New dg-algebras generalize Brauer graph algebras, with applications to stability conditions and quadratic differentials.
problem Generalizing Brauer graph algebras to new dg-algebras.
method Derived categories, mixed-angulations of surfaces, stability conditions, and quadratic differentials.
result Spaces of stability conditions on derived categories of these algebras are described in terms of spaces of quadratic differentials.
Paper describes stability conditions on contraction algebra derived categories.
problem Stability conditions on contraction algebra derived categories.
method Description of full space of stability conditions on derived category.
result Stability manifold is universal cover of hyperplane arrangement.
We apply a recent theorem of Li and the first author to give some criteria for the K-stability of Fano varieties in terms of anticanonical Q-divisors. First, we propose a condition in terms of certain anticanonical Q-divisors of given Fano variety, which we conjecture to be equivalent to the K-stability. We prove that …
The thesis explores stability conditions and metrics in differential geometry.
problem Understanding extremal objects in differential geometry.
method Introduces and analyzes Z-critical metrics and optimal symplectic connections. result Proves a correspondence between existence of metrics and stability conditions.
For each integer n≥2 we describe the space of stability conditions on the derived category of the n-dimensional Ginzburg algebra associated to the A2 quiver. The form of our results points to a close relationship between these spaces and the Frobenius-Saito structure on the unfolding space of the A2 singul…
Study compares different stability notions in Kähler geometry.
problem Comparing various stability notions in Kähler geometry.
method Introduce and study geodesic stability on rays with specific singularity types.
result Equivalence of some stability notions under certain conditions.
Maths and physics explore a nonlinear equation on Kahler manifolds.
problem Solving a nonlinear geometric PDE on Kahler manifolds.
method Discuss physical origin and recent progress towards solution.
result Proved a new Chern number inequality in dimension 3.
Surveying recent work on Kähler metrics and algebraic variety stability.
problem Understanding canonical Kähler metrics on algebraic varieties.
method Analyzing recent developments in algebraic geometry.
result Relation between canonical Kähler metrics and stability in algebraic geometry.
Proves constant scalar curvature Kähler metrics are very general.
problem Existence of constant scalar curvature Kähler metrics on smooth polarized varieties.
method Combining uniform arc K-stability and algebraic properties in families.
result The constant scalar curvature Kähler locus is very general.
Homology of partition algebras matches symmetric group homology under certain conditions.
problem Understanding homology of partition algebras and comparing it to symmetric groups.
method Inductive resolution and high acyclicity arguments, parallel to earlier work on Brauer algebras.
result Homology of partition algebras is isomorphic to symmetric group homology under specific conditions.
Stable algebraic filters improve neural network performance.
problem Improving neural network stability to deformations.
method Analyzed stability of algebraic filters and neural networks under deformations of the homomorphism.
result Stable algebraic filters have frequency responses whose derivative is inversely proportional to frequency.
Study higher rank deformed Hermitian-Yang-Mills equations for stable vector bundles.
problem Stability conditions for higher rank vector bundles over complex manifolds.
method Establish equivalence between dHYM equations and Z-stability. result Equivalence between dHYM solutions and Z-stability for vortex type bundles. Extends K-stability theory to projective klt pairs with a big anticanonical class.
problem Behavioral pathologies in K-stability for projective klt pairs with a big anticanonical class.
method Extends K-stability theory to projective klt pairs with a big anticanonical class, observing that K-semistability forces a klt anticanonical model with the same stability property.
result K-semistability forces projective klt pairs with a big anticanonical class to have a klt anticanonical model with the same stability property.
Contact Lie algebras have specific properties related to stabilizers and invariant polynomials.
problem Characterizing contact Lie algebras and their stabilizers.
method Analyzing algebraic Lie algebras of index 1 and their orbits.
result Contact Lie algebras have specific properties related to stabilizers and invariant polynomials.
The paper provides presentations for mapping class groups and cluster automorphism groups of surfaces.
problem Presentations of mapping class groups of surfaces stabilizing boundaries.
method Gave presentations of mapping class groups of marked surfaces stabilizing boundaries.
result Presented cluster automorphism groups of cluster algebras from surfaces.
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.
