Matthew Badger | Department of Mathematics | University of Connecticut

NSF DMS 1650546      March 2017 – February 2022

Project Description

Principal Investigator: Matthew Badger

Background

Geometric measure theory is a field of mathematics that evolved from investigations in the 1920s and 1930s into the structure of sets in the plane with finite length. The term "measure" refers to an abstract generalization of length, area, or volume, which assigns a size value to every mathematical set. Traditional outlets for geometric measure theory have expanded in recent decades. The widespread utility and current use of geometric measure theory in different areas of analysis justifies its continued development. The research component of this project seeks to advance our understanding about underlying structures of general measures and to develop new techniques that will expand the toolbox that geometric measure theory provides for researchers in adjacent areas of analysis and geometry. On the educational front, this project will support a network of early career researchers whose research involves nonsmooth analysis, including graduate students and postdoctoral researchers who work in a number areas. Principal activities by the PI include organizing a Workshop for Postdocs in Fall 2017 and a Conference for Graduate Students with Mini-Courses in Spring 2019. The two conferences will be linked: postdoctoral participants from the workshop will be invited to give mini-courses for graduate students in the follow-up conference. The PI will further integrate research and education by organizing an analysis learning seminar and mentoring two postdoctoral researchers at the PI's home institution.

Objectives

This project focuses on a constellation of questions about the structure of Radon measures in Euclidean space. The underlying theme is that general measures may be understood in terms of their behavior with respect to lower dimensional sets such as finite length curves in the plane and finite area surfaces in space. This point-of-view originated in the 1920s and 1930s through investigations by A.S. Besicovitch, which compared and contrasted properties of finite length sets with properties of rectifiable curves. Later contributions by A.P. Morse and J.F. Randolph, H. Federer, P. Mattila, and D. Preiss from the 1940s through the 1980s produced a rich theory of qualitative rectifiability of measures in Euclidean space that are absolutely continuous with respect to Hausdorff measures; a quantitative theory of rectifiability for Ahlfors regular measures emerged in the 1990s through the work of G. David and S. Semmes. The proposed research seeks to broaden our understanding of different notions of rectifiability of measures in the absence of background regularity hypotheses from past investigations. Specifically, the PI will look for characterizations of Radon measures which are carried by countable families of Hölder continuous curves, Lipschitz graphs, or Lipschitz continuous images of linear subspaces. This goal requires integration of techniques from modern harmonic analysis and quantitative geometric measure theory. The PI will explore approaches based on the PI's work with R. Schul, which characterized Radon measures that are carried by countable families of rectifiable curves, as well as approaches based on G. David and T. Toro's extension of the Reifenberg algorithm and approaches based on K. Rajala's quasiconformal uniformization theorem.

News Articles

UConn College of Liberal Arts and Science News

UConn Mathematics Blog

Events

Analysis Learning Seminar (Spring 2017 forward)

Nonsmooth Analysis: A Workshop for Postdocs (November 2017)

Geometric and Harmonic Analysis 2019: A Conference for Graduate Students (March 2019)

Research Products

Identifying 1-rectifiable measures in Carnot groups
(arXiv:2109.06753)

(with Sean Li and 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, extending a theorem of the first author and R. Schul in Euclidean space, for an arbitrary locally finite Borel measure in an arbitrary Carnot group, we develop tests that identify the part of the measure that is carried by rectifiable curves and the part of the measure that is singular to rectifiable curves. Our main result is entwined with an extension of the Analyst's Traveling Salesman Theorem, which characterizes subsets of rectifiable curves in R² (P. Jones, 1990), in Rⁿ (K. Okikolu, 1992), or in an arbitrary Carnot group (the second author) in terms of local geometric least squares data called Jones' β-numbers. In a secondary result, we implement the Garnett-Killip-Schul construction of a doubling measure in Rⁿ that charges a rectifiable curve in an arbitrary complete, quasiconvex, doubling metric space.

Status: preprint, submitted.

Hausdorff dimension of caloric measure
(arXiv:2108.12340)

(with Alyssa Genschaw)

We examine caloric measures on general domains in Rⁿ⁺¹ = RⁿxR (space x times) from the perspective of geometric measure theory. On one hand, we give a direct proof of a consequence of a theorem of Taylor and Watson (1985) that the lower parabolic Hausdorff dimension of caloric measure is at least n and is absolutely continuous with respect to the n-dimensional parabolic Hausdorff measure. On the other hand, we prove that the upper parabolic Hausdorff dimension of caloric measure is at most n+2-β_n, where β_n>0 depends only on n. Analogous bounds for harmonic measures were first shown by Nevanlinna (1934) and Bourgain (1987). Heuristically, we show that the density of obstacles in a cube needed to make it unlikely that a Brownian motion started outside of the cube exits a domain near the center of the cube must be chosen according to the ambient dimension.
In the course of the proof, we give a caloric measure analogue of Bourgain's alternative: with certain dimensional constraints on the size and position of cubes, for any closed set E in Rⁿ, either (i) the part of E in Q has relatively large caloric measure in Q minus E for every pole in F or (ii) the part of E in Q_* has relatively small ρ-dimensional parabolic Hausdorff content for every n < ρ ≤ n+2, where Q is a cube, F is a subcube of Q aligned at the center of the top time-face, and Q_* is a subcube of Q that is close to, but separated backwards-in-time from F. Further, we supply a version of the strong Markov property for caloric measures.

