ are all exceptional surgeries. Study on rational projective planes with small index singularities.
problem Existence and classification of rational homology projective planes with small index quotient singularities.
method Topological and smooth obstructions analysis, classification of singularities.
result Classification of quotient singularities for rational homology projective planes with indices up to three.
No characterizing slopes for multi-component links.
problem No characterizing slopes for multi-component links.
method Demonstrated through infinitely many examples of non-homeomorphic complements.
result Infinitely many links with non-homeomorphic complements.
The paper finds left orderable slopes for specific double twist knots.
problem Identifying left orderable slopes for double twist knots.
method Analyzing the fundamental group of 3-manifolds obtained by surgery on double twist knots.
result Explicit intervals of rational numbers are left orderable slopes for certain double twist knots.
The paper studies slopes for knot fillings with left-orderable fundamental groups.
problem Understanding slopes for knot fillings with left-orderable fundamental groups.
method Using the Riley polynomial and root analysis, the paper computes and conjectures on slopes.
result The paper computes the range of rational slope r for left-orderable fillings of two-bridge knots. 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.
Investigates intrinsic Lipschitz sections in nonlinear quotient maps.
problem Analyzing intrinsic Lipschitz sections in non-linear quotient maps.
method Introduced Leibniz formula for intrinsic slope under weaker conditions, used properties of intrinsic dilations in Carnot groups, and provided conditions for sum of sections.
result Found conditions for sum of intrinsically Lipschitz sections in Carnot groups of step 2.
Paper finds new ball quotients from curve products.
problem Finding rational deformations of surfaces.
method Using cocompact lattices in PU(2,1).
result Shows existence of new ball quotients for surfaces.
The paper solves the dHYM equation on rational homogeneous varieties using Lie theory.
problem Solving the deformed Hermitian Yang-Mills equation on rational homogeneous varieties.
method Using Lie theory to describe the Lagrangian phase and characterize solutions.
result Characterization of all supercritical and hypercritical homogeneous solutions of the dHYM equation.
We prove that polarised manifolds that admit a constant scalar curvature Kähler (cscK) metric satisfy a condition we call slope semistability. That is, we define the slope μ for a projective manifold and for each of its subschemes, and show that if X is cscK then μ(Z)≤μ(X) for all subschemes Z. This gives man…
Simplified presentation of symplectic fillings of lens spaces.
problem Symplectic fillings of lens spaces.
method Visual presentation of rational blowdown algorithm using triangulations of convex polygons.
result Organization of symplectic fillings into a graph.
We show that the properties of admitting a co-oriented taut foliation and having a left-orderable fundamental group are equivalent for rational homology 3-sphere graph manifolds and relate them to the property of not being a Heegaard-Floer L-space. This is accomplished in several steps. First we show how to detect fa…
The center of a quotient group of piecewise linear homeomorphisms is trivial.
problem Understanding the structure of a specific group of homeomorphisms.
method Analyzing a quotient of piecewise linear homeomorphisms of the real line.
result The center of the quotient group is trivial.
Equivalence proven between uniformizing varieties and tensors, generalizing uniformization results.
problem Characterizing complex-projective varieties with klt singularities and ample canonical divisors.
method Constructing a uniformizing variation of Hodge structure from slope zero tensors and vice versa.
result Generalization of uniformization results to singular settings, including quotients of tube domains.
The paper introduces Slope Conjecture which relates the degree of the Jones polynomial of a knot and its parallels with the slopes of incompressible surfaces in the knot complement. More precisely, we introduce two knot invariants, the Jones slopes (a finite set of rational numbers) and the Jones period (a natural numb…
Montgomery-Yang problem predicts that every pseudofree differentiable circle action on the 5-dimensional sphere has at most 3 non-free orbits. Using a certain one-to-one correspondence, Kollár formulated the algebraic version of the Montgomery-Yang problem: every projective surface S with the second Betti number $b_2…
Torus decomposition shows foliation detected slopes for glued knot manifolds.
problem Detecting foliation detected slopes in glued knot manifolds.
method Torus decomposition and foliation analysis.
result Gluing knot manifolds identifies rational boundary slopes.