For a polarized algebraic manifold (X,L), let T be an algebraic torus in the group of all holomorphic automorphisms of X. Then strong relative K-stability will be shown to imply asymptotic relative Chow-stability. In particular, by taking T to be trivial, we see that asymptotic Chow-stability follows from stron…
New criterion found for Hermitian-Yang-Mills metrics on non-compact Kähler manifolds.
problem Existence of Hermitian-Yang-Mills metrics on non-compact Kähler manifolds.
method Algebraic criterion and stability condition introduced.
result New stability condition is both sufficient and necessary for the existence of Hermitian-Yang-Mills metrics.
Equivalence proven between algebraic stability and geometric stability.
problem Equivalence of algebraic and geometric stability criteria.
method Algebraic proof of equivalence, existence and uniqueness of minimal centers.
result Existence and uniqueness of minimal optimal destabilizing centers.
New stability theorem for nonorientable surfaces mapping class groups.
problem Stability of homology groups of mapping class groups of nonorientable surfaces.
method Galatius--Kupers--Randal-Williams framework of cellular E2-algebras. result New best known stability range for homology of nonorientable surfaces.
The paper develops stability criteria for real reductive Lie groups acting on manifolds.
problem Analyzing stability of real reductive Lie group actions on manifolds.
method Introduced a gradient map and maximal weight function to characterize stability conditions.
result Characterized stability, semistability, and polystability using numerical criteria.
Classifies geodesic vectors in low-dimensional Lie algebras.
problem Stability of geodesic vectors in Lie algebras.
method Complete classification of Lyapunov stable and unstable geodesic vectors.
result Classification for metric Lie algebras of dimension 3 and 4.
Study investigates metrics on manifolds with stable curvature conditions.
problem Investigating the space of Riemannian metrics with surgery stable curvature conditions.
method Utilized surgery stability condition and Gromov-Lawson construction.
result Homotopy type of the space of metrics is invariant under surgeries.
We investigate principal G-bundles on a compact Kähler manifold, where G is a complex algebraic group such that the connected component of it containing the identity element is reductive. Defining (semi)stability of such bundles, it is shown that a principal G-bundle EG admits an Einstein-Hermitian connection …
The study of special Lagrangian classes and semistable Mukai vectors on K3 surfaces.
problem Counting special Lagrangian classes and semistable Mukai vectors for K3 surfaces.
method Analyzing flat surfaces and K3 surfaces, using asymptotics and stability conditions.
result Exact leading term in the asymptotics of the number of semistable Mukai vectors.
Torelli groups' homology is finitely generated in stable range.
problem Whether the homology groups of Torelli subgroups are finitely generated in stable range.
method Using unipotency condition and Tavgen's theorem.
result Homology groups are finitely generated in stable range.
The paper classifies test configurations and derives a criterion for uniform K-stability of certain algebraic varieties.
problem Uniform K-stability of G-varieties of complexity 1. method Classification of G-equivariant normal test configurations via combinatorial data and derivation of a criterion for uniform K-stability. result Derivation of a criterion for uniform K-stability in terms of combinatorial data.
Paper studies homology and cohomology of Temperley-Lieb algebra TL_n(a).
problem Homology and cohomology of Temperley-Lieb algebra TL_n(a).
method Homological stability and computation of stable homology. Use of chain complex of 'planar injective words'.
result Vanishing of homology and cohomology up to degree (n-2) under certain conditions.
Investigates J-equation on holomorphic vector bundles over Kähler manifolds.
problem Analyzes properties and solutions of J-equation on holomorphic vector bundles. method Introduces and studies J-equation, provides algebraic and numerical criteria. result Provides an algebraic condition (asymptotic J-stability) and a numerical criterion for vortex bundles. We propose new types of canonical metrics on Kähler manifolds, called coupled Kähler-Einstein metrics, generalizing Kähler-Einstein metrics. We prove existence and uniqueness results in the cases when the canonical bundle is ample and when the manifold is Kähler-Einstein Fano. In the Fano case we also prove that existe…
Paper generalizes K-stability to non-algebraic spaces for Kähler metrics.
problem Existence of constant scalar curvature Kähler metrics on non-algebraic spaces.
method Developed a non-Archimedean theory of complex spaces and applied it to Kähler manifolds.
result Proved K-stability implies existence of unique constant scalar curvature Kähler metric.
In this paper we present some conditions for the (strong) stabilizability of an n-D Quantum MIMO system P(X). It contains two parts. The first part is to introduce the n-D Quantum MIMO systems where the coefficients vary in the algebra of Q-meromorphic functions. Then we introduce some conditions for the stabilizabilit…
Proves existence of Hermitian-Einstein metrics on stable bundles.
problem Analytic stability of vector bundles and instantons/monopoles.
method Analyzes curvature decay and proves existence of Hermitian-Einstein metrics.
result Existence of Hermitian-Einstein metrics on analytically stable bundles.
