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.
Study stabilizes Morse-Bott cohomology for equivariant manifolds.
problem Equivariant cohomology of manifolds with group actions.
method Stabilization technique to construct Morse-Bott functions.
result Realization of equivariant transversality and orientability.
The paper defines new knot genera and finds bounds for stabilization distances.
problem Finding bounds for stabilization distances of symmetric surfaces.
method Defining new knot genera and using them to find bounds.
result Constructs unknotted symmetric 2-spheres without symmetric 3-ball bounds.
We present an equivariant Liapunov stability criterion for dynamical systems with symmetry. This result yields a simple proof of the energy-momentum-Casimir stability analysis of relative equilibria of equivariant Hamiltonian systems.
The paper proves stability in compact finite dimensional Alexandrov spaces using equivariant Gromov--Hausdorff convergence.
problem Stability in compact finite dimensional Alexandrov spaces.
method Equivariant Gromov--Hausdorff convergence and almost commutative diagrams.
result Stability result in compact finite dimensional Alexandrov spaces.
The paper introduces μK-stability for polarized schemes and develops equivariant calculus.
problem The existence of μ-cscK metrics and their stability. method Develops equivariant calculus and introduces μ-character to study μK-stability. result Derives μ-Futaki invariant and an equivariant first Chern class for general test configurations. New approach to Z-stability and critical metrics on Kähler manifolds.
problem Determining Z-stability and existence of Z-critical metrics on Kähler manifolds. method Equivariant localisation applied to integrals over test configurations.
result Existence of Z-critical metrics is equivalent to Z-stability. Investigates harmonic self-maps' stability on cohomogeneity one manifolds.
problem Stability of harmonic self-maps on cohomogeneity one manifolds.
method Systematic study of Jacobi equation for harmonic self-maps.
result Explicit solutions for specific cases show identity map's stability.
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. Exotic diffeomorphism survives stabilizations on a contractible 4-manifold.
problem Finding exotic diffeomorphisms on contractible 4-manifolds.
method Developed a Pin(2) × Z_2-equivariant refinement for computing Seiberg-Witten Floer homotopy types.
result Constructed an exotic diffeomorphism that survives two stabilizations.
Equivalence proven between divisorial stability and quotient log divisorial stability.
problem Equivalence of divisorial stability and log divisorial stability under finite group actions.
method Interpolation technique and equivariant divisorial stability construction.
result Equivariant divisorial stability of a polarized variety is equivalent to log divisorial stability of its quotient.
We show that G-equivariant K-semistability (resp. G-equivariant K-polystability) implies K-semistability (resp. K-polystability) for log Fano pairs when G is a finite group.
Stability of singularity formation in Yang-Mills fields in higher dimensions.
problem Stability of self-similar blowup profiles for Yang-Mills equations in (1+d)-dimensions. method Analysis of explicitly known equivariant self-similar blowup solution and small equivariant perturbations.
result Global-in-space asymptotic stability of the self-similar blowup solution for Yang-Mills equations in (1+d)-dimensions for d≥5. Generalized stability theorem for compact manifolds with boundary.
problem Stability of manifolds with boundary.
method Equivariant μ-bubbles technique.
result Yamabe invariant of compact manifolds with boundary is positive if and only if the invariant of the manifold times S^1 is positive.
The paper develops methods for calculating equivariant homology from Morse functions.
problem Calculating equivariant homology from equivariant Morse functions.
method Alter equivariant Morse functions to stable ones, use generic equivariant metrics, and analyze the Morse spectral sequence.
result Equivariant Morse functions induce a filtration that computes equivariant homology.
The paper constructs non-smoothable actions on spin 4-manifolds.
problem Non-smoothability of Z/p-actions on indefinite spin 4-manifolds. method Constructs examples of non-smoothable actions using equivariant κ-invariants and calculations of η-invariants. result Non-smoothable actions remain non-smoothable under certain stabilizations.
We prove a criterion for K-stability of a Q-Fano spherical variety with respect to equivariant special test configurations, in terms of its moment polytope and some combinatorial data associated to the open orbit. Combined with the equivariant version of the Yau-Tian-Donaldson conjecture for Fano manifolds p…
Study shows equivariant Khovanov homotopy types are equivalent.
problem Understanding equivariant structures in Khovanov homotopy types.
method Investigates group actions on homotopy coherent diagrams to prove equivalence.
result Equivariant Khovanov homotopy types are equivariantly stably homotopy equivalent.
