Matthew Badger
Department of Mathematics
Stony Brook University
Stony Brook, NY 11794-3651
Office: Math Tower 4-117
E-mail: badger (( a t )) math.sunysb.edu
| Spring 2013 Office Hours | |
| M | By Appointment |
| T | By Appointment |
| W | By Appointment |
| H | By Appointment |
| F | By Appointment |
Office hours are held
in Math Tower 4-117.
Mathematics
I am a James H. Simons Instructor and NSF Postdoctoral Fellow at Stony Brook University, studying geometric measure theory (GMT) and its applications.
I completed my Ph.D. at the University of Washington. My thesis advisor was Tatiana Toro.
Here is a random (well not quite random, but rather useful) fact from GMT. "Tangent measures to tangent measures are tangent measures":
If μ is a measure on Rn and ν ∈ Tan(μ, x) is a tangent measure, then Tan(ν, 0) ⊂ Tan(μ,x).
Upcoming Event
Harmonic Analysis, Partial Differential Equations and Geometric Measure Theory: AMS Special Session at the Joint Mathematics Meeting in San Diego, January 10, 2013.
Teaching
[Course Schedules: Current Semester | Next Semester]
Fall 2013
MAT 638: Brownian Motion and Harmonic Measure
Fall 2012
MAT 211: Introduction to Linear Algebra
Spring 2012
MAT 322: Analysis in Several Dimensions
Research
I am currently using tools from geometric measure theory to study harmonic measure on non-tangentially accessible (NTA) domains in dimensions three and higher.
Here is a related picture. There exist homogeneous harmonic polynomials of degree 3, e.g.
x2(y-z) + y2(z-x) + z2(x-y) - xyz
whose zero sets divide the 2-sphere into two components.
Here is another example of this phenomenon. A zero set of a homogeneous harmonic polynomial of degree 5 that separates space into two components (cross your eyes to see a stereographic picture) is:
500x4y-1000x2y3+100y5 -5(x4+y4)z+10(x2+y2)z3+2z5
Publications and Preprints
[Statistics] Newest preprints/papers are listed first.
- Beurling's criterion and extremal metrics for Fuglede modulus (arXiv:1207.5277)
- 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.
- Status: Accepted, To appear in Ann. Acad. Sci. Fenn. Math.
- Quasisymmetry and rectifiability of quasispheres (arXiv:1201.1581)
- (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.
- Status: Accepted, To appear in Trans. Amer. Math. Soc.
- 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.
Dissertation
PhD Thesis: Harmonic Polynomials and Free Boundary Regularity for Harmonic Measure from Two Sides. Defended on May 5, 2011.
Presentations
Selected slides from recent research talks:
- Quasispheres and Bi-Lipschitz Parameterizations
- Perspectives in Analysis, Philadelphia, September 2012
- Free Boundary Regularity for Harmonic Measure from Two Sides
- Slides from talk at 2011 Joint Meetings, Special Session on Harmonic Analysis and PDEs.
- 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)
Miscellaneous
- Bee Sting
- North American history in Ontario County, NY
- KGS Online <link to>
- Play the Game of Go! (Baduk)
- Sage <link to>
- Open Source Mathematics Software