arXiv research

Study linear differential operators on special manifolds.

Lectures on linearized Kapustin-Witten equations on half-line.

Study on stochastic approximation with Polyak-Ruppert averaging for linear systems.

Theoretical validation of linear PCA and ICA for accurate nonlinear BSS.

The paper analyzes SGD with dropout regularization in linear models, proving asymptotic properties and providing inference tools.

The paper proves a discrete positive mass theorem for graphs.

Dynamic hedging of an European option under a general local volatility model with small linear transaction costs is studied. A continuous control version of Leland's strategy that asymptotically replicates the payoff is constructed. An associated central limit theorem of hedging error is proved. The asymptotic error va…

The paper proves positive energy-momentum theorems for charged AdS initial data sets.

Study on Einstein manifolds linking stability and rigidity.

The main purpose of this monograph is to give an elementary and self-contained account of the existence of asymptotically hyperbolic Einstein metrics with prescribed conformal infinities sufficiently close to that of a given asymptotically hyperbolic Einstein metric with nonpositive curvature. The proof is based on an …

The paper derives deterministic inequalities for M-estimators using Banach fixed point theorem.

Deep random feature models are analyzed for their performance with exact asymptotic expressions.

LinUCB algorithm handles adaptive sampling biases for linear bandits.

The paper proves a new inequality for 3-manifolds with noncompact boundaries.

We use planar coordinates as well as hyperbolic coordinates to separate the de Sitter spacetime into two parts. These two ways of cutting the de Sitter give rise to two different spatial infinities. For spacetimes which are asymptotic to either half of the de Sitter spacetime, we are able to provide definitions of the …

New proof of Riemannian Penrose Inequality for manifolds with corners

The paper studies asymptotic expansions of operators related to Bochner-Schrödinger on Riemannian manifolds.

Explicit mass bound for 3D asymptotically flat manifolds using harmonic functions.

Corrects errors in previous work on spectral asymptotics in elasticity.

We prove a comparison theorem on the first Neumann eigenvalue on Bakry-Emery manifolds. Examples are constructed to illustrate the sharpness of the result. A linear explicit lower bound is also proved. We also discuss the asymptotic sharpness of such a result.

Study bounds noise level in linear regression with dependent data.

The paper applies potential theory to conformal geometry, proving theorems and dimension estimates.

Study Q-learning with averaging for reinforcement learning, proving efficient inference and error bounds.

Study shows normal distribution in divisor counts of random sections on complex manifolds.

A key challenge for modern Bayesian statistics is how to perform scalable inference of posterior distributions. To address this challenge, variational Bayes (VB) methods have emerged as a popular alternative to the classical Markov chain Monte Carlo (MCMC) methods. VB methods tend to be faster while achieving comparabl…

Proves positive mass theorems for specific types of curved spaces.

Study on linear regression with dependent covariates, proving universality and error characterization.

Proves positive mass theorem for specific manifold types.

The study proves a central limit theorem for Gaussian holomorphic sections on Kähler manifolds.

Geometric arguments show simplicial arrangements with few double points can't have an irreducible cubic curve dual.

Paper improves CLT and bootstrap approximations for LSA with decreasing step size.

We introduce a geometric property complementary-finite asymptotic dimension (coas- dim). Similar with asymptotic dimension, we prove the corresponding coarse invariant theorem, union theorem and Hurewicz-type theorem.

Proves spacetime positive mass theorem in all dimensions.

Paper extends positive energy theorem to anti-de Sitter spacetimes.

Proves positive mass theorem for hyperbolic manifolds with ends.

We prove the equality case of the Penrose inequality in all dimensions for asymptotically flat hypersurfaces. It was recently proven by G. Lam that the Penrose inequality holds for asymptotically flat graphical hypersurfaces in Euclidean space with non-negative scalar curvature and with a minimal boundary. Our main the…

A singularity theorem based on asymptotic volume growth

Solves Jang equation for hyperboloidal data, proving positive mass theorem.

Abstract reviews hyperbolic positive energy theorems.

The positive energy theorem is proven for certain spacetimes with irregular curvature.

Study volume growth and asymptotic cones of nonnegative Ricci curvature manifolds.

Paper proposes a bias-constrained deep learning approach to non-linear estimation.

Proves density and mass theorems for specific initial data sets.

We prove an asymptotic analog of the classical Hurewicz theorem on mappings which lower dimension. This theorem allows us to find sharp upper bound estimates for the asymptotic dimension of groups acting on finite dimensional metric spaces and allows us to prove a useful extension theorem for asymptotic dimension. As a…

Study of Dirichlet minimizers on manifolds with boundary and their asymptotic behavior.

Study the positive mass theorem for certain asymptotic manifolds.

We formulate and prove the Lorentzian version of the positive mass theorems with arbitrary negative cosmological constant for asymptotically AdS spacetimes. This work is the continuation of the second author's recent work on the positive mass theorem on asymptotically hyperbolic 3-manifolds.

The study proves Strichartz and spectral projection theorems on specific types of curved surfaces.