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Publications (45)

math.CA2016

Interpolation of data by smooth non-negative functions

Charles Fefferman, Arie Israel, Garving K. Luli

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 problem…

stat.ML2025

Reconstruction of Manifold Distances from Noisy Observations

Charles Fefferman, Jonathan Marty, Kevin Ren

We consider the problem of reconstructing the intrinsic geometry of a manifold from noisy pairwise distance observations. Specifically, let $M$ denote a diameter 1 d-dimensional ma…

math.OC2024

Almost Optimal Agnostic Control of Unknown Linear Dynamics

Jacob Carruth, Maximilian F. Eggl, Charles Fefferman +1

We consider a simple control problem in which the underlying dynamics depend on a parameter $a$ that is unknown and must be learned. We study three variants of the control problem:…

math.AP2010

Turning waves and breakdown for incompressible flows

Angel Castro, Diego Cordoba, Charles Fefferman +2

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…

math.CA2015

Finiteness Principles for Smooth Selection

Charles Fefferman, Arie Israel, Garving K. Luli

In this paper we prove finiteness principles for $C^{m} (R^{n}, R^{D})$ -selection, and for $C^{m - 1, 1} (R^{n}, R^{D})$ -se…

math.DG2001

Q-Curvature and Poincare Metrics

Charles Fefferman, C. Robin Graham

This article presents a new definition of Branson's Q-curvature in even-dimensional conformal geometry. We derive the Q-curvature as a coefficient in the asymptotic expansion of th…

stat.ME2026

Denoising data using convex relaxations

Charles Fefferman, Aalok Gangopadhyay, Matti Lassas +2

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 var…

math.ST2023

Fitting a manifold to data in the presence of large noise

Charles Fefferman, Sergei Ivanov, Matti Lassas +1

We assume that $M_{0}$ is a $d$ -dimensional $C^{2, 1}$ -smooth submanifold of $R^{n}$ . Let $K_{0}$ be the convex hull of $M_{0},$ and $B_{1}^{n} (0)$ be the unit ball. We assume that $ M_0 \subse…

math.DG2025

Reconstruction and interpolation of manifolds II: Inverse problems with partial data for distances observations and for the heat kernel

Charles Fefferman, Sergei Ivanov, Matti Lassas +2

We consider how a closed Riemannian manifold $M$ and its metric tensor $g$ can be approximately reconstructed from local distance measurements. Moreover, we consider an inverse pro…

math.DG2019

Reconstruction and interpolation of manifolds I: The geometric Whitney problem

Charles Fefferman, Sergei Ivanov, Yaroslav Kurylev +2

We study the geometric Whitney problem on how a Riemannian manifold $(M, g)$ can be constructed to approximate a metric space $(X, d_{X})$ . This problem is closely related to manifold…

math.FA2023

Linear extension operators for Sobolev spaces on radially-symmetric binary trees

Charles Fefferman, Bo'az Klartag

Let $1 < p < \infty$ and suppose that we are given a function $f$ defined on the leaves of a weighted tree. We would like to extend $f$ to a function $F$ defined on the entire tree…

math.AP2019

Splash singularities for the free boundary Navier-Stokes equations

Angel Castro, Diego CÃ³rdoba, Charles Fefferman +2

In this paper, we prove the existence of smooth initial data for the 2D free boundary incompressible Navier-Stokes equations, for which the smoothness of the interface breaks down…

math.AP2011

Splash singularity for water waves

Angel Castro, Diego CÃ³rdoba, Charles Fefferman +2

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 t…

math.AP2012

Finite time singularities for water waves with surface tension

Angel Castro, Diego CÃ³rdoba, Charles Fefferman +2

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 singulari…

math.AP2012

Breakdown of smoothness for the Muskat problem

Angel Castro, Diego Cordoba, Charles Fefferman +1

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…

math.DG2008

The ambient metric

Charles Fefferman, C. Robin Graham

This paper provides details of the construction, properties and some applications of the ambient metric associated to a conformal class of metrics on a smooth manifold. Existence a…

math.OC2021

Controlling Unknown Linear Dynamics with Bounded Multiplicative Regret

Jacob Carruth, Maximilian F. Eggl, Charles Fefferman +2

We consider a simple control problem in which the underlying dynamics depend on a parameter that is unknown and must be learned. We exhibit a control strategy which is optimal to w…

math.AP2001

On the Collapse of Tubes Carried by 3D Incompressible Flows

Diego Cordoba, Charles Fefferman

We povide a test for numerical simulations for the collapse of regular tubes carried by a 3D incompressible flow. In particular, we obtain necessary conditions for 3D Euler to have…

