arXiv research

Surveying recent work on Kähler metrics and algebraic variety stability.

New dg-algebras generalize Brauer graph algebras, with applications to stability conditions and 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.

New stability theorem for nonorientable surfaces mapping class groups.

Classifies geodesic vectors in low-dimensional Lie algebras.

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.

Paper studies homology and cohomology of Temperley-Lieb algebra TL_n(a).

Stable algebraic filters improve neural network performance.

Paper generalizes K-stability to non-algebraic spaces for Kähler metrics.

Proves constant scalar curvature Kähler metrics are very general.

Develops a new method to study algebraic tangent cones of sheaves using valuations.

Paper translates train track concepts to cluster algebras for pseudo-Anosov mapping classes.

Homology of partition algebras matches symmetric group homology under certain conditions.

Enhances Hamiltonian systems stability through generalized double bracket vector fields.

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.

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.

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.

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.

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.

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.

Characterizes pseudo-Anosov mapping classes on general marked surfaces.

Geometric formalism views optimization algorithms as discrete connections, revealing their algebraic curvature and flatness properties.

Decomposes J-energy into simpler intersection numbers for stability analysis.

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.

Builds geometric structures for algebraic groups over real closed fields.

Study on stability of Einstein metrics on symmetric spaces.

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.

Under the assumption of asymptotic relative Chow-stability for polarized algebraic manifolds $(M, L)$, a series of weighted balanced metrics $ω_m$, $m \gg 1$, called polybalanced metrics, are obtained from complete linear systems $|L^m|$ on $M$. Then the asymptotic behavior of the weights as $m \to \infty$ 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 …

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…

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.

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…

We present classical and recent results on Kähler-Einstein metrics on compact complex manifolds, focusing on existence, obstructions and relations to algebraic geometric notions of stability (K-stability). These are the notes for the SMI course "Kähler-Einstein metrics" given by C.S. in Cortona (Italy), May 2017. The m…