Research interests

• Nonlinear partial differential equations (PDEs)

• Calculus of variations

• Applied mathematics

I am interested in applying comprehensive techniques from nonlinear PDEs and calculus of variations to understand complex singularity structures in certain physical systems, including superconductors, liquid crystals, thin film blisters, convection pattern formations and some systems described by hyperbolic conservation laws. The studies of these systems are highly interdisciplinary. The mathematical studies of such problems require the development of new mathematical tools, and these studies further foster the fundamental understanding in related fields of sciences. My work currently evolves along three lines:

• The first line of my research is centered around a second order singular perturbation problem, called the Aviles-Giga functional, in connection with the theory of smectic liquid crystals and thin film blisters. A similar energy functional,  called the Regularized Cross-Newell energy, occurs in convection pattern formations. I am interested in the variational analysis of these problems, leading to better understanding of the singularity structures in the physical systems. These problems lie at the interface between variational problems and scalar conservation laws, and as such my interests also extend to scalar conservation laws.

• The second line of my focus explores the connections between certain notions of convexity, including polyconvexity, quasiconvexity and rank-one convexity, and regularity properties of nonlinear PDEs that can be formulated as differential inclusions. The geometric aspects carried by these various notions of convexity have fascinating connections to uniqueness and regularity of certain nonlinear PDEs, including certain hyperbolic conservation laws.

• The third line of my research concerns three-dimensional superconductivity models. My work has been focusing on an anisotropic model called the Lawrence-Doniach model, which models highly anisotropic superconductors with layered structure. I am also interested in the variational analysis to understand the singularity structures in these superconductors. However the techniques are very different from those used in the analysis of the Aviles-Giga functional due to the different nature of singularities in these systems.

Invited talks at conferences and workshops

• Quantitative rigidity of elliptic differential inclusions in two dimensions. Special Session on Analysis and Applications of Nonlinear Elliptic and Parabolic Equations, 13th AIMS Conference on Dynamical Systems, Differential Equations and Applications, June 2023.

• Compactness and regularity for a generalized Aviles-Giga functional. HIM Workshop: Topological and geometrical aspects in complex materials, March 2023. (https://youtu.be/ukmZJb5xIpI)

• Compactness and regularity for a generalized Aviles-Giga functional. Special Session on Topics in PDEs and Harmonic Analysis, AMS Fall Eastern Sectional Meeting