**Matthew Badger**

University of Connecticut

Department of Mathematics

341 Mansfield Road, U-1009

Storrs, CT 06269-1009

**Office**: Monteith 326

**Email**: firstname.lastname _at_ uconn.edu

Spring 2020 Office Hours | |

M | Drop-in |

Tu | Drop-in |

W | Not Available |

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Office hours are held in

Monteith 326

I am an Associate Professor of Mathematics. I study the **geometry of sets and measures** in Euclidean space and general metric spaces using a mixture of geometric measure theory, harmonic analysis and quasiconformal analysis.

My recent work has focused on the question of how to construct good parameterizations of connected sets by Lipschitz and Hölder continuous maps.

Quick Links: [Curriculum Vitae | Teaching | Research]

In Fall 2019, the Analysis and Probability Seminar meets Fridays at 1:30pm. This is a research seminar featuring speakers from around the world. Organized by Scott Zimmerman.

In Fall 2019, we are continuing the Analysis Learning Seminar. Meeting time is Fridays 3:30-4:30. This seminar will primarily feature introductory talks by UConn graduate students and postdocs. All graduate students and advanced math majors who are interested in analysis, geometry, and probability are welcome to attend! Organized by me and Marco Carfagnini

2019 Northeast Analysis Network: September 21-22, 2019.

Geometric and Harmonic Analysis: a Conference for Graduate Students: March 29-31, 2019.

Nonsmooth Analysis: a Workshop for Postdocs: November 9-11, 2017.

[Math Course Schedules: Current Semester]

Math 5120: Complex Analysis

Math 2210Q (Sections 2 and 3): Applied Linear Algebra

I am happy to advise Ph.D. students who would like to carry out research at the interface of analysis and geometry.

A non-technical description of my research with Raanan Schul on rectifiable measures can be found here.

Here is a picture related to my "Harmonic polynomials..." and "Flat points..." papers. The zero sets of homogeneous harmonic polynomials in x,y,z of odd degree may separate space into two components (cross your eyes to see a stereographic picture):

500x^{4}y-1000x^{2}y^{3}+100y^{5}
-5(x^{4}+y^{4})z+10(x^{2}+y^{2})z^{3}+2z^{5}=0 intersecting the unit sphere

- NSF DMS 1901256
- Analysis Program. Conference Grant. Northeast Analysis Network. 2019 – 2023
- NSF DMS 1650546
- Analysis Program. CAREER Award. 2017 – 2022
- NSF DMS 1500382
- Analysis Program. Standard Grant. 2015 – 2018
- NSF DMS 1203497
- 2012 NSF Mathematical Sciences Postdoctoral Research Fellowship

[Statistics] Newest preprints/papers are listed first.

- Hölder connectedness and parameterization of iterated function systems

(arXiv:1910.08850) - (
*with*Vyron Vellis) - [Click to Show/Hide Abstract]
- 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. In our primary result, 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. We then 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.
**Status:**preprint, submitted.- 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) - [Click to Show/Hide Abstract]
- 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:**accepted. To appear in Rev. Mat. Iberoam.- Hölder curves and parameterizations in the Analyst's Traveling Salesman theorem

(arXiv:1806.01197 | Published Version) - (
*with*Lisa Naples and Vyron Vellis) - [Click to Show/Hide Abstract]
- 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}.

**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) - [Click to Show/Hide Abstract]
- 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) - [Click to Show/Hide Abstract]
- 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.**29**(2019), no. 2, 1153-1192. doi:10.1007/s12220-018-0034-2- Multiscale analysis of 1-rectifiable measures II: characterizations

