Sean McCurdy
Overview
My research is in Geometric Measure Theory (GMT). Broadly speaking, GMT is the study of the relationship between the geometry of sets and measures and their measure-theoretic properties. It often works at the intersection of Calculus of Variations, Partial Differential Equations, Geometric Analysis, Harmonic Analysis, and Metric Analysis.
GMT is the natural home for many geometric problems in the Calculus of Variations such as Minimal Surfaces, Isoperimetric problems, Brakke Mean Curvature Flow, and Free-boundary problems. It also has a 'pure' branch which does fundamental research into the geometry of sets and measures. This fundamental research has found applications in many branches of Analysis.
Here is a recent article describing some of the history and recent applications of GMT.
Currently, my research touches upon several branches of GMT.
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Here in Taiwan, I am engaged in the study of varifold regularity under the mentorship of Ulrich Menne (National Taiwan Normal University). Varifolds are a generalization of surfaces. See this article for a brief introduction to the concept of varifolds.
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I also have a collaboration with Kuan-Ting Yeh (University of Washington, Seattle) studying symmetrization inequalities, Isoperimetric problems, and the fine properties of Ehrhard symmetrization when the underlying measure is not an isotropic Gaussian function.
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I also have a long-standing interest in the structure of singular sets arising from Free boundary problems. As a graduate student, I studied two-phase free-boundary problems for harmonic measure. Later as a Post-Doctoral Fellow at Carnegie Mellon University, I studied the singular structure of local minimizers of degenerate Alt-Caffarelli functionals inspired by the theory of water waves.
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Here are some slides from a talk I gave summarizing some of my work on these degenerate Alt-Caffarelli functionals.
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I also have an on-going collaboration with Matthew Badger (University of Connecticut, Storrs) studying the quantitative geometry of rectifiable curves in Banach spaces.