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.
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.
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…
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.
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.
This paper gives a description of the full space of Bridgeland stability conditions on the bounded derived category of a contraction algebra associated to a 3-fold flop. The main result is that the stability manifold is the universal cover of a naturally associated hyperplane arrangement, which is known to be simplicia…
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.
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.
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.
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.
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.
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.
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.
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.
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.
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…
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.
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.
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 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.
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.
This is a survey article, based on the author's lectures in the 2015 AMS Summer Research Institute in Algebraic Geometry, and to appear in the Proceedings.
The paper connects Kähler-Ricci shrinkers to Fano fibrations in algebraic geometry.
problem Understanding the relationship between Kähler-Ricci shrinkers and Fano fibrations.
method Using birational algebraic geometry, the paper proves properties of Kähler-Ricci shrinkers and formulates conjectures relating them to Fano fibrations.
result The existence of Kähler-Ricci shrinkers is conjectured to be related to K-stability of polarized Fano fibrations.
In this paper, by introducing a wider class of one-parameter group actions for test configurations, we have a stronger form of the definition of K-stability. This allows us to obtain some key step of my preceding work in proving that constant scalar curvature polarization implies K-stability for polarized algebraic man…
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.
Petr Novotný and Jiřĺ Hrivnák \cite{Nov} investigated generalize the concept of Lie derivations via certain complex parameters and obtained various Lie and Jordan operator algebras as well as two one- parametric sets of linear operators. Moreover, they established the structure and properties of (α,β,γ)−derivations o…
New approach proves K-stability of Fano varieties.
problem Proving K-stability of Fano varieties.
method Developed a general approach using admissible flags.
result Proved K-stability of smooth Fano hypersurfaces of index two.
Characterizes pseudo-Anosov mapping classes on general marked surfaces.
problem Stability of mapping classes on marked surfaces.
method Cluster algebraic description and reduction procedure of mapping classes.
result Characterizes pseudo-Anosov mapping classes in terms of uniform sign stability.
Geometric formalism views optimization algorithms as discrete connections, revealing their algebraic curvature and flatness properties.
problem Understanding and optimizing the behavior of iterative optimization algorithms.
method Introducing a geometric and operator-theoretic formalism where optimization algorithms are encoded by coupled channels (drift and diffusion) whose algebraic curvature measures the deviation from ideal reversibility.
result Flat connections correspond to methods whose updates commute up to higher order, achieving minimal numerical dissipation and preserving stability.
Decomposes J-energy into simpler intersection numbers for stability analysis.
problem Analyzing J-stability in algebraic geometry.
method Proves a decomposition formula for J-energy and shows equivalence of stability conditions.
result Equivalence of J-stability and K-stability for surfaces under pseudoeffective conditions.
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 …
We algebraically prove K-stability of polarized Calabi-Yau varieties and canonically polarized varieties with mild singularities. In particular, the} "stable varieties" introduced by Kollar-Shepherd-Barron and Alexeev, which form compact moduli space, are proven to be K-stable although it is well known that they are \t…
We apply the barcodes of persistent homology theory to the Chekanov-Eliashberg algebra of a Legendrian submanifold to deduce displacement energy bounds for arbitrary Legendrians. We do not require the full Chekanov-Eliashberg algebra to admit an augmentation as we linearize the algebra only below a certain action level…
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.
Builds geometric structures for algebraic groups over real closed fields.
problem Characterizing and decomposing algebraic groups over specific valued fields.
method Real algebraic geometry to construct and analyze affine buildings.
result Computed stabilizers and obtained group decompositions.
Study on stability of Einstein metrics on symmetric spaces.
problem Stability of Einstein-Hilbert functional on compact symmetric spaces.
method Classification of irreducible representations and use of Casimir eigenvalues.
result Proves stability of Einstein metrics on quaternionic and Cayley projective plane, instability on other quaternionic Grassmannians.
In this paper, we shall show that a polarized algebraic manifold is K-stable if the polarization class admits a Kaehler metric of constant scalar curvature. This generalizes the results of Chen-Tian, Donaldson and Stoppa. (Parts of the arguments are based on a forthcoming paper "A stronger concept of K-stability." )
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.
Under the assumption of asymptotic relative Chow-stability for polarized algebraic manifolds (M,L), a series of weighted balanced metrics ωm, m≫1, called polybalanced metrics, are obtained from complete linear systems ∣Lm∣ on M. Then the asymptotic behavior of the weights as m→∞ will be stud…
We consider dynamical stability for a modified Ricci flow equation whose stationary solutions include Einstein and Ricci soliton metrics. Our focus is on homogeneous metrics on non-compact manifolds. Following the program of Guenther, Isenberg, and Knopf, we define a class of weighted little Hölder spaces with certain …
In this paper, we address the stability of a broad class of discrete-time hypercomplex-valued Hopfield-type neural networks. To ensure the neural networks belonging to this class always settle down at a stationary state, we introduce novel hypercomplex number systems referred to as real-part associative hypercomplex nu…
Homologies of Jones and partition algebras match cyclic and symmetric groups.
problem Matching homologies of specific algebras to cyclic and symmetric groups.
method Proving isomorphisms between algebras' homologies and group homologies.
result Homologies of Jones and partition algebras are isomorphic to cyclic and symmetric groups.
We make a systematic study of the Hilbert-Mumford criterion for different notions of stability for polarised algebraic varieties (X,L); in particular for K- and Chow stability. For each type of stability this leads to a concept of slope μ for varieties and their subschemes; if (X,L) is semistable then $μ(Z)\leμ(X…
We study the stability of coassociative 4-folds with conical singularities under perturbations of the ambient G_2 structure by defining an integer invariant of a coassociative cone which we call the stability index. The stability index of a coassociative cone is determined by the spectrum of the curl operator acting on…
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.