Matthew Badger | Department of Mathematics | University of Connecticut

Rectifiability and Fine Geometry of Sets, Radon Measures, Harmonic Functions, and Temperatures

NSF DMS 2154047 [Award Info on NSF Fastlane]

Principal Investigator: Matthew Badger

Award Dates: August 2022 – July 2025

Award Abstract

Geometric measure theory provides an analytical toolkit to detect hidden structure in high-dimensional data sets and to describe non-smooth phenomena such as the formation of corners and singularities in soap bubble clusters. The term measure refers to an abstract generalization of length, area, and volume, which assigns a notion of size to each mathematical set. The pervasiveness of measures within contemporary mathematics notwithstanding, there are presently only a few tools available to analyze measures in regimes with low regularity. The proposed investigation seeks to develop novel and robust quantitative methods to study the geometry of general sets and measures in the absence of traditional simplifying hypotheses. The project will explore applications of these tools and methods to the analysis of harmonic functions and temperatures (solutions to the heat equation) in domains with rough boundary, both in ideal settings and inside non-homogenous media. The project will also provide training to graduate research assistants at the University of Connecticut and to visiting Ph.D. students working in geometric measure theory, harmonic analysis, and partial differential equations.

The project will develop four interrelated threads of research on the fine geometry of sets, the structure and regularity of measures, and the solutions of partial differential equations. The first line of inquiry pursues questions about subsets of rectifiable curves in infinite dimensional Banach spaces, with a concrete goal of solving the Analyst's Traveling Salesman Problem in a non-Hilbert setting. A second line of inquiry concerns the parameterization problem for Lipschitz images of the plane and the classification of 2-rectifiable Radon measures. Additional work will focus on the structure of measures supported on the graphs of Hölder continuous functions. A third direction of study will involve improved upper bounds on the Hausdorff dimension of harmonic and caloric measures on general domains, along with the relationship between the static and time-dependent cases. The final strand of the research program will confront contemporary challenges and initiate new inquiries in the theory of non-variational free boundary problems for harmonic and elliptic measures with two or more phases.

Research Products

#27 On the number of nodal domains of homogeneous caloric polynomials
(with Cole Jeznach)
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Status: preprint, submitted.
#26 Square packings and rectifiable doubling measures
(with Raanan Schul)
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Status: preprint, submitted.
#24 Slowly vanishing mean oscillations: non-uniqueness of blow-ups in a two-phase free boundary problem
(arXiv:2210.17531 | Published Version)
(with Max Engelstein and Tatiana Toro)
Dedicado a Carlos Kenig, un gran maestro y amigo en conmemoración de sus 70 años
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Status: accepted, to appear in Vietnam J. Math.
#23 Subsets of rectifiable curves in Banach spaces II: universal estimates for almost flat arcs
(with Sean McCurdy)
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Status: accepted, to appear in Illinois Journal of Mathematics
#22 Lower bounds on Bourgain's constant for harmonic measure
(with Alyssa Genschaw)
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Status: preprint, submitted.
#21 Identifying 1-rectifiable measures in Carnot groups
(with Sean Li and Scott Zimmerman)
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Status: accepted, to appear in Anal. Geom. Metr. Spaces
#18 Subsets of rectifiable curves in Banach spaces I: sharp exponents in traveling salesman theorems
(with Sean McCurdy)
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Note: While revising an earlier draft of the manuscript, we identified a mistake in Schul's 2007 proof of the necessary conditions in the traveling salesman theorem in infinite-dimensional Hilbert space. We show how to correct the error in a minimal way, leaving the outline of virtually all of the original proofs intact.
Status: accepted, to appear in Illinois Journal of Mathematics
Date of Freshest Content: January 30, 2024.