NSF DMS 1650546 March 2017 – February 2022

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.

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.

UConn College of Liberal Arts and Science News

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)

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

(arXiv:1807.08002) - (
*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. **Status:**preprint, submitted.- Hölder curves and parameterizations in the Analyst's Traveling Salesman theorem

(arXiv:1806.01197) - (
*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] → l
^{2}, 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 l^{2}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}.

**Status:**preprint, submitted.- Generalized rectifiability of measures and the identification problem

(arXiv:1803.10022) - 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.**Status:**Accepted. To appear in Complex Anal. Synerg.- 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
**R**^{n}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., 2018, 40 pp. doi:10.1007/s12220-018-0034-2

Last updated: August 6, 2018