math.ST2022

Fitting a manifold of large reach to noisy data

Charles Fefferman, Sergei Ivanov, Matti Lassas +1

Let $M \subset R^{n}$ be a $C^{2}$ -smooth compact submanifold of dimension $d$ . Assume that the volume of $M$ is at most $V$ and the reach (i.e. the norm…

math.DG2012

Juhl's Formulae for GJMS Operators and Q-Curvatures

Charles Fefferman, C. Robin Graham

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

math.AP2015

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

Angel Castro, Diego Cordoba, Charles Fefferman +1

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 s…

math.DG2003

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

Charles Fefferman, Kengo Hirachi

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 constructi…

math.OC2023

Controlling Unknown Linear Dynamics with Almost Optimal Regret

Jacob Carruth, Maximilian F. Eggl, Charles Fefferman +1

Here and in a companion paper, we consider a simple control problem in which the underlying dynamics depend on a parameter $a$ that is unknown and must be learned. In this paper, w…

math.AP2012

Finite time singularities for the free boundary incompressible Euler equations

Angel Castro, Diego CÃ³rdoba, Charles Fefferman +2

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 pro…

math.FA2022

A property of ideals of jets of functions vanishing on a set

Charles Fefferman, Ary Shaviv

For a set $E \subset R^{n}$ that contains the origin we consider $I^{m} (E)$ -- the set of all $m^{th}$ degree Taylor approximations (at the origin) of $C^{m}$ functions on…

math.AP2001

Growth of solutions for QG and 2D Euler equations

Diego Cordoba, Charles Fefferman

We study the rate of growth of sharp fronts of the Quasi-geostrophic equation and 2D incompressible Euler equations.. The development of sharp fronts are due to a mechanism that pi…

math.FA2016

Finitness Principles for Smooth Selection

Charles Fefferman, Arie Israel, Garving K. Luli

In this paper we prove finiteness principles for $C^{m} (R^{n}, R^{D})$ -selection, and for $C^{m - 1, 1} (R^{n}, R^{D})$ -se…

math.CA2024

Sobolev extension in a simple case

Marjorie Drake, Charles Fefferman, Kevin Ren +1

In this paper, we establish the existence of a bounded, linear extension operator $T : L^{2, p} (E) \to L^{2, p} (R^{2})$ when $1 < p < 2$ and $E$ is a finite subset of $\mathbb{R}^2…

math.AP2011

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

Angel Castro, Diego Cordoba, Charles Fefferman +2

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 th…

math.CA2021

$C^{2}$ Interpolation with Range Restriction

Charles Fefferman, Fushuai Jiang, Garving K. Luli

Given $- \infty < I^» < I^ < \infty$ , $E \subset R^{n}$ finite, and $f : E \to [I^», I^]$ , how can we extend $f$ to a $C^{m} (R^{n})$ function $F$ such that $ Î»\le…

math.CA2019

Generators for the $C^{m}$ -closures of Ideals

Charles Fefferman, Garving K. Luli

Let $R$ denote the ring of real polynomials on $R^{n}$ . Fix $m \geq 0$ , and let $A_{1}, \dots, A_{M} \in R$ . The $C^{m}$ -closure of $\left( A_{1},\cdots…

math.FA2018

Sharp finiteness principles for Lipschitz selections

Charles Fefferman, Pavel Shvartsman

Let $(M, I ¨ )$ be a metric space and let $Y$ be a Banach space. Given a positive integer $m$ , let $F$ be a set-valued mapping from $M$ into the family of all compact convex subsets o…

math.ST2013

Testing the Manifold Hypothesis

Charles Fefferman, Sanjoy Mitter, Hariharan Narayanan

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 algor…

math.CA2019

Solutions to a System of Equations for $C^{m}$ Functions

Charles Fefferman, Garving K. Luli

Fix $m \geq 0$ , and let $A = (A_{ij} (x))_{1 \leq i \leq N, 1 \leq j \leq M}$ be a matrix of semialgebraic functions on $R^{n}$ or on a compact subset $E…

math.OC2023

Optimal Agnostic Control of Unknown Linear Dynamics in a Bounded Parameter Range

Jacob Carruth, Maximilian F. Eggl, Charles Fefferman +1

Here and in a follow-on paper, we consider a simple control problem in which the underlying dynamics depend on a parameter $a$ that is unknown and must be learned. In this paper, w…

math.PR2019

Reconstruction of a Riemannian manifold from noisy intrinsic distances

Charles Fefferman, Sergei Ivanov, Matti Lassas +1

We consider reconstruction of a manifold, or, invariant manifold learning, where a smooth Riemannian manifold $M$ is determined from intrinsic distances (that is, geodesic distance…

math.DS2021

Invariant Curves for Degenerate Hyperbolic Maps of the Plane

Charles Fefferman

We prove existence and uniqueness of an unstable manifold for a degenerate hyperbolic map of the plane arising in statistics.

