**Matthew Badger**

Department of Mathematics

Stony Brook University

Stony Brook, NY 11794-3651

**Office**: Math Tower 4-117

**E-mail**: badger _at_ math.sunysb.edu

Spring 2014 Office Hours | |

M | 2:30 - 4:00 |

Tu | By Appointment |

W | 2:30 - 4:00 |

Th | By Appointment |

F | By Appointment |

Office hours are held

in Math Tower 4-117.

Teaching and Research

I am a James H. Simons Instructor and NSF Postdoctoral Fellow at Stony Brook University. 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 Fall 2014, I will join the faculty at the University of Connecticut.

Complex Analysis, Probability and Metric Geometry: AMS Special Session at the Southeastern Spring 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.

[Course Schedules: Current Semester | Next Semester]

MAT 550: Real Analysis II

MAT 638: Brownian Motion and Harmonic Measure

MAT 211: Introduction to Linear Algebra

MAT 322: Analysis in Several Dimensions

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

[Statistics] Newest preprints/papers are listed first.

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

(arXiv:1403.2991) - (
*with*Jonas Azzam and Tatiana Toro) - 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. **Status:**Preprint, Submitted.- Multiscale analysis of 1-rectifiable measures: necessary conditions

(arXiv:1307.0804) - (
*with*Raanan Schul) - 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. **Status:**Preprint, Submitted.- Beurling's criterion and extremal metrics for Fuglede modulus

(arXiv:1207.5277 | Published Version) - 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) - 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) - 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) - 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) - 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:

- Multiscale Analysis of 1-Rectifiable Measures
- AMS Fall Southeastern Sectional Meeting, Special Session on Harmonic Analysis and PDE, Louisville, October 2013
- Quasispheres and Bi-Lipschitz Parameterizations
- Perspectives in Analysis, Philadelphia, September 2012
- Free Boundary Regularity for Harmonic Measure from Two Sides
- Joint Mathematics Meetings, Special Session on Harmonic Analysis and PDEs, New Orleans, January 2011
- Lipschitz Approximation to Corkscrew Domains
- Rainwater Seminar on February 16, 2010. (PDF)
- Tangent Measures and Harmonic Polynomials
- Short talk on June 19, 2009 at CRM. (PDF)

Date of Freshest Content: August 7, 2014