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papers

Publications (28)

math.CA2020

Radon measures and Lipschitz graphs

Matthew Badger, Lisa Naples

For all $1 \leq m \leq n - 1$ , we investigate the interaction of locally finite measures in $R^{n}$ with the family of $m$ -dimensional Lipschitz graphs. For instance, we charact…

math.CA2014

Local set approximation: Mattila-Vuorinen type sets, Reifenberg type sets, and tangent sets

Matthew Badger, Stephen Lewis

We investigate the interplay between the local and asymptotic geometry of a set $A \subseteq R^{n}$ and the geometry of model sets $\mathcal{S} \subset \mathcal{P}(\mathbb{R…

math.CA2017

Rectifiability and elliptic measures on 1-sided NTA domains with Ahlfors-David regular boundaries

Murat Akman, Matthew Badger, Steve Hofmann +1

Let $I^c ◯ \subset R^{n + 1}$ , $n \geq 2$ , be 1-sided NTA domain (aka uniform domain), i.e. a domain which satisfies interior Corkscrew and Harnack Chain conditions, and assume t…

math.MG2026

Tangents to Lipschitz and Sobolev images

Matthew Badger, Jared Krandel, Vyron Vellis

We develop geometric versions of Rademacher and Calderon type differentiability theorems in two categories. A special case of our results is that for any Lipschitz or continuous $W…

math.CA2017

Structure of sets which are well approximated by zero sets of harmonic polynomials

Matthew Badger, Max Engelstein, Tatiana Toro

The zero sets of harmonic polynomials play a crucial role in the study of the free boundary regularity problem for harmonic measure. In order to understand the fine structure of th…

math.CA2012

Flat points in zero sets of harmonic polynomials and harmonic measure from two sides

Matthew Badger

We obtain quantitative estimates of local flatness of zero sets of harmonic polynomials. There are two alternatives: at every point either the zero set stays uniformly far away fro…

math.CA2012

Quasisymmetry and rectifiability of quasispheres

Matthew Badger, James T. Gill, Steffen Rohde +1

We obtain Dini conditions with "exponent 2" that guarantee that an asymptotically conformal quasisphere is rectifiable. In particular, we show that for any e>0 integrability of (es…

math.CA2022

Subsets of rectifiable curves in Banach spaces II: universal estimates for almost flat arcs

Matthew Badger, Sean McCurdy

We prove that in any Banach space the set of windows in which a rectifiable curve resembles two or more straight line segments is quantitatively small with constants that are indep…

math.CA2014

Multiscale analysis of 1-rectifiable measures: necessary conditions

Matthew Badger, Raanan Schul

We repurpose tools from the theory of quantitative rectifiability to study the qualitative rectifiability of measures in $R^{n}$ , $n \geq 2$ . To each locally finite Borel measur…

math.AP2021

Åojasiewicz Inequalities and Generic Smoothness of Nodal Sets of Solutions to Elliptic PDE

Matthew Badger, Max Engelstein, Tatiana Toro

In this article, we prove that for a broad class of second order elliptic PDEs, including the Laplacian, the zero sets of solutions to the Dirichlet problem are smooth for "generic…

math.CA2012

Beurling's criterion and extremal metrics for Fuglede modulus

Matthew Badger

We formulate a necessary and sufficient condition for an admissible metric to be extremal for the Fuglede p-modulus of a system of measures. When p=2, this characterization general…

math.CA2014

Quasiconformal planes with bi-Lipschitz pieces and extensions of almost affine maps

Jonas Azzam, Matthew Badger, Tatiana Toro

A quasiplane $f (V)$ is the image of an $n$ -dimensional Euclidean subspace $V$ of $R^{N}$ ( $1 \leq n \leq N - 1$ ) under a quasiconformal map $f : R^{N} \to R^{N}$ . We give s…

math.CA2022

Lower bounds on Bourgain's constant for harmonic measure

Matthew Badger, Alyssa Genschaw

For every $n \geq 2$ , Bourgain's constant $b_{n}$ is the largest number such that the (upper) Hausdorff dimension of harmonic measure is at most $n - b_{n}$ for every domain in $\mathbb{R…

math.CA2010

Null sets of harmonic measure on NTA domains: Lipschitz approximation revisited

Matthew Badger

We show the David-Jerison construction of big pieces of Lipschitz graphs inside a corkscrew domain does not require its surface measure be upper Ahlfors regular. Thus we can study…

math.AP2009

Harmonic polynomials and tangent measures of harmonic measure

Matthew Badger

We show that on an NTA domain if each tangent measure to harmonic measure at a point is a polynomial harmonic measure then the associated polynomials are homogeneous. Geometric inf…

math.AP2022

Slowly vanishing mean oscillations: non-uniqueness of blow-ups in a two-phase free boundary problem

Matthew Badger, Max Engelstein, Tatiana Toro

In Kenig and Toro's two-phase free boundary problem, one studies how the regularity of the Radon-Nikodym derivative $h = d I ¨ ^{-} / d I ¨ ^{+}$ of harmonic measures on complementary NTA domai…

math.CA2022

Subsets of rectifiable curves in Banach spaces I: sharp exponents in traveling salesman theorems