math.FA2019

Efficient Algorithms for Approximate Smooth Selection

Charles Fefferman, Bernat Guillen Pegueroles

In this paper we provide efficient algorithms for approximate $C^{m} (R^{n}, R^{D}) -$ selection. In particular, given a set $E$ , constants $M_{0} > 0$ and $0 <Ï\…

math.FA2022

Classification of implication-closed ideals in certain rings of jets

Charles Fefferman, Ary Shaviv

For a set $E \subset R^{n}$ that contains the origin we consider $I^{m} (E)$ -- the set of all $m^{th}$ degree Taylor approximations (at the origin) of $C^{m}$ functions on…

math.AP2014

Structural stability for the splash singularities of the water waves problem

Angel Castro, Diego CÃ³rdoba, Charles Fefferman +2

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 g…

math.AP2006

Regularity of coupled two-dimensional nonlinear Fokker-Planck and Navier-Stokes systems

Peter Constantin, Charles Fefferman, Edriss Titi +1

We consider systems of particles coupled with fluids. The particles are described by the evolution of their density, and the fluid is described by the Navier-Stokes equations. The…

math.CA2011

Continuous linear combinations of polynomials

Charles Fefferman, JÃ¡nos KollÃ¡r

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

math.AP2001

Scalars convected by a 2D incompressible flow

Diego Cordoba, Charles Fefferman

We provide a test for numerical simulations, for several two dimensional incompressible flows, that appear to develop sharp fronts. We show that in order to have a front the veloci…

math.FA2017

Sharp finiteness principles for Lipschitz selections: long version

Charles Fefferman, Pavel Shvartsman

Let $(M, I ¨ )$ be a metric space and let $Y$ be a Banach space. Given a positive integer $m$ , let $F$ be a set-valued mapping from $M$ into the family of all co…

math.AP2013

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

Charles Fefferman, Alexandru D. Ionescu, Victor Lie

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 o…

Publications (45)

Interpolation of data by smooth non-negative functions

Reconstruction of Manifold Distances from Noisy Observations

Almost Optimal Agnostic Control of Unknown Linear Dynamics

Turning waves and breakdown for incompressible flows

Finiteness Principles for Smooth Selection

Q-Curvature and Poincare Metrics

Denoising data using convex relaxations

Fitting a manifold to data in the presence of large noise

Reconstruction and interpolation of manifolds II: Inverse problems with partial data for distances observations and for the heat kernel

Reconstruction and interpolation of manifolds I: The geometric Whitney problem

Linear extension operators for Sobolev spaces on radially-symmetric binary trees

Splash singularities for the free boundary Navier-Stokes equations

Splash singularity for water waves

Finite time singularities for water waves with surface tension

Breakdown of smoothness for the Muskat problem

The ambient metric

Controlling Unknown Linear Dynamics with Bounded Multiplicative Regret

On the Collapse of Tubes Carried by 3D Incompressible Flows

Fitting a manifold of large reach to noisy data

Juhl's Formulae for GJMS Operators and Q-Curvatures

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

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

Controlling Unknown Linear Dynamics with Almost Optimal Regret

Finite time singularities for the free boundary incompressible Euler equations

A property of ideals of jets of functions vanishing on a set

Growth of solutions for QG and 2D Euler equations

Finitness Principles for Smooth Selection

Sobolev extension in a simple case

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

C2 Interpolation with Range Restriction

Generators for the Cm-closures of Ideals

Sharp finiteness principles for Lipschitz selections

Testing the Manifold Hypothesis

Solutions to a System of Equations for Cm Functions

Optimal Agnostic Control of Unknown Linear Dynamics in a Bounded Parameter Range

Reconstruction of a Riemannian manifold from noisy intrinsic distances

Invariant Curves for Degenerate Hyperbolic Maps of the Plane

Efficient Algorithms for Approximate Smooth Selection

Classification of implication-closed ideals in certain rings of jets

Structural stability for the splash singularities of the water waves problem

Regularity of coupled two-dimensional nonlinear Fokker-Planck and Navier-Stokes systems

Continuous linear combinations of polynomials

Scalars convected by a 2D incompressible flow

Sharp finiteness principles for Lipschitz selections: long version

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

$C^{2}$ Interpolation with Range Restriction

Generators for the $C^{m}$ -closures of Ideals

Solutions to a System of Equations for $C^{m}$ Functions