Source author record

Charles Fefferman

Explore this author’s publications, collaborators and research topics. This catalog record has not been claimed by its author.

ResearcherUnclaimed source record

math.AP math.CA math.DG math-ph math.FA math.MP math.ST physics.flu-dyn Statistics Theory Machine Learning math.AG math.NA math.OC Methodology

Catalog footprint

Research overview

18works

14topics

4close collaborators

Actions

Is this your work?

Open graph Browse papers

preprint2026arXiv

Denoising data using convex relaxations

We study the problem of denoising observations $Y_i=X_i+Z_i$ , where the latent variables $X_i$ are sampled from a low-dimensional manifold in $\mathbb{R}^n$ and the noise variables $Z_i$ are isotropic Gaussian. We propose a convex-relaxation estimator that first reduces dimension by principal component analysis and then projects the observations onto the convex hull of the projected latent manifold. We construct a statistical oracle that estimates its supporting hyperplanes from empirical Gaussian tail probabilities of the noisy sample. Under a lower-mass condition on the latent distribution, we prove finite-sample guarantees for the oracle and derive error bounds for the resulting denoiser. The analysis combines risk bounds for least-squares projection under convex constraints with entropy bounds for convex hulls. We also verify the assumptions of the framework for a Cryo-Electron Microscopy observation model by establishing suitable covering number and Lipschitz estimates for the associated group action and imaging operators.

preprint2022arXiv

Fitting a manifold of large reach to noisy data

Let ${\mathcal M}\subset {\mathbb R}^n$ be a $C^2$ -smooth compact submanifold of dimension $d$ . Assume that the volume of ${\mathcal M}$ is at most $V$ and the reach (i.e. the normal injectivity radius) of ${\mathcal M}$ is greater than $τ$ . Moreover, let $μ$ be a probability measure on ${\mathcal M}$ whose density on ${\mathcal M}$ is a strictly positive Lipschitz-smooth function. Let $x_j\in {\mathcal M}$ , $j=1,2,\dots,N$ be $N$ independent random samples from distribution $μ$ . Also, let $ξ_j$ , $j=1,2,\dots, N$ be independent random samples from a Gaussian random variable in ${\mathbb R}^n$ having covariance $σ^2I$ , where $σ$ is less than a certain specified function of $d, V$ and $τ$ . We assume that we are given the data points $y_j=x_j+ξ_j,$ $j=1,2,\dots,N$ , modelling random points of ${\mathcal M}$ with measurement noise. We develop an algorithm which produces from these data, with high probability, a $d$ dimensional submanifold ${\mathcal M}_o\subset {\mathbb R}^n$ whose Hausdorff distance to ${\mathcal M}$ is less than $Cdσ^2/τ$ and whose reach is greater than $cτ/d^6$ with universal constants $C,c > 0$ . The number $N$ of random samples required depends almost linearly on $n$ , polynomially on $σ^{-1}$ and exponentially on $d$ .

preprint2016arXiv

Finitness Principles for Smooth Selection

preprint2016arXiv

Interpolation of data by smooth non-negative functions

We prove a finiteness principle for interpolation of data by nonnegative Cm functions. Our result raises the hope that one can start to understand constrained interpolation problems in which e.g. the interpolating function F is required to be nonnegative.

preprint2015arXiv

Finiteness Principles for Smooth Selection

In this paper we prove finiteness principles for $C^{m}\left( \mathbb{R}^{n}, \mathbb{R}^{D}\right)$ -selection, and for $C^{m-1,1}\left( \mathbb{R}^{n}, \mathbb{R}^{D}\right)$ -selection, in particular providing a proof for a conjecture of Brudyni-Shvartsman (1994) on Lipschitz selections for the case when the domain is $X = \mathbb{R}^n$ . Our results raise the hope that one can start to understand constrained interpolation problems in which e.g. the interpolating function $F$ is required to be nonnegative everywhere.

preprint2015arXiv

Splash singularities for the one-phase Muskat problem in stable regimes

This paper shows finite time singularity formation for the Muskat problem in a stable regime. The framework we found is with a dry region, where the density and the viscosity are set equal to $0$ (the gradient of the pressure is equal to $(0,0)$ ) in the complement of the fluid domain. The singularity is a splash-type: a smooth fluid boundary collapses due to two different particles evolve to collide at a single point. This is the first example of a splash singularity for a parabolic problem.

preprint2014arXiv

Structural stability for the splash singularities of the water waves problem

In this paper we show a structural stability result for water waves. The main motivation for this result is that we would like to exhibit a water wave whose interface starts as a graph and ends in a splash. Numerical simulations lead to an approximate solution with the desired behaviour. The stability result will conclude that near the approximate solution to water waves there is an exact solution.

preprint2013arXiv

On the absence of "splash" singularities in the case of two-fluid interfaces

We show that "splash" singularities cannot develop in the case of locally smooth solutions of the two-fluid interface in two dimensions. More precisely, we show that the scenario of formation of singularities discovered by Castro-Córdoba-Fefferman-Gancedo-Gómez-Serrano in the case of the water waves system, in which the interface remains locally smooth but self-intersects in finite time, is completely prevented in the case of two-fluid interfaces with positive densities.

