arXiv research

Sharp $L^\infty$ estimates proved for complex Monge-Ampère equations.

New proof of generalized Chow-Rashevskii theorem for non-linear systems.

We present a general existence proof for a wide class of non-linear elliptic equations which can be applied to problems with barrier conditions without specifying any assumptions guaranteeing the uniqueness or local uniqueness of particular solutions. As an application we prove the existence of closed hypersurfaces wit…

Study shows long-term solutions for complex equations on curved spaces.

New proof of Riemannian Penrose Inequality for manifolds with corners

Quantile Temporal-Difference learning proved convergent with proof.

We prove the global non-linear stability, without symmetry assumptions, of slowly rotating charged black holes in de Sitter spacetimes in the context of the initial value problem for the Einstein-Maxwell equations: If one perturbs the initial data of a slowly rotating Kerr-Newman-de Sitter (KNdS) black hole, then in a …

Two proofs of Kalman Theorem using flows of vector fields.

Proves energy estimates for tensorial wave equations, decoupling components for stability proof.

Sharp L∞ estimates for fully non-linear elliptic equations on compact complex manifolds.

Uniform bounds for Green's function on Kähler manifolds derived from complex Monge-Ampère equations.

We address the problem of causal discovery from data, making use of the recently proposed causal modeling framework of modular structural causal models (mSCM) to handle cycles, latent confounders and non-linearities. We introduce σ-connection graphs (σ-CG), a new class of mixed graphs (containing undirected, bidirected…

New proof of $L^\infty$ estimates for Monge-Ampère and Hessian equations on nef classes.

This paper studies semiparametric contextual bandits, a generalization of the linear stochastic bandit problem where the reward for an action is modeled as a linear function of known action features confounded by an non-linear action-independent term. We design new algorithms that achieve $\tilde{O}(d\sqrt{T})$ regret …

This survey paper contains an elementary exposition of Casson and Rivin's technique for finding the hyperbolic metric on a 3-manifold M with toroidal boundary. We also survey a number of applications of this technique. The method involves subdividing M into ideal tetrahedra and solving a system of gluing equations to f…

Study $p$-parabolicity on graphs using various energy functionals.

Proves stability of Schwarzschild black holes without symmetry assumptions.

New proof shows origin-centred balls are unique solutions to curvature problems.

This paper is motivated by the non-linear stability problem for the expanding region of Kerr de Sitter cosmologies in the context of Einstein's equations with positive cosmological constant. We show that under dynamically realistic assumptions the conformal Weyl curvature of the spacetime decays towards future null inf…

Essential curves in certain graph manifolds imply non-linear groups.

We prove in this paper the linear stability of the celebrated Schwarzschild family of black holes in general relativity: Solutions to the linearisation of the Einstein vacuum equations around a Schwarzschild metric arising from regular initial data remain globally bounded on the black hole exterior and in fact decay to…

Method learns causal effects from multiple interventions in presence of unobserved confounders.

This work develops methods to analyze data on curved spaces using deep learning.

The fundamental tool in the classification of orthogonal coordinate systems in which the Hamilton-Jacobi and other prominent equations can be solved by a separation of variables are second order Killing tensors which satisfy the Nijenhuis integrability conditions. The latter are a system of three non-linear partial dif…

Alternative proof and extension of curvature estimates for minimal immersions.

This paper tackles efficient and scalable estimation of a complex model involving stochastic linear combinations of non-linear regressions.

The paper constructs special Lagrangian n-folds in arbitrary dimensions.

On a Riemannian metric-measure space, we establish an Alexandrov-Bakelman-Pucci type measure estimate connecting Bakry-Émery Ricci curvature lower bound, modified Laplacian and the measure of certain special sets. We apply this estimate to prove Harnack inequalities for the modified Laplacian operator and fully non-lin…

The paper classifies solitons for a specific type of flow.

New deep learning method simplifies parameter estimation design.

We show that a stationary solution of the Einstein-Maxwell equations which is close to a non-degenerate Reissner-Nordström-de Sitter solution is in fact equal to a slowly rotating Kerr-Newman-de Sitter solution. The proof uses the non-linear stability of the Kerr-Newman-de Sitter family of black holes for small angular…

In this paper, we present C-ADAM, the first adaptive solver for compositional problems involving a non-linear functional nesting of expected values. We proof that C-ADAM converges to a stationary point in $\mathcal{O}(δ^{-2.25})$ with $δ$ being a precision parameter. Moreover, we demonstrate the importance of our resul…

New methods prove controllability of non-linear systems, extending classical results.

We discuss inverse resonance scattering for the Laplacian on a rotationally symmetric manifold $M = (0,\infty) \times Y$ whose rotation radius is constant outside some compact interval. The Laplacian on $M$ is unitarily equivalent to a direct sum of one-dimensional Schrödinger operators with compactly supported potenti…

Classifies non-linear Fredholm maps linking to stable homotopy groups of spheres.

Stability of black holes proven in full subextremal range with positive cosmological constant.

Paper uses non-linear dimension reduction for better economic forecasting.

We show how the tangent functor extends from ordinary smooth maps to "microformal morphisms" (also called "thick morphisms") of supermanifolds. Microformal morphisms generalize ordinary maps and correspond to formal canonical relations between the cotangent bundles specified by generating functions depending on positio…

Study shows solutions to certain equations form smooth manifolds.

We study the Futaki invariant and the Mabuchi K-energy of a Kähler manifold $M$ using the Deligne pairing technique developed in earlier papers. We first prove a rather simple characterization of the Futaki character: The Futaki character on a Q-Fano variety is the eigenvalue of the action of $Aut(M)$ on $Chow(M)$, the…

Optimal algorithms for non-linear ridge bandits reduce burn-in cost.

Uniform bounds derived for fully non-linear equations.

TaCo prevents non-linear classifiers from detecting sensitive attributes.

Paper models non-linear dynamics from time series data.

SSL framework identifies non-linear systems without labeled data.

Optimizes non-linear outcomes from summed contributions.

WeSpeR speeds up non-linear shrinkage for high-dimensional weighted covariance.

New model for Knightian uncertainty with jumps.