We provide a sufficient condition for polarisations of Fano varieties to be K-stable in terms of Tian's alpha invariant, which uses the log canonical threshold to measure singularities of divisors in the linear system associated to the polarisation. This generalises a result of Odaka-Sano in the anti-canonically polari…
Geometric perspective on unique solution in matrix completion with a deterministic pattern.
problem Identifying unique solutions in matrix completion with a specific pattern of observed entries.
method Geometric and algebraic analysis, focusing on the well-posedness condition and local stability.
result A sufficient condition for local uniqueness of matrix completion solutions, called the well-posedness condition.
Develops a new method to study algebraic tangent cones of sheaves using valuations.
problem Analyzing tangent cones of torsion-free sheaves on algebraic varieties.
method Introduces a slope stability theory and uses it to define a canonical tangent cone for quasi-regular valuations.
result Shows the existence of a canonical tangent cone for torsion-free sheaves, up to equivalence.
Paper translates train track concepts to cluster algebras for pseudo-Anosov mapping classes.
problem Understanding pseudo-Anosov mapping classes on surfaces.
method Using Goncharov--Shen's potential function, the paper translates train track concepts into cluster algebra language.
result Proves sign stability of general pseudo-Anosov mapping classes.
Paper analyzes stability of discrete-time hypercomplex-valued Hopfield neural networks.
problem Stability of discrete-time hypercomplex-valued Hopfield neural networks.
method Introduces real-part associative hypercomplex number systems and B-projection functions to ensure stability. result Stability analysis of several discrete-time hypercomplex-valued Hopfield-type neural networks confirmed.
The study of multisymplectic structures using Spencer cohomology.
problem Integrability of multisymplectic structures.
method Applying Spencer cohomology to identify multisymplectic structures as G-structures and giving conditions for integrability. result Conditions for a multisymplectic form to admit a chart with constant coefficients.
Enhances Hamiltonian systems stability through generalized double bracket vector fields.
problem Stabilizing already stable points in Hamiltonian systems.
method Generalized double bracket vector fields on Poisson manifolds with pseudo-Riemannian metrics.
result Enhanced equilibria stability through dissipation terms.
Study bubbling Kahler metrics using algebraic geometry.
problem Analyzing the degeneration of Kahler metrics with Euclidean volume growth.
method Algebraic construction of birational modifications to simplify degenerations, comparing with analytic constructions.
result Provide a framework to compare algebraic and analytic approaches to bubbling phenomena.
We define a new notion of "b-stability" for a polarised algebraic variety, adapted to the existence problem for Kahler-Einstein metrics on Fano manifolds.
Characterizes pseudo-Anosov mapping classes using cluster algebra techniques.
problem Characterize pseudo-Anosov mapping classes purely in terms of shear coordinates.
method Uses cluster algebraic generalization and tropical cluster transformations.
result Algebraic entropies of cluster transformations match topological entropy.
Decomposes Q-Fano Kähler-Einstein varieties into simpler components.
problem Understanding the structure of Q-Fano Kähler-Einstein varieties.
method Proves decomposition theorem using algebraically integrable foliations and stability conditions.
result Q-Fano Kähler-Einstein varieties decompose into simpler components.
In this paper, improving a preceding work, we obtain asymptotic polybalanced kernels associated to extremal Kaehler metrics on polarized algebraic manifolds. As a corollary, we have a stronger asymptotic relative Chow-polystability for extremal Kaehler polarized algebraic manifolds. Finally, related to the Yau-Tian-Don…
We formulate a stability conjecture for the coefficients of the colored Jones polynomial of a knot, colored by irreducible representations in a fixed ray of a simple Lie algebra, and verify it for all torus knots and all simple Lie algebras of rank 2. Our conjecture is motivated by a structure theorem for the degree …
The Kobayashi-Hitchin correspondence is established for mini-holomorphic bundles on certain 3-folds.
problem Establishing a correspondence between mini-holomorphic bundles and HE-monopoles.
method Defining mini-holomorphic bundles, Dirac-type singularities, and admissible BHE-metrics.
result A special Hermitian metric (admissible BHE-metric) exists on slope stable mini-holomorphic bundles.
In this note we define the stabilizer group of any adjoint-invariant l-form on a complex simple Lie algebra. This result partially extend a previous result by Kable.