arXiv research

Quantifies scalar curvature under $C^0$ convergence, proving a refined version in all dimensions.

New bounds for knot distances using Khovanov homology.

We give a refined upper bound for the hyperbolic volume of an alternating link in terms of the first three and the last three coefficients of its colored Jones polynomial.

Let $T$ be a triangulation of a Riemann surface. We show that the 1-skeleton of $T$ may be oriented so that there is a global bound on the outdegree of the vertices. Our application is to construct extremal metrics on triangulations formed from $T$ by attaching new edges and vertices and subdividing its faces. Such ref…

In a previous paper we constructed a spectrum-level refinement of Khovanov homology. This refinement induces stable cohomology operations on Khovanov homology. In this paper we show that these cohomology operations commute with cobordism maps on Khovanov homology. As a consequence we obtain a refinement of Rasmussen's …

Paper refines PAC-Bayes bounds for bandit problems.

New proofs and refined theorems on bounded cohomology.

Upper bounds on nullhomotopy volumes in nilpotent spaces are refined.

The paper refines and extends batched kernelized bandits, improving regret bounds and introducing a robust setting.

The concept of refinement from probability elicitation is considered for proper scoring rules. Taking directions from the axioms of probability, refinement is further clarified using a Hilbert space interpretation and reformulated into the underlying data distribution setting where connections to maximal marginal diver…

Improved lower bound for first Dirichlet eigenvalue using variance refinement.

New IDS algorithm refines parameter norm bounds for better bandit performance.

A refinement of Bennett's inequality is introduced which is strictly tighter than the classical bound. The new bound establishes the convergence of the average of independent random variables to its expected value. It also carefully exploits information about the potentially heterogeneous mean, variance, and ceiling of…

We formulate large $N$ duality of $\mathrm{U}(N)$ refined Chern-Simons theory with a torus knot/link in $S^3$. By studying refined BPS states in M-theory, we provide the explicit form of low-energy effective actions of Type IIA string theory with D4-branes on the $Ω$-background. This form enables us to relate refined C…

This article studies the achievable guarantees on the error rates of certain learning algorithms, with particular focus on refining logarithmic factors. Many of the results are based on a general technique for obtaining bounds on the error rates of sample-consistent classifiers with monotonic error regions, in the real…

Let a $A$ be the 1-skeleton of a triangulated topological annulus. We establish bounds on the combinatorial modulus of a refinement $A'$, formed by attaching new vertices and edges to $A$, that depend only on the refinement and not on the structure of $A$ itself. This immediately applies to showing that a disk triangul…

Refines knot defect measurement in 3D and 4D.

This is part 2 of a 3-part article where we provide an $O(n^2)$-algorithm to produce a surgery presentation of a 3-manifold induced by a gem with a resolution. In this part we produce a sequence of colored simplicial 2-complexes which are inverses and dual to the sequence of gems produced in the first part. The refinem…

The paper refines the three-page index for links, proving a new bound and characterizing specific links.

Given a simplicial complex $K$, we consider several notions of geometric complexity of embeddings of $K$ in a Euclidean space ${\mathbb R}^d$: thickness, distortion, and refinement complexity (the minimal number of simplices needed for a PL embedding). We show that any $n$-complex with $N$ simplices which topologically…

For certain classes of knots we define geometric invariants called higher-order genera. Each of these invariants is a refinement of the slice genus of a knot. We find lower bounds for the higher-order genera in terms of certain von Neumann $ρ$-invariants, which we call higher-order signatures. The higher-order genera o…

Proposes a method to refine PDE-driven high-dimensional rare-event simulation.

Study Kähler-Einstein potentials on stable varieties near singularities

A framework to boost the efficiency of Bayesian inference in probabilistic programs is introduced by embedding a sampler inside a variational posterior approximation. We call it the refined variational approximation. Its strength lies both in ease of implementation and automatically tuning of the sampler parameters to …

Based on work of Rasmussen, we construct a concordance invariant associated to the knot Floer complex, and exhibit examples in which this invariant gives arbitrarily better bounds on the 4-ball genus than the Ozsvath-Szabo tau invariant.

The paper improves confidence regions for band-limited functions using tighter norm bounds and majority voting.

Improved analysis for clipped gradient methods in nonsmooth convex optimization under heavy-tailed noise.

New vanishing theorems for harmonic and pluriharmonic functions on Kähler and quaternionic Kähler manifolds.

Improved analysis of UCBVI algorithm with better empirical performance.

New invariants refine link homology, showing large genus differences.

We prove a new lower bound for the first eigenvalue of the Dirac operator on a compact Riemannian spin manifold by refined Weitzenböck techniques. It applies to manifolds with harmonic curvature tensor and depends on the Ricci tensor. Examples show how it behaves compared to other known bounds.

Relying on the recent work of Liu-Székelyhidi we give a weak asymptotic estimate for the Bergman kernels of polarized Kähler manifolds with Ricci lower bound and Sobolev constant upper bound. We will also give a simple proof for the partial $C^0$ estimate along the (generalized) Kähler-Ricci flow on Fano manifolds.

Enhances Ricci flow theorem with scalar curvature bound.

Geometrically refines Cramér-Rao bound using extrinsic manifold curvature.

This paper provides a general result on controlling local Rademacher complexities, which captures in an elegant form to relate the complexities with constraint on the expected norm to the corresponding ones with constraint on the empirical norm. This result is convenient to apply in real applications and could yield re…

We analyze an upper bound on the curvature of a Riemannian manifold, using "root-Ricci" curvature, which is in between a sectional curvature bound and a Ricci curvature bound. (A special case of root-Ricci curvature was previously discovered by Osserman and Sarnak for a different but related purpose.) We prove that our…

We prove an explicit and sharp upper bound for the Castelnuovo-Mumford regularity of an FI-module V in terms of the degrees of its generators and relations. We use this to refine a result of Putman on the stability of homology of congruence subgroups, extending his theorem to previously excluded small characteristics a…

We obtain some improved essentially sharp Kakeya-Nikodym estimates for eigenfunctions in two-dimensions. We obtain these by proving stronger related microlocal estimates involving a natural decomposition of phase space that is adapted to the geodesic flow.

UCB-V algorithm improves on UCB for MAB problems with variance estimates.

New CRB derived for curved models using extrinsic geometry.

We refine and generalize several interpolation inequalities bounding the $L^p$ norm of a probability density with respect to the reference measure $μ$ by its Sobolev norm and the Kantorovich distance to $μ$ on a smooth weighted Riemannian manifold satisfying $CD(0, \infty)$ condition.

We present in this note a lower bound for the Calabi functional in a given Kähler class. This yields an integral inequality for constant scalar curvature metrics, which can be viewed as a refined version of Yau's Chern number inequality.

LMC achieves sqrt(d) dependence in sampling error, improving previous bounds.

New inequality for refined knot invariants in a specific space.

Proves convergence of gradient Ricci shrinkers with uniform bounds.

Algorithm captures and refines features for efficient lifelong learning.

This paper refines bounds on random walk speed in Teichmüller space.

Optimistic Hedge achieves optimal regret bounds in two-player zero-sum games.