The main aim of this paper is to study existence and stability properties of rotationally symmetric proper biharmonic maps between two m-dimensional models (in the sense of Greene and Wu). We obtain a complete classification of rotationally symmetric, proper biharmonic conformal diffeomorphisms in the special case th…
Solves Tian's stabilization problem for toric Fano manifolds.
problem Tian's stabilization problem for equivariant global log canonical thresholds.
method Expressed complex singularity exponents in terms of support and gauge functions from convex geometry.
result First general result on Tian's problem.
GNNs generalize CNNs for graph data, showing equivariance and stability.
problem Processing signals on graphs.
method Graph convolutional filters, nonlinearities, stacked layers.
result GNNs converge to graphon neural networks under graph convergence.
Stability of Wasserstein spaces under various convergence types.
problem Stability and finiteness of Wasserstein spaces over singular and non-singular spaces.
method Gromov--Hausdorff convergence and equivariant Gromov--Hausdorff convergence.
result Analogue of Perelman's stability theorem on Wasserstein spaces.
We provide several results on the existence of metrics of non-negative sectional curvature on vector bundles over certain cohomogeneity one manifolds and homogeneous spaces up to suitable stabilization. Beside explicit constructions of the metrics, this is achieved by identifying equivariant structures upon these vecto…
Explicit encoding of group actions in deep features makes it possible for convolutional neural networks (CNNs) to handle global deformations of images, which is critical to success in many vision tasks. This paper proposes to decompose the convolutional filters over joint steerable bases across the space and the group …
New example of non-smooth isotopy after stabilization in 4-manifolds.
problem Non-smooth isotopy of Dehn twist after stabilization.
method Pin(2)-equivariant Bauer-Furuta invariant.
result Dehn twist is not smoothly isotopic even after stabilization.
Neural networks adapt to any input dimensionality.
problem Learning functions with inputs of varying dimensions.
method Equivariant neural networks using representation stability.
result Training on fixed dimensions allows extension to any input.
Study knot Floer homology to create concordance invariants and slice genus bounds.
problem Developing concordance invariants using knot Floer homology.
method Using knot Floer homology, define and analyze equivariant concordance invariants.
result Showed a family of strongly invertible slice knots with arbitrarily large equivariant slice genus.
Constructs a functor for equivariant smooth h-cobordisms.
problem Defines a functor for equivariant smooth h-cobordisms.
method Constructs an (∞,1)-functor mapping smooth G-manifolds to spaces of equivariant h-cobordisms. result The functor structure is subtle and relies on new ideas.
Proves minimization for Kähler manifolds with automorphisms.
problem Minimizing CM degree for Kähler manifolds with automorphisms.
method Equivariant CM minimization conjecture, asymptotic filtration Chow stability.
result Strict minimization by extremal filling.
For (X,L) a polarized toric variety and G a torus of automorphisms of (X,L), denote by Y the GIT quotient X/G. We define a family of fully faithful functors from the category of torus equivariant reflexive sheaves on Y to the category of torus equivariant reflexive sheaves on X. We show, under a genericity assumption o…
It is conjectured that to test the K-polystability of a polarised variety it is enough to consider test-configurations which are equivariant with respect to a torus in the automorphism group. We prove partial results towards this conjecture. We also show that it would give a new proof of the K-polystability of constant…
In this paper, for a compact Lie group action,we prove the anomaly formula and the functoriality of the equivariant Bismut-Cheeger eta forms with perturbation operators when the equivariant family index vanishes. In order to prove them, we extend the Melrose-Piazza spectral section and its main properties to the equiva…
We study the heat flow for Yang-Mills connections on Rd×SO(d). It is well-known that in dimensions 5≤d≤9 this model admits homothetically shrinking solitons, i.e., self-similar blowup solutions, with an explicit example given by Weinkove \cite{Wei04}. We prove the nonlinear asymptotic stab…
Classifies K-stable Fano varieties and finds new examples.
problem Classifying K-stable Fano varieties and their properties.
method Classification and analysis of Gorenstein Fano bi-equivariant compactifications.
result Several explicit examples of K-stable Fano varieties and their properties.