(arXiv:1602.03823 | Published Version) - (
*with*Raanan Schul) - [Click to Show/Hide Abstract]
- A measure is 1-rectifiable if there is a countable union of finite length curves whose complement has zero measure. We characterizae 1-rectifiable Radon measures in
*n*-dimensional Euclidean space for all*n*≥2 in terms of positivity of the lower density and finiteness of a geometric square function, which loosely speaking, records in an*L*^{2}gauge the extent to which the measure admits approximate tangent lines, or has rapidly growing density rations, along its support. In contrast with the classical theorems of Besicovitch, Morse and Randolph, and Moore, we do not assume an*a priori*relationship between the measure and 1-dimensional Hausdorff measure. We also characterize purely 1-unrectifiable Radon measures, i.e. locally finite measures that give measure zero to every finite length curve. Characterizations of this form were originally conjectured to exist by P. Jones. Along the way, we develop an*L*^{2}variant of P. Jones' traveling salesman construction, which is of indepenedent interest. **Citation:**M. Badger, R. Schul,*Multiscale analysis of 1-rectifiable measures II: characterizations*, Anal. Geom. Metr. Spaces**5**(2017), no. 1, 1-39.**Related:**H. Martikainen and T. Orponen (arXiv:1604.04091) have constructed a finite measure in the plane with bounded density-normalized L^{2}Jones function and vanishing lower 1-density. This implies that our use of β^{**}in Theorem D is sharp and answers a question we posed following Theorem E.- Structure of sets which are well approximated by zero sets of harmonic polynomials

(arXiv:1509.03211 | Published Version) - (
*with*Max Engelstein and Tatiana Toro) - [Click to Show/Hide Abstract]
- 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 these free boundaries a detailed study of the singular points of these zero sets is required. In this paper we study how "degree
*k*points" sit inside zero sets of harmonic polynomials in**R**^{n}of degree*d*(for all*n*≥ 2 and 1 ≤*k*≤*d*) and inside sets that admit arbitrarily good local approximations by zero sets of harmonic polynomials. We obtain a general structure theorem for the latter type of sets, including sharp Hausdorff and Minkowski dimension estimates on the singular set of "degree*k*points" (*k*≥ 2) without proving uniqueness of blowups or aid of PDE methods such as monotonicity forumlas. In addition, we show that in the presence of a certain topological separation condition, the sharp dimension estimates improve and depend on the parity of*k*. An application is given to the two-phase free boundary regularity problem for harmonic measure below the continuous threshold introduced by Kenig and Toro. **Citation:**M. Badger, M. Engelstein, T. Toro,*Structure of sets which are well approximated by zero sets of harmonic polynomials*, Anal. PDE**10**(2017), no. 6, 1455-1495.- Rectifiability and elliptic measures on 1-sided NTA domains with Ahflors-David regular boundaries

(arXiv:1507.02039 | Published Version) - (
*with*Murat Akman, Steve Hofmann, and José María Martell) - [Click to Show/Hide Abstract]
- Consider a 1-sided NTA domain (aka uniform domain) in
**R**^{n+1}, n≥2, i.e. a domain which satisfies interior Corkscrew and Harnack Chain conditions, and assume the boundary of the domain is n-dimensional Ahflors-David regular. We characterize the rectifiability of the boundary in terms of absolute continuiuty of surface measure with respect to harmonic measure. We also show that these are equivalent to the fact that the boundary can be covered H^{n}-a.e. by a countable union of portions of boundaries of bounded chord-arc subdomains adn to the fact that the boundary possesses exterior corkscrew points in a qualitiative way H^{n}-a.e. Our methods apply to harmonic measure and also to elliptic measures associated with real symmetric second order divergence form elliptic operators with locally Lipschitz coefficients whose derivatives satisfy a natural qualitative Carleson condition. **Citation:**M. Akman, M. Badger, S. Hofmann, J.M. Martell,*Rectifiability and elliptic measures on 1-sided NTA domains with Ahlfors-David regular boundaries*, Trans. Amer. Math. Soc.**369**(2017), no. 8, 2017, 5711-5745.- Two sufficient conditions for rectifiable measures

(arXiv:1412.8357 | Published Version) - (
*with*Raanan Schul) - [Click to Show/Hide Abstract]
- We identify two sufficient conditions for locally finite Borel measures on
**R**^{n}to give full mass to a countable family of Lipschitz maps of**R**^{m}. The first condition, extending a prior result of Pajot, is a sufficient test in terms of L^{p}affine approximability for a locally finite Borel measure μ on**R**^{n}satisfying the global regularity hypothesis limsup_{r↓0}μ(B(x,r))/r^{m}< ∞ at μ almost every x to be m-rectifiable in the sense above. The second condition is an assumption on the growth rate of the 1-density that ensures a locally finite Borel measure μ on**R**^{n}with lim_{r↓0}μ(B(x,r))/r=∞ at μ almost every x in**R**^{n}is 1-rectifiable. **Citation:**M. Badger, R. Schul,*Two sufficient conditions for rectifiable measures*, Proc. Amer. Math. Soc.**144**(2016), 2445-2454.- Local set approximation: Mattila-Vuorinen type sets, Reifenberg type sets, and tangent sets