preprint2013arXiv

Testing the Manifold Hypothesis

The hypothesis that high dimensional data tend to lie in the vicinity of a low dimensional manifold is the basis of manifold learning. The goal of this paper is to develop an algorithm (with accompanying complexity guarantees) for fitting a manifold to an unknown probability distribution supported in a separable Hilbert space, only using i.i.d samples from that distribution. More precisely, our setting is the following. Suppose that data are drawn independently at random from a probability distribution $P$ supported on the unit ball of a separable Hilbert space $H$ . Let $G(d, V, τ)$ be the set of submanifolds of the unit ball of $H$ whose volume is at most $V$ and reach (which is the supremum of all $r$ such that any point at a distance less than $r$ has a unique nearest point on the manifold) is at least $τ$ . Let $L(M, P)$ denote mean-squared distance of a random point from the probability distribution $P$ to $M$ . We obtain an algorithm that tests the manifold hypothesis in the following sense. The algorithm takes i.i.d random samples from $P$ as input, and determines which of the following two is true (at least one must be): (a) There exists $M \in G(d, CV, \fracτ{C})$ such that $L(M, P) \leq C ε.$ (b) There exists no $M \in G(d, V/C, Cτ)$ such that $L(M, P) \leq \fracε{C}.$ The answer is correct with probability at least $1-δ$ .

preprint2012arXiv

Breakdown of smoothness for the Muskat problem

In this paper we show that there exist analytic initial data in the stable regime for the Muskat problem such that the solution turns to the unstable regime and later breaks down i.e. no longer belongs to $C^4$ .

preprint2012arXiv

Finite time singularities for the free boundary incompressible Euler equations

In this paper, we prove the existence of smooth initial data for the 2D free boundary incompressible Euler equations (also known for some particular scenarios as the water wave problem), for which the smoothness of the interface breaks down in finite time into a splash singularity or a splat singularity.

preprint2012arXiv

Finite time singularities for water waves with surface tension

Here we consider the 2D free boundary incompressible Euler equation with surface tension. We prove that the surface tension does not prevent a finite time splash or splat singularity, i.e. that the curve touches itself either in a point or along an arc. To do so, the main ingredients of the proof are a transformation to desingularize the curve and a priori energy estimates.

preprint2012arXiv

Juhl's Formulae for GJMS Operators and Q-Curvatures

Direct proofs are given of Juhl's formulae for GJMS operators and Q-curvatures starting from the original construction of GJMS.

preprint2011arXiv

Continuous linear combinations of polynomials

We give necessary and sufficient existence criteria, and methods for finding, continuous solutions of linear equations whose coefficients are polynomials.

preprint2011arXiv

Rayleigh-Taylor breakdown for the Muskat problem with applications to water waves

The Muskat problem models the evolution of the interface given by two different fluids in porous media. The Rayleigh-Taylor condition is natural to reach the linear stability of the Muskat problem. We show that the Rayleigh-Taylor condition may hold initially but break down in finite time. As a consequence of the method used, we prove the existence of water waves turning.

preprint2011arXiv

Splash singularity for water waves

We exhibit smooth initial data for the 2D water wave equation for which we prove that smoothness of the interface breaks down in finite time. Moreover, we show a stability result together with numerical evidence that there exist solutions of the 2D water wave equation that start from a graph, turn over and collapse in a splash singularity (self intersecting curve in one point) in finite time.

preprint2010arXiv

Turning waves and breakdown for incompressible flows

We consider the evolution of an interface generated between two immiscible incompressible and irrotational fluids. Specifically we study the Muskat and water wave problems. We show that starting with a family of initial data given by $(\al,f_0(\al))$, the interface reaches a regime in finite time in which is no longer a graph. Therefore there exists a time $t^*$ where the solution of the free boundary problem parameterized as $(\al,f(\al,t))$ blows-up: $\|\da f\|_{L^\infty}(t^*)=\infty$. In particular, for the Muskat problem, this result allows us to reach an unstable regime, for which the Rayleigh-Taylor condition changes sign and the solution breaks down.

preprint2003arXiv

Ambient metric construction of Q-curvature in conformal and CR geometries

We give a geometric derivation of Branson's Q-curvature in terms of the ambient metric associated with conformal structures; it naturally follows from the ambient metric construction of conformally invariant operators and can be applied to a large class of invariant operators. This procedure can be also applied to CR geometry and gives a CR analog of the Q-curvature; it then turns out that the Q-curvature gives the coefficient of the logarithmic singularity of the Szego kernel of 3-dimensional CR manifolds.

Inspect adjacent papers, topics, institutions and collaborators without losing the researcher page.

Building this map preview

BZPEER is loading the nearby papers, people, topics and institutions for this page.

Charles Fefferman

Research overview

Is this your work?

18 publications

Denoising data using convex relaxations

Fitting a manifold of large reach to noisy data

Finitness Principles for Smooth Selection

Interpolation of data by smooth non-negative functions

Finiteness Principles for Smooth Selection

Splash singularities for the one-phase Muskat problem in stable regimes

Structural stability for the splash singularities of the water waves problem

On the absence of "splash" singularities in the case of two-fluid interfaces

Testing the Manifold Hypothesis

Breakdown of smoothness for the Muskat problem

Finite time singularities for the free boundary incompressible Euler equations

Finite time singularities for water waves with surface tension

Juhl's Formulae for GJMS Operators and Q-Curvatures

Continuous linear combinations of polynomials

Rayleigh-Taylor breakdown for the Muskat problem with applications to water waves

Splash singularity for water waves

Turning waves and breakdown for incompressible flows

Ambient metric construction of Q-curvature in conformal and CR geometries

See the researcher in context

Building this map preview