The effectiveness of Convolutional Neural Networks (CNNs) has been substantially attributed to their built-in property of translation equivariance. However, CNNs do not have embedded mechanisms to handle other types of transformations. In this work, we pay attention to scale changes, which regularly appear in various t…
Characterizes periodic elements in Artin-Tits groups via stability conditions.
problem Understanding periodic elements in Artin-Tits groups.
method Dynamical characterization via 2-Calabi-Yau category and stability conditions.
result An element is periodic if and only if it has a fixed point in the stability manifold.
Study equivariant isotopy in higher dimensions, finding exceptions.
problem Under what conditions are equivariant isotopic diffeomorphisms also equivariantly isotopic?
method Construct an invariant valued in the homology of an infinite cover to distinguish isotopic from equivariantly isotopic diffeomorphisms.
result Many equivariant diffeomorphisms are isotopic but not equivariantly isotopic in higher dimensions.
Study of strongly invertible Legendrian links in contact 3-space.
problem Characterizing and understanding strongly invertible Legendrian links.
method Equivariant analogs of basic results for strongly invertible and Legendrian links.
result Existence of maximal equivariant Thurston-Bennequin number for strongly invertible links.
Stable blowup solutions found for supercritical Yang-Mills equations.
problem Understanding blowup solutions for supercritical Yang-Mills equations.
method Investigated equivariant self-similar blowup solutions and their stability.
result Stability of blowup mechanism for odd dimensions greater than or equal to 5.
We present applications of the notion of isomorphic vector fields to the study of nonlinear stability of relative equilibria. Isomorphic vector fields were introduced by Hepworth [Theory Appl. Categ. 22 (2009), 542-587] in his study of vector fields on differentiable stacks. Here we argue in favor of the usefulness of …
We propose a new model for digital pathology segmentation, based on the observation that histopathology images are inherently symmetric under rotation and reflection. Utilizing recent findings on rotation equivariant CNNs, the proposed model leverages these symmetries in a principled manner. We present a visual analysi…
The paper simplifies K-stability conditions for spherical varieties.
problem K-stability of polarized spherical varieties.
method Expressed K-stability in combinatorial terms, provided sufficient conditions.
result G-uniform K-stability provides a checkable condition for existence of constant scalar curvature metrics.
Study proves Yau-Tian-Donaldson conjecture for generalized Kähler-Ricci solitons.
problem Proving Yau-Tian-Donaldson conjecture for generalized Kähler-Ricci solitons.
method Analyzing Monge-Ampère equations corresponding to generalized and twisted Kähler-Ricci g-solitons, proving stability conditions.
result Existence of solutions is equivalent to equivariantly uniform Θ-twisted g-Ding-stability.
We consider Fano manifolds admitting an algebraic torus action with general orbit of codimension one. Using a recent result of Datar and Szekelyhidi, we effectively determine the existence of Kahler-Ricci solitons for those manifolds via the notion of equivariant K-stability. This allows us to give new examples of Kahl…
Mabuchi solitons generalize Kähler-Einstein metrics on Fano manifolds, which constitute a Yau-Tian-Donaldson type correspondence with relative Ding stability. Comparing with Kähler-Ricci solitons, there is a distinct necessary condition for the existence. We show this condition can be implied by the uniformly relative …
In this paper we describe a 1-dimensional variational approach to the analytical construction of equivariant biharmonic maps. Our goal is to provide a direct method which enables analysts to compute directly the analytical conditions which guarantee biharmonicity in the presence of suitable symmetries. In the second pa…
New optimizer designs respect symmetry, improving deep learning models.
problem Optimizers lack respect for symmetry in neural networks.
method Introduce symmetry-compatible principle for optimizer design.
result Symmetry-compatible optimizers improve model performance.
Paper computes stability of Q-Fano spherical varieties using test configurations and Futaki invariants.
problem Stability of Q-Fano spherical varieties.
method Test configurations, Futaki invariants, intersection numbers.
result Equivalence of stability criteria and existence of Kähler-Ricci g-solitons.