(arXiv:1409.7851 | Published Version) - (
*with*Stephen Lewis) - [Click to Show/Hide Abstract]
- We investigate the interplay between the local and asymptotic geometry of a set
*A*in**R**^{n}and the geometry of model sets, which approximate*A*locally uniformly on small scales. The framework for local set approximation developed in this paper unifies and extends ideas of Jones, Mattila and Vuorinen, Reifenberg, and Preiss. We indicate several applications of this framework to variational problems that arise in geometric measure theory and partial differential equations. For instance, we show that the singular part of the support of an*(n-1)*-dimensional asymptotically optimally doubling measure in**R**^{n}(*n*≥4) has upper Minkowski dimension at most*n-4*. **Note:**The arXiv version of the paper has outdated numbering. The published version is open access and is the authoritative version.**Citation:**M. Badger, S. Lewis,*Local set approximation: Mattila-Vuorinen type sets, Reifenberg type sets, and tangent sets*, Forum Math. Sigma**3**(2015), e24, 63 pp.- Quasiconformal planes with bi-Lipschitz pieces and extensions of almost affine maps

(arXiv:1403.2991 | Published Version) - (
*with*Jonas Azzam, and Tatiana Toro) - [Click to Show/Hide Abstract]
- A quasiplane is the image of an n-dimensional Euclidean subspace of
**R**^{N}(1 ≤ n ≤ N-1) under a quasiconformal map of**R**^{N}. We give sufficient conditions in terms of the weak quasisymmetry constant of the underlying map for a quasiplane to be a bi-Lipschitz n-manifold and for a quasiplane to have big pieces of bi-Lipschitz images of**R**^{n}. One main novelty of these results is that we analyze quasiplanes in arbitrary codimension N-n. To establish the big pieces criterion, we prove new extension theorems for "almost affine" maps, which are of independent interest. This work is related to investigations by Tukia and Väisälä on extensions of quasisymmetric maps with small distortion. **Citation:**J. Azzam, M. Badger, T. Toro,*Quasiconformal planes with bi-Lipschitz pieces and extensions of almost affine maps*, Adv. Math.**275**(2015), 195-259.- Multiscale analysis of 1-rectifiable measures: necessary conditions

(arXiv:1307.0804 | Published Version) - (
*with*Raanan Schul) - [Click to Show/Hide Abstract]
- We repurpose tools from the theory of quantitative rectifiability to study the qualitative rectifiability of measures in
**R**^{n}, n > 2. To each locally finite Borel measure μ, we associate a function tJ_{2}(μ,x) which uses a weighted sum to record how closely the mass of μ is concentrated on a line in the triples of dyadic cubes containing x. We show that tJ_{2}(μ,x) < ∞ μ-a.e. is a necessary condition for μ to give full mass to a countable family of rectifiable curves. This confirms a conjecture of Peter Jones from 2000. A novelty of this result is that no assumption is made on the upper Hausdorff density of the measure. Thus we are able to analyze generic 1-rectifiable measures that are mutually singular with the 1-dimensional Hausdorff measure. **Citation:**M. Badger, R. Schul,*Multiscale analysis of 1-rectifiable measures: necessary conditions*, Math. Ann.**361**(2015), no. 3-4, 1055-1072.- Beurling's criterion and extremal metrics for Fuglede modulus

(arXiv:1207.5277 | Published Version) - [Click to Show/Hide Abstract]
- For each 1 ≤ p < ∞, 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 generalizes Beurling's criterion, a sufficient condition for an admissible metric to be extremal for the extremal length of a planar curve family. In addition, we prove that every non-negative Borel function in Euclidean space with positive and finite p-norm is extremal for the p-modulus of some curve family.
**Citation:**M. Badger,*Beurling's criterion and extremal metrics for Fuglede modulus*, Ann. Acad. Sci. Fenn. Math.**38**(2013), 677-689.- Quasisymmetry and rectifiability of quasispheres