Matthew Badger, Sean McCurdy

The Analyst's Traveling Salesman Problem is to find a characterization of subsets of rectifiable curves in a metric space. This problem was introduced and solved in the plane by Jo…

math.MG2020

HÃ¶lder parameterization of iterated function systems and a self-affine phenomenon

Matthew Badger, Vyron Vellis

We investigate the HÃ¶lder geometry of curves generated by iterated function systems (IFS) in a complete metric space. A theorem of Hata from 1985 asserts that every connected attr…

math.CA2019

HÃ¶lder curves and parameterizations in the Analyst's Traveling Salesman theorem

Matthew Badger, Lisa Naples, Vyron Vellis

We investigate the geometry of sets in Euclidean and infinite-dimensional Hilbert spaces. We establish sufficient conditions that ensure a set of points is contained in the image o…

math.MG2017

Multiscale analysis of 1-rectifiable measures II: characterizations

Matthew Badger, Raanan Schul

A measure is 1-rectifiable if there is a countable union of finite length curves whose complement has zero measure. We characterize 1-rectifiable Radon measures $I^¼$ in $n$ -dimensi…

math.CA2019

Generalized rectifiability of measures and the identification problem

Matthew Badger

One goal of geometric measure theory is to understand how measures in the plane or higher dimensional Euclidean space interact with families of lower dimensional sets. An important…

math.CA2018

Geometry of measures in real dimensions via HÃ¶lder parameterizations

Matthew Badger, Vyron Vellis

We investigate the influence that $s$ -dimensional lower and upper Hausdorff densities have on the geometry of a Radon measure in $R^{n}$ when $s$ is a real number between $0…

math.CA2015

Two sufficient conditions for rectifiable measures

Matthew Badger, Raanan Schul

We identify two sufficient conditions for locally finite Borel measures on $R^{n}$ to give full mass to a countable family of Lipschitz images of $R^{m}$ . The first c…

math.MG2023

Identifying 1-rectifiable measures in Carnot groups

Matthew Badger, Sean Li, Scott Zimmerman

We continue to develop a program in geometric measure theory that seeks to identify how measures in a space interact with canonical families of sets in the space. In particular, ex…

math.AP2019

Regularity of the singular set in a two-phase problem for harmonic measure with HÃ¶lder data

Matthew Badger, Max Engelstein, Tatiana Toro

In non-variational two-phase free boundary problems for harmonic measure, we examine how the relationship between the interior and exterior harmonic measures of a domain $Î©\subset…

math.AP2025

On the number of nodal domains of homogeneous caloric polynomials

Matthew Badger, Cole Jeznach

We investigate the minimum and maximum number of nodal domains across all time-dependent homogeneous caloric polynomials of degree $d$ in $R^{n} \times R$ (space $\…

math.MG2025

Square packings and rectifiable doubling measures

Matthew Badger, Raanan Schul

We prove that for all integers $2 \leq m \leq d - 1$ , there exists doubling measures on $R^{d}$ with full support that are $m$ -rectifiable and purely $(m - 1)$ -unrectifiable in th…

math.CA2023

Hausdorff dimension of caloric measure

Matthew Badger, Alyssa Genschaw

We examine caloric measures $I ¨ $ on general domains in $R^{n + 1} = R^{n} \times R$ (space $\times$ time) from the perspective of geometric measure theory. On…

Publications (28)

Radon measures and Lipschitz graphs

Local set approximation: Mattila-Vuorinen type sets, Reifenberg type sets, and tangent sets

Rectifiability and elliptic measures on 1-sided NTA domains with Ahlfors-David regular boundaries

Tangents to Lipschitz and Sobolev images

Structure of sets which are well approximated by zero sets of harmonic polynomials

Flat points in zero sets of harmonic polynomials and harmonic measure from two sides

Quasisymmetry and rectifiability of quasispheres

Subsets of rectifiable curves in Banach spaces II: universal estimates for almost flat arcs

Multiscale analysis of 1-rectifiable measures: necessary conditions

Åojasiewicz Inequalities and Generic Smoothness of Nodal Sets of Solutions to Elliptic PDE

Beurling's criterion and extremal metrics for Fuglede modulus

Quasiconformal planes with bi-Lipschitz pieces and extensions of almost affine maps

Lower bounds on Bourgain's constant for harmonic measure

Null sets of harmonic measure on NTA domains: Lipschitz approximation revisited

Harmonic polynomials and tangent measures of harmonic measure

Slowly vanishing mean oscillations: non-uniqueness of blow-ups in a two-phase free boundary problem

Subsets of rectifiable curves in Banach spaces I: sharp exponents in traveling salesman theorems

HÃ¶lder parameterization of iterated function systems and a self-affine phenomenon

HÃ¶lder curves and parameterizations in the Analyst's Traveling Salesman theorem

Multiscale analysis of 1-rectifiable measures II: characterizations

Generalized rectifiability of measures and the identification problem

Geometry of measures in real dimensions via HÃ¶lder parameterizations

Two sufficient conditions for rectifiable measures

Identifying 1-rectifiable measures in Carnot groups

Regularity of the singular set in a two-phase problem for harmonic measure with HÃ¶lder data

On the number of nodal domains of homogeneous caloric polynomials

Square packings and rectifiable doubling measures

Hausdorff dimension of caloric measure

Åojasiewicz Inequalities and Generic Smoothness of Nodal Sets of Solutions to Elliptic PDE