Status: accepted, to appear in Amer. J. Math.

Radon measures and Lipschitz graphs
(arXiv:2007.08503 | Published Version)

(with Lisa Naples)

For all 1≤m≤n-1, we investigate the interaction of locally finite measures in Rⁿ with the family of m-dimensional Lipschitz graphs. For instance, we characterize Radon measures μ, which are carried by Lipschitz graphs in the sense that there exist graphs Γ₁, Γ₂, ... such that μ(Rⁿ\Γ_i)=0, using only countably many evaluations of the measure. This problem in geometric measure theory was classically studied within smaller classes of measures, e.g. for the restrictions of m-dimensional Hausdorff measure H^m to E in Rⁿ with 0<H^m(E)<∞. However, an example of Csörnyei, Käenmäki, Rajala, and Suomala shows that classical methods are insufficient to detect when a general measure charges a Lipschitz graph. To develop a characterization of Lipschitz graph rectifiability for arbitrary Radon measures, we look at the behavior of coarse doubling ratios of the measure on dyadic cubes that intersect conical annuli. This extends a characterization of graph rectifiability for pointwise doubling measures by Naples by mimicking the approach used in the characterization of Radon measures carried by rectifiable curves by Badger and Schul.

Citation: M. Badger, L. Naples, Radon measures and Lipschitz graphs, Bull. London Math. Soc. (2021), 16 pages. https://doi.org/10.1112/blms.12473

Subsets of rectifiable curves in Banach spaces I: sharp exponents in traveling salesman theorems
(arXiv:2002.11878)

(with 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 Jones in 1990 and subsequently solved in higher-dimensional Euclidean spaces by Okikiolu in 1992 and in the infinite-dimensional Hilbert space ℓ₂ by Schul in 2007. In this paper, we establish sharp extensions of Schul's necessary and sufficient conditions for a bounded set E ⊂ ℓ_p to be contained in a rectifiable curve from p=2 to 1<p<∞. While the necessary and sufficient conditions coincide when p=2, we demonstrate that there is a strict gap between the necessary condition and sufficient condition when p≠2. We also identify and correct technical errors in the proof by Schul. This investigation is partly motivated by recent work of Edelen, Naber, and Valtorta on Reifenberg-type theorems in Banach spaces and complements work of Hahlomaa and recent work of David and Schul on the Analyst's TSP in general metric spaces.

Note: While revising an earlier draft of the manuscript, we identified a mistake in Schul's 2007 proof of the necessary conditions in the traveling salesman theorem in infinite-dimensional Hilbert space. We show how to correct the error in a minimal way, leaving the outline of virtually all of the original proofs intact.

Status: accepted, to appear in Illinois Journal of Mathematics

Rectifiability of pointwise doubling measures in Hilbert space
(arXiv:2002.07570)

(by Lisa Naples)

In geometric measure theory, there is interest in studying the interaction of measures with rectifiable sets. Here, we extend a theorem of Badger and Schul in Euclidean space to characterize rectifiable pointwise doubling measures in Hilbert space. Given a measure, we construct a multiresolution family of windows, and then we use a weighted Jones' function to record how well lines approximate the distribution of mass in each window. We show that when is rectifiable, the mass is sufficiently concentrated around a lines at each scale and that the converse also holds. Additionally, we present an algorithm for the construction of a rectifiable curve using appropriately chosen d-nets. Throughout, we discuss how to overcome the fact that in infinite dimensional Hilbert space there may be infinitely many d-separated points, even in a bounded set. Finally, we prove a characterization for pointwise doubling measures carried by Lipschitz graphs.

Status: preprint, submitted.

Hölder parameterization of iterated function systems and a self-affine phenomenon
(arXiv:1910.08850)

(with Vyron Vellis)

We investigate the Hölder geometry of curves generated by iterated function systems (IFS) in a complete metric space. A theorem of Hata from 1985 asserts that every connected attractor of an IFS is locally connected and path-connected. First we give a quantitative strengthening of Hata's theorem. We first prove that every connected attractor of an IFS is 1/s-Hölder path-connected, where s is the similarity dimension of the IFS. Then we show that every connected attractor of an IFS is parameterized by a 1/a-Hölder curve for all a>s. At the endpoint, a=s, a theorem of Remes from 1998 already established that connected self-similar sets in Euclidean space that satisfy the open set condition are parameterized by 1/s-Hölder curves. In a secondary result, we show how to promote Remes' theorem to self-similar sets in complete metric spaces, but in this setting require the attractor to have positive s-dimensional Hausdorff measure in lieu of the open set condition. To close the paper, we determine sharp Hölder exponents of parameterizations in the class of connected self-affine Bedford-McMullen carpets and build parameterizations of self-affine sponges. An interesting phenomenon emerges in the self-affine setting. While the optimal parameter s for a self-similar curve in Rⁿ is at most n, the optimal parameter s for a self-affine curve in Rⁿ may be strictly greater than n.