(arXiv:1201.1581 | Published Version) - (
*with*James T. Gill, Steffen Rohde, and Tatiana Toro) - [Click to Show/Hide Abstract]
- We obtain Dini conditions with "exponent 2" that guarantee that an asymptotically conformal quasisphere is rectifiable. We also establish estimates for the weak quasisymmetry constant of a global K-quasiconformal map in neighborhoods with maximal dilitation close to 1.
**Citation:**M. Badger, J.T. Gill, S. Rohde, T. Toro,*Quasisymmetry and rectifiability of quasispheres*, Trans. Amer. Math. Soc.**366**(2014), no. 3, 1413-1431.- Flat points in zero sets of harmonic polynomials and harmonic measure from two sides

(arXiv:1109.1427 | Published Version) - [Click to Show/Hide Abstract]
- 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 from a hyperplane in the Hausdorff distance at all scales or the zero set becomes locally flat on small scales with arbitrarily small constant. An application is given to a free boundary problem for harmonic measure from two sides, where blow-ups of the boundary are zero sets of harmonic polynomials.
**Citation:**M. Badger,*Flat points in zero sets of harmonic polynomials and harmonic measure from two sides*, J. London Math. Soc.**87**(2013), no. 1, 111-137.- Null sets of harmonic measure on NTA domains: Lipschitz approximation revisited

(arXiv:1003.4547 | Published Version) - [Click to Show/Hide Abstract]
- 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 absolute continuity of harmonic measure and surface measure on NTA domains of locally finite perimeter using Lipschitz approximations. A partial analogue of the F. and M. Riesz Theorem for simply connected planar domains is obtained for NTA domains in space. As a consequence every Wolff snowflake has infinite surface measure.
**Citation:**M. Badger,*Null sets of harmonic measure on NTA domains: Lipschitz approximation revisited*, Math. Z.**270**(2012), no. 1-2, 241-262.- Harmonic polynomials and tangent measures of harmonic measure

(arXiv:0910.2591 | Published Version) - [Click to Show/Hide Abstract]
- 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 information for solutions of a two-phase free boundary problem studied by Kenig and Toro is derived.
**Citation:**M. Badger,*Harmonic polynomials and tangent measures of harmonic measure*, Rev. Mat. Iberoam.**27**(2011), no. 3, 841-870.

PhD Thesis: Harmonic Polynomials and Free Boundary Regularity for Harmonic Measure from Two Sides. Defended on May 5, 2011.

Selected slides from research talks and colloquiua, in reverse chronological order:

- Rectifiability of Measures: the Identification Problem
- AMS Special Session in GMT and PDE. Joint Mathemathematics Meetings. Denver, CO 2020
- Hölder parameterizations of Bedford-McMullen carpets and connected IFS
- AMS Special Session. Analysis and Probability on Metric Spaces and Fractals. Madison, WI. September 2019
- Open Problems about Curves, Sets, and Measures (Version 2)
- PCMI Research Program Seminar. July 2018. Updated version of Talk at ORAM 2018.
- Open Problems about Curves, Sets, and Measures
- 8th Ohio River Analysis Meeting, Lexington, March 2018.
- Geometry of Radon measures via Hölder parameterizations
- Geometric Measure Theory, Warwick, July 2017.
- Structure theorems for Radon measures
- Analysis on Metric Spaces, Pittsburgh, March 2017.
- Singular Points for Two-Phase Free Boundary Problems for Harmonic Measure
- SIAM Minisymposium on New Trends in Elliptic PDE, December 2015.
- What is Nonsmooth Analysis?
- An introductory colloquium (joint presentation with Vasileios Chousionis) for the UConn Special Semester in Nonsmooth Analysis. September 2015.

- Brownian Motion Demo
- HTML 5 simulation of Brownian motion exiting a domain.
- Bee Sting
- North American history in Ontario County, NY
- Matthew Ward <link to>
- Fiction and non-fiction by Connecticut writer Matthew Ward.

Date of Freshest Content: January 17, 2020