Research

Interests

Geometric Analysis; Affine and Projective Geometric Structures; Teichmüller Theory; Affine Differential Geometry

Research Articles

• Limits of Cubic Differentials and Buildings (with Andrea Tamburelli and Michael Wolf), submitted, 2022.
• Coordinates on the Augmented Moduli Space of Convex RP² Structures (with Tengren Zhang), Journal of the London Mathematical Society, 2021.
• The Moduli Spaces of Equivariant Minimal Surfaces in RH³ and RH⁴ via Higgs Bundles (with Ian McIntosh), Geometriae Dedicata, 2019.
• Equivariant Minimal Surfaces in CH² and Their Higgs Bundles (with Ian McIntosh), Asian Journal of Mathematics, 2019.
• Convex RP² Structures and Cubic Differentials Under Neck Separation, Journal of Differential Geometry, 2019.
• Approximate Yang-Mills-Higgs Metrics on Flat Higgs Bundles over an Affine Manifold (with Indranil Biswas and Matthias Stemmler), Communications in Analysis and Geometry, 2014.
• The Vortex Equations on Affine Manifolds (with Indranil Biswas and Matthias Stemmler), Transactions of the AMS, 2014.
• Holomorphic Cubic Differentials and Minimal Lagrangian Surfaces in CH² (with Zheng Huang and Marcello Lucia), Mathematical Research Letters, 2013.
• Affine Yang-Mills-Higgs Metrics (with Indranil Biswas and Matthias Stemmler), Journal of Symplectic Geometry, 2013.
• Minimal Lagrangian Surfaces in CH² and Representations of Surface Groups into SU(2,1) (with Ian McIntosh), Geometriae Dedicata, 2013.
• Hermitian-Einstein Connections on Principal Bundles over Flat Affine Manifolds (with Indranil Biswas), International Journal of Mathematics, 2012.
• Affine Hermitian-Einstein Metrics, Asian Journal of Mathematics, 2009.
• Limits of Solutions to a Parabolic Monge Ampère Equation (with Mao-Pei Tsui), in Proceeding of the International Conference on Geometric Analysis, NCTS (Taipei, Taiwan; June, 2007), 2009.
• Ancient Solutions of the Affine Normal Flow (with Mao-Pei Tsui), Journal of Differential Geometry, 2008.
• Flat Metrics, Cubic Differentials and Limits of Projective Holonomies, Geometriae Dedicata, 2007.
• Affine Manifolds, SYZ Geometry and the “Y” Vertex (with Shing-Tung Yau and Eric Zaslow), Journal of Differential Geometry, 2005, with erratum, 2008.
• Singular Semi-Flat Calabi-Yau Metrics on S², Communications in Analysis and Geometry, 2005.
• The Compactification of the Moduli Space of Convex RP² Surfaces, I, Journal of Differential Geometry, 2004.
• Affine Spheres and Kähler-Einstein Metrics, Mathematical Research Letters, 2002.
• Riemannian Metrics on Locally Projectively Flat Manifolds, American Journal of Mathematics, 2002.
• Affine Spheres and Convex RPⁿ Manifolds, American Journal of Mathematics, 2001.
• Dissertation, Harvard University, 1999.

 

Survey Articles

• On the Donaldson-Uhlenbeck-Yau Correspondence, Notices of the International Congress of Chinese Mathematicians, 2019.
• Cubic Differentials in the Differential Geometry of Surfaces (with Ian McIntosh), in Handbook of Teichmüller Theory (Vol. 6), 2016.
• Flat Bundles on Affine Manifolds (with Indranil Biswas and Matthias Stemmler), Arabian Journal of Mathematics, 2013.
• Cheng and Yau’s Work on the Monge-Ampère Equation and Affine Geometry (with Xu-Jia Wang and Deane Yang), in Geometry and Analysis (Vol. I), 2010.
• Survey on Affine Spheres, in Handbook of Geometric Analysis (Vol. II), 2010.