NSF Mathematical Sciences Postdoctoral Research Fellowship

Project Description

Principal Investigator: Matthew Badger

This award was made as part of the FY 2012 Mathematical Sciences Postdoctoral Research Fellowships Program. These fellowships support a research and training plan at a host institution in the mathematical sciences, including applications to other disciplines. The title of the research fellowship awarded to Matthew Badger was Geometry of Measures and Analysis on Rough Domains. The first host institution for the fellowship was Stony Brook University, and the sponsoring scientist was Dr. Raanan Schul.

In Fall 2014, the PI changed the host institution for the award to University of Connecticut with sponsoring scientist Dr. Maria Gordina.

Research Products

Two sufficient conditions for rectifiable measures
(arXiv:1412.8357 | Published Version)

(with Raanan Schul)

We identify two sufficient conditions for locally finite Borel measures on Rⁿ 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 Rⁿ 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 Rⁿ with lim_r↓0 μ(B(x,r))/r=∞ at μ almost every x in Rⁿ is 1-rectifiable.

Citation: M. Badger, R. Schul, Two sufficient conditions for rectifiable measures, Proc. Amer. Math. Soc. (2015), 10 pages, DOI 10.1090/proc/12881.

Local set approximation: Mattila-Vuorinen type sets, Reifenberg type sets, and tangent sets
(arXiv:1409.7851 | Published Version)

(with Stephen Lewis)

We investigate the interplay between the local and asymptotic geometry of a set A in Rⁿ 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 Rⁿ (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)

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 Rⁿ. 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)

We repurpose tools from the theory of quantitative rectifiability to study the qualitative rectifiability of measures in Rⁿ, n > 2. To each locally finite Borel measure μ, we associate a function tJ₂(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 tJ₂(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)

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.