**Matthew Badger**

University of Connecticut

Department of Mathematics

196 Auditorium Rd, U-3009

Storrs, CT 06269-3009

**Office**: MSB 204

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

Spring 2016 Office Hours | |

M | 10:40 - 11:40 |

Tu | By Appointment |

W | 10:40 - 11:40 |

Th | By Appointment |

F | 10:40 - 11:40 |

Office hours are held in

Math Sciences Building 204

(Gant Science Complex)

I am an Assistant Professor of Mathematics at UCONN. I study the **geometry of sets and measures** using a mixture of geometric measure theory, harmonic analysis and quasiconformal analysis.

Quick Links: [Curriculum Vitae | Teaching | Research]

In Spring 2016, the Analysis and Probability seminar meets most Fridays at 3:25pm. Co-organized with Joe Chen.

Geometric Aspects of Harmonic Analysis: AMS Special Session at the Fall Eastern Sectional Meeting in Brunswick, Maine, September 24 and 25, 2016.

Geometric Measure Theory and Its Applications: AMS Special Session at the Spring Eastern Sectional Meeting in Stony Brook, March 19 and 20, 2016.

Complex Analysis, Probability and Metric Geometry: AMS Special Session at the Spring Southeastern Sectional Meeting in Knoxville, March 22 and 23, 2014.

Harmonic Analysis, Partial Differential Equations and Geometric Measure Theory: AMS Special Session at the Joint Mathematics Meeting in San Diego, January 10, 2013.

[Math Course Schedules: Current Semester]

Math 2410, Section 11: Elementary Differential Equations

Math 5120: Complex Analysis

Math 4110 / 5110: Introduction to Modern Analysis

Math 1131Q: Calculus I

Math 4110 / 5110: Introduction to Modern Analysis

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 1500382
- Analysis Program. Continuing Grant.
- NSF DMS 1203497
- 2012 NSF Mathematical Sciences Postdoctoral Research Fellowship

[Statistics] Newest preprints/papers are listed first.

- Multiscale analysis of 1-rectifiable measures II: characterizations

(arXiv:1602.03823) - (
*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. **Status:**Preprint, Submitted.**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) - (
*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. **Status:**Preprint, Submitted.- Rectifiability and elliptic measures on 1-sided NTA domains with Ahflors-David regular boundaries

(arXiv:1507.02039) - (
*with*Murat Akman, José María Martell, and Steve Hofmann) - [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. **Status:**Accepted. To appear in Trans. Amer. Math. Soc.- 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. Iberoamericana**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:

- Rectifiable and Purely Unrectifiable Measures in the Absence of Absolute Continuity
- JMM 2016 (Seattle), Special Session on Analysis and Geometry in Nonsmooth Metric Measure Spaces, January 2016.
- Singular Points for Two-Phase Free Bounbdary 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.
- Quasiconformal Planes and Bi-Lipschitz Parameterizations
- Ahlfors-Bers VI, October 2014.
- Multiscale Analysis of 1-Rectifiable Measures
- AMS Fall 2013 Southeastern Sectional Meeting, Special Session on Harmonic Analysis and PDE, Louisville, October 2013.
- Quasispheres and Bi-Lipschitz Parameterizations
- Perspectives in Analysis, Philadelphia, September 2012.
- Harmonic Measure from Two Sides (and Tools from Geometric Measure Theory)
- Simons Postdoctoral Fellowship Meeting, April 2012.
- Harmonic Measure in Space (Brownian Motion Demonstrations)
- Dynamics seminar at Stony Brook University, September 2011.
- Free Boundary Regularity for Harmonic Measure from Two Sides
- JMM 2011 (New Orleans), Special Session on Harmonic Analysis and PDEs, January 2011.
- Lipschitz Approximation to Corkscrew Domains
- Rainwater Seminar at University of Washington, February 2010.
- Tangent Measures and Harmonic Polynomials
- Short talk at CRM, June 2009.

Date of Freshest Content: April 15, 2016