Solves open problems for fully nonlinear elliptic equations on manifolds.
problem Solving fully nonlinear elliptic equations on manifolds.
method Analytic slope invariant and Nakai-Moishezon criterion.
result Solves open problems including hessian and hessian quotient equations.
The genus of satellite tunnel number one knots and torti-rational knots is computed using the tools introduced by Floyd and Hatcher. An implementation of an algorithm is given to compute genus and slopes of minimal genus Seifert surfaces for such knots.
A knot κ in S3 is persistently foliar if, for each non-trivial boundary slope, there is a co-oriented taut foliation meeting the boundary of the knot complement transversely in a foliation by curves of that slope. For rational slopes, these foliations may be capped off by disks to obtain a co-oriented taut foliati…
We define the slope of a colored link in an integral homology sphere, associated to admissible characters on the link group. Away from a certain singular locus, the slope is a rational function which can be regarded as a multivariate generalization of the Kojima--Yamasaki η-function. It is the ratio of two Conway pot…
Let M be a compact nonnegatively curved Riemannian manifold admitting an isometric action by a compact Lie group G in a way that the quotient space M/G has nonempty boundary. Let π:M→M/G denote the quotient map and B be any boundary stratum of M/G. Via a specific soul co…
Constructs Lefschetz fibrations with slopes near 2.
problem Finding Lefschetz fibrations with slopes close to 2.
method Constructs minimal, simply connected genus-g Lefschetz fibrations over the sphere.
result The infimum and supremum of slopes are not achievable.
A 3-manifold is foliar if it supports a codimension-one co-oriented taut foliation. Suppose M is an oriented 3-manifold with connected boundary a torus, and suppose M contains a properly embedded, compact, oriented, surface R with a single boundary component that is Thurston norm minimizing in $H_2(M, \partial M)…
We consider projective rational strong Calabi dream surfaces: projective smooth rational surfaces which admit a constant scalar curvature Kähler metric for every Kähler class. We show that there are only two such rational surfaces, namely the projective plane and the quadric surface. In particular, we show that all rat…
Paper proves structure for compact Kähler manifolds with pseudo-effective tangent bundles.
problem Compact Kähler manifolds with pseudo-effective tangent bundles.
method Smooth or locally constant rationally connected fibration onto a quotient of a compact complex torus.
result Compact Kähler manifolds with pseudo-effective tangent bundles admit a fibration structure.
The study bounds slopes for Dehn fillings of two-bridge knots with hyperbolic representations.
problem Bounding slopes for Dehn fillings of two-bridge knots with hyperbolic representations.
method Combining the Riley polynomial with Khoi's surgery-slope formula, and analyzing meridian and longitude translation parameters.
result The set of surgery slopes admitting hyperbolic PSL(2,R) representations is bounded. We show that, under some technical conditions, the Strong Slope Conjecture proposed by Kalfagianni and Tran is closed under connect sums and cabling. As an application, we establish the Strong Slope Conjecture for graph knots.
In 1985 D.Sullivan had introduced a dictionary between two domains of complex dynamics: iterations of rational functions on the Riemann sphere and Kleinian groups. The latters are discrete subgroups of the group of conformal automorphisms of the Riemann sphere. This dictionary motivated many remarkable results in both …
Let F be an incompressible, meridionally incompressible and not boundary-parallel surface with boundary in the complement of an algebraic tangle (B,T). Then F separates the strings of T in B and the boundary slope of F is uniquely determined by (B,T) and hence we can define the slope of the algebraic tang…
Study compares Kähler quotients of torus actions under varying moment maps.
problem Comparing Kähler quotients of torus actions under varying moment maps.
method Analyzes the transformation of Kähler quotients as moment maps change, proving bimeromorphic transformations and desingularizations.
result Each nondegenerate singular Kähler quotient has a partial and rational desingularization.
We show that any closed biquotient with finite fundamental group admits metrics of positive Ricci curvature. Also, let M be a closed manifold on which a compact Lie group G acts with cohomogeneity one, and let L be a closed subgroup of G which acts freely on M. We show that the quotient N := M/L carries metrics of nonn…
Paper studies invariants of knots using logarithmic Gauss maps and character varieties.
problem Understanding invariants of knots using logarithmic Gauss maps and character varieties.
method Develops a homological point of view on the slope using non-abelian representations.
result Defines a rational function on the character variety that unifies various known invariants.