Status: preprint, submitted.

Regularity of the singular set in a two-phase problem for harmonic measure with Hölder data
(arXiv:1807.08002 | Published Version)

(with Max Engelstein and 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 in n-dimensional Euclidean space influences the geometry of its boundary. This type of free boundary problem was initially studied by Kenig and Toro in 2006 and was further examined in a series of separate and joint investigations by several authors. The focus of the present paper is on the singular set in the free boundary, where the boundary looks infinitesimally like zero sets of homogeneous harmonic polynomials of degree at least 2. We prove that if the Radon-Nikodym derivative of the exterior harmonic measure with respect to the interior harmonic measure has a H\"older continuous logarithm, then the free boundary admits unique geometric blowups at every singular point and the singular set can be covered by countably many C^1,β submanifolds of dimension at most n-3. This result is partly obtained by adapting tools such as Garofalo and Petrosyan's Weiss type monotonicity formula and an epiperimetric inequality for harmonic functions from the variational to the non-variational setting.

Citation: M. Badger, M. Engelstein, T. Toro, Regularity of the singular set in a two-phase problem for harmonic measure with Hölder data, Rev. Mat. Iberoam. 36 (2020), no. 5, 1375–1408. https://doi.org/10.4171/rmi/1170

Hölder curves and parameterizations in the Analyst's Traveling Salesman theorem
(arXiv:1806.01197 | Published Version)

(with Lisa Naples and 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 of a (1/s)-Hölder continuous map f:[0,1] → l², with s>1. Our results are motivated by and generalize the "sufficient half" of the Analyst's Traveling Salesman Theorem, which characterizes subsets of rectifiable curves in R^N or l² in terms of a quadratic sum of linear approximation numbers called Jones' beta numbers. The original proof of the Analyst's Traveling Salesman Theorem depends on a well-known metric characterization of rectifiable curves from the 1920s, which is not available for higher-dimensional curves such as Hölder curves. To overcome this obstacle, we reimagine Jones' non-parametric proof and show how to construct parameterizations of the intermediate approximating curves f_k([0,1]). We then find conditions in terms of tube approximations that ensure the approximating curves converge to a Hölder curve. As an application, we provide sufficient conditions to guarantee fractional rectifiability of pointwise doubling measures in R^N.

Citation: M. Badger, L. Naples, V. Vellis, Hölder curves and parameterizations in the Analyst's Traveling Salesman theorem, Adv. Math. 349 (2019), 564-647. doi:10.1016/j.aim.2019.04.011

Generalized rectifiability of measures and the identification problem
(arXiv:1803.10022 | Published Version)

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 dichotomy arises between the class of rectifiable measures, which give full measure to a countable union of the lower dimensional sets, and the class of purely unrectifiable measures, which assign measure zero to each distinguished set. There are several commonly used definitions of rectifiable and purely unrectifiable measures in the literature (using different families of lower dimensional sets such as Lipschitz images of subspaces or Lipschitz graphs), but all of them can be encoded using the same framework. In this paper, we describe a framework for generalized rectifiability, review a selection of classical results on rectifiable measures in this context, and survey recent advances on the identification problem for Radon measures that are carried by Lipschitz or Hölder or C^1,α images of Euclidean subspaces, including theorems of Azzam-Tolsa, Badger-Schul, Badger-Vellis, Edelen-Naber-Valtorta, Ghinassi, and Tolsa-Toro.

Note: This survey paper is based on a talk at the Northeast Analysis Network Conference held in Syracuse, New York in September 2017.

Citation: M. Badger, Generalized rectifiability of measures and the identification problem, Complex Anal. Synerg. 5 (2019), 2.

Geometry of measures in real dimensions via Hölder parameterizations
(arXiv:1706.07846 | Published Version)

(with Vyron Vellis)

We investigate the influence that s-dimensional lower and upper Hausdorff densities have on the geometry of a Radon measure in Rⁿ when s is a real number between 0 and n. This topic in geometric measure theory has been extensively studied when s is an integer. In this paper, we focus on the non-integer case, building upon a series of papers on s-sets by Martín and Mattila from 1988 to 2000. When 0<s<1, we prove that measures with almost everywhere positive and lower density and finite upper density are carried by countably many bi-Lipschitz curves. When 1≤s<n, we identify conditions on the lower density that ensure the measure is either carried by or singular to (1/s)-Hölder curves. The latter results extend part of the recent work of Badger and Schul, which examined the case s=1 (Lipschitz curves) in depth. Of further interest, we introduce Hölder and bi-Lipschitz parameterization theorems for Euclidean sets with "small" Assouad dimension.

Citation: M. Badger, V. Vellis, Geometry of measures in real dimensions via Hölder parameterizations, J. Geom. Anal. 29 (2019), no. 2, 1153-1192. doi:10.1007/s12220-018-0034-2