Proves criteria for uniform K-stability of log Fano pairs.
problem Determining conditions for uniform K-stability of log Fano pairs.
method Analyzes criteria and proves equivalence to β-invariant having a positive lower bound.
result Uniform K-stability is equivalent to β-invariant having a positive lower bound.
Uniformizes klt pairs using bounded symmetric domains.
problem Characterizing klt pairs uniformizable by bounded symmetric domains.
method Determines conditions for uniformization using Miyaoka-Yau-type inequalities.
result Characterizations of orbifold quotients of polydisc and classical bounded symmetric domains.
We show relationships between uniform K-stability and plt blowups of log Fano pairs. We see that it is enough to evaluate certain invariants defined by volume functions for all plt blowups in order to test uniform K-stability of log Fano pairs. We also discuss the uniform K-stability of two log Fano pairs under crepant…
Equality in Miyaoka-Yau inequality implies uniformization of Klt pairs.
problem Understanding uniformization of Klt pairs under equality in Miyaoka-Yau inequality.
method Analyzing Kähler klt pairs with specific conditions and using orbifold Miyaoka-Yau inequality.
result Orbifold universal cover is either the unit ball or affine space.
Generalizes uniformization to algebraic correspondences.
problem Uniformizing non-homeomorphic genus zero orbifolds.
method Constructs algebraic correspondences to simultaneously uniformize orbifolds.
result Realizes Teichmüller space of a punctured sphere in correspondences.
Paper discusses K-stability of pairs and its implications.
problem K-stability of pairs
method Analyzes stable pairs and proves implications.
result K-stability implies CM-stability.
Uniform foliations with Reeb components on 3-manifolds.
problem Characterizing uniform foliations with Reeb components.
method Study of foliations on 3-manifolds with infinite fundamental group.
result Examples and results on behavior of foliations.
Proves finitely generated associated graded rings for valuations on log Fano pairs.
problem Stability thresholds of log Fano pairs.
method Proves finite generation of associated graded rings for valuations.
result Log Fano pairs are uniformly K-stable if their stability threshold is less than a certain value.
Study proves uniform ellipticity implies uniform polyconvexity for anisotropic energy functionals.
problem Investigating uniform ellipticity and polyconvexity for anisotropic geometric energy functionals.
method Proves a variant of a recent result using real polyhedral chains.
result Uniform ellipticity of an anisotropic energy functional implies uniform polyconvexity of the integrand.
We construct a pair of transverse genuine laminations on an atoroidal 3-manifold admitting transversely orientable uniform 1-cochain. The laminations are induced by the uniform 1-cochain and they are indeed the "straightening" of the coarse laminations defined in [Ca], by using minimal surface techniques. Moreover, whe…
Paper introduces 'zippers' for constructing universal circles.
problem Constructing universal circles for hyperbolic 3-manifolds.
method Introduces zippers to directly construct universal circles.
result New and direct way to construct universal circles.
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.
The purpose of the present paper is to set up a formalism inspired from non-Archimedean geometry to study K-stability. We first provide a detailed analysis of Duistermaat-Heckman measures in the context of test configurations, characterizing in particular the trivial case. For any normal polarized variety (or, more gen…
Shows uniform K-stability is open in Q-Gorenstein families of Q-Fano varieties.
problem Detecting uniform K-stability in families of Q-Fano varieties.
method Examined the behavior of the stability threshold in families and showed it is lower semicontinuous.
result Uniform K-stability is a Zariski open condition in Q-Gorenstein families of Q-Fano varieties.
PCA++ improves robustness to background noise in contrastive learning.
problem Recovering shared signal subspaces from positive pairs in high-dimensional data with structured background noise.
method PCA++ uses hard uniformity-constrained contrastive learning to enforce identity covariance on projected features.
result PCA++ outperforms standard PCA and alignment-only PCA+ in simulations and real-world datasets.
Minimal action on universal circle for foliations on 3-manifolds.
problem Action of fundamental group on universal circle of foliations.
method Analyzes uniform foliations on 3-manifolds, proving minimality and transitivity.
result Action is minimal and transitive on pairs of different points.
This work optimizes alignment and uniformity of features on a hypersphere for better downstream performance.
problem Improving the performance of contrastive representation learning.
method Identifying and optimizing alignment and uniformity of features on a hypersphere.
result Directly optimizing alignment and uniformity leads to comparable or better performance than contrastive learning.
We give uniform, explicit, and simple face-pairing descriptions of all the branched cyclic covers of the 3-sphere, branched over two-bridge knots. Our method is to use the bi-twisted face-pairing constructions of Cannon, Floyd, and Parry; these examples show that the bi-twist construction is often efficient and natural…
Proves results on K-stability using arcs and Mabuchi functional.
problem K-stability in Fano manifolds and uniform K-polystability.
method Arcs and numerical criterion for stability of pairs.
result Characterizes coercivity of Mabuchi functional in terms of K-polystability.
Improved sample complexity for Gaussian Mixture Models using Pair Correlation Factor.
problem Understanding the sample complexity of Gaussian Mixture Models.
method Introducing Pair Correlation Factor (PCF) to measure clustering of component means and improving sample complexity bounds.
result The Pair Correlation Factor (PCF) more accurately determines the difficulty of parameter recovery in Gaussian Mixture Models.
The study shows that random surfaces built from polygons converge to a Poisson-Dirichlet partition.
problem Understanding geometric properties of random surfaces constructed from polygons.
method Uniformly pairing polygon sides to form surfaces, analyzing degree sequences and geometric properties using probabilistic techniques.
result Several geometric properties of the graph are universal, converging to a Poisson-Dirichlet partition as no∞. We consider a class X of continuous functions on [0,1] that is of interest from two different perspectives. First, it is closely related to sets of functions that have been studied as generalizations of the Takagi function. Second, each function in X admits a linear pathwise quadratic variatio…
Extends Poincaré-Lefschetz duality to pairs of ∞-categories.
problem Generalizing Poincaré-Lefschetz duality to ∞-categories.
method Introduces Poincaré duality pairs of ∞-categories and uses them to study various diagrams of spaces.
result Unified treatment of Wall's Poincaré ads and iterated Poincaré cobordisms.
We will show that for a polynomially contractible manifold of bounded geometry and of polynomial volume growth every coarse and rough cohomology class pairs continuously with the K-theory of the uniform Roe algebra. As an application we will discuss non-vanishing of rough index classes of Dirac operators over such mani…
Pessimistic Minimax Value Iteration finds efficient NE policies from offline data.
problem Finding an approximate Nash equilibrium in offline Markov games with non-uniform coverage.
method Pessimistic Minimax Value Iteration (PMVI) constructs pessimistic value function estimates and solves NEs.
result Established a nearly minimax optimal result for offline Markov games with function approximation.
Study on signatures of positive braids with bounds derived.
problem Understanding signatures of positive braids and their invariants.
method Derived lower bounds for Levine-Tristram signatures, and upper and lower bounds on signature ratios.
result Established bounds on signatures of positive braids, uniformly valid across monoids.
Let Σ_g be a closed orientable surface of genus g \geq 2 and τa graph on Σ_g with one vertex which lifts to a triangulation of the universal cover. We have shown that the cross ratio parameter space \mathcal{C}_τassociated with τ, which can be identified with the set of all pairs of a projective structure and a circle …
Unified bounds for sketched bilinear forms in machine learning and statistics.
problem Uniform bounds on sketched bilinear forms for modern analyses.
method Generic chaining and new techniques for handling suprema over pairs of sets.
result Improved convergence bounds for sketched Federated Learning and bandit algorithms.
Uniform-in-time analysis for Stein Variational Gradient Descent across various metrics.
problem Understanding long-term behavior of finite-particle systems in relation to their mean-field limits.
method Developed uniform-in-time propagation-of-chaos results for continuous-time SVGD using cutoff strategies and finite-dimensional theories.
result Uniform-in-time propagation-of-chaos bounds in various metrics, including Langevin kernel Stein discrepancy, Wasserstein-1, and Wasserstein-2 distances.
Study heat flow on collapsing K3 surfaces, handling conic singularities.
problem Analyzing heat flow on K3 surfaces as they collapse.
method Using semi-flat product approximations and conic-renormalized bilinear functionals.
result Heat operators converge to base Laplacian on regular locus.
Solves modified conjecture for Fano manifolds using Ding stability.
problem Finding Kähler-Einstein metrics on Fano manifolds.
method Interprets Ding semistability and solves modified conjecture.
result Solves modified conjecture for coupled Kähler-Einstein metrics on Fano manifolds.
Uniform heat kernel and diffusion bridge asymptotics for sub-Riemannian geometry.
problem Analyzing sub-Riemannian heat kernels and their derivatives on incomplete manifolds.
method Localized asymptotic analysis, focusing on minimizing geodesics and the non-abnormal cut locus.
result Uniform bounds and expansions for heat kernels and their derivatives on compacts, including the diffusion bridge measure.
The complex analytic methods have found a wide range of applications in the study of multiplicity-free representations. This article discusses, in particular, its applications to the question of restricting highest weight modules with respect to reductive symmetric pairs. We present a number of multiplicity-free branch…
The paper connects arithmetic invariants of hyperbolic 3-manifolds.
problem Understanding the arithmetic properties of hyperbolic 3-manifolds.
method Analyzes profinite completions and algebraic invariants of fundamental groups.
result Uniform lattices with isomorphic profinite completions have identical arithmetic properties.
The paper proves a mapping from a space of holonomy varieties to Teichmüller spaces, with a non-empty discrete intersection.
problem Intersection of Poincaré holonomy varieties and their properties.
method Holomorphic mapping and branched covering proof.
result Intersection of arbitrary Poincaré holonomy varieties is a non-empty discrete set.
Tian initiated the study of incomplete Kähler-Einstein metrics on quasi-projective varieties with cone-edge type singularities along a divisor, described by the cone-angle 2π(1−α) for α∈(0,1). In this paper we study how the existence of such Kähler-Einstein metrics depends on α. We show that in the negative s…
New groups with distinct Dehn functions and properties.
problem Finding groups with different Dehn functions.
method Created specific Lie groups and Carnot graded groups.
result Groups with uniform lattices have different asymptotic cones and Dehn functions.
The problem of biclustering consists of the simultaneous clustering of rows and columns of a matrix such that each of the submatrices induced by a pair of row and column clusters is as uniform as possible. In this paper we approximate the optimal biclustering by applying one-way clustering algorithms independently on t…
Origamis described using Schottky groups for surfaces of genus g ≥ 1.
problem Describing origamis by Schottky groups for Riemann surfaces.
method Using geometrical structural picture and Klein-Maskit combination theorems.
result Provided a geometrical structural picture of origami-Schottky groups.
The study examines subgroups of torus mapping class group generated by Dehn twists powers.
problem Characterizing subgroups generated by powers of Dehn twists.
method Using the ping pong lemma and geometric intersection numbers.
result Subgroups can be free groups, direct products, or have specific ranks.
We examine the L2-topology of the gauge orbits over a closed Riemann surface. We prove a subtle local slice theorem based on the div-curl Lemma of harmonic analysis, and deduce local pathwise connectedness and local uniform quasiconvexity of the gauge orbits. Using these, we generalize compactness results for anti-s…
The set of Clifford bundles of bounded geometry over open manifolds can be endowed with a metrizable uniform structure. For one fixed bundle E we define the generalized component $\gencomp (E)$ as the set of Clifford bundles E′ which have finite distance to E. If D, D′ are the associated generalized Dirac ope…
We produce examples of taut foliations of hyperbolic 3-manifolds which are R-covered but not uniform --- ie the leaf space of the universal cover is R, but pairs of leaves are not contained in bounded neighborhoods of each other. This answers in the negative a conjecture of Thurston `Three-manifolds, foliations and cir…
A new kernel-based CI test improves on existing methods.
problem Testing conditional independence (CI) in a broad range of dependencies.
method Regression-model-agnostic kernel-based CI test using reproducing kernel Hilbert spaces.
result GKCM outperforms state-of-the-art CI tests in simulations.
Signed pairwise interactions conflate uniqueness, redundancy, and synergy
problem Signed pairwise interactions conflate uniqueness, redundancy, and synergy
method Stochastic Hi-Fi
result Stochastic Hi-Fi recovers structure missed by scalar baselines
SDP approach recovers communities in multilayer hypergraphs from aggregated similarity matrices.
problem Community recovery in multilayer hypergraphs using aggregated similarity matrices.
method Semidefinite programming (SDP) approach.
result Information-theoretic conditions for exact recovery in both assortative and disassortative cases.
New uniform K-theory and Poincare duality established for manifolds.
problem Developing a new framework for K-theory and K-homology.
method Constructing uniform K-homology, defining external and cap products, proving homotopy invariance and Poincare duality.
result Established Poincare duality between uniform K-theory and uniform K-homology on spin-c manifolds.
Develops a universal Hermitian projective calculus for complex hyperbolic two-space
problem Complex hyperbolic geometry
method Algebraic invariant calculus
result Denominator-cleared identities for various geometric quantities