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.

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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.

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Here is a recent article describing some of the history and recent applications of GMT.

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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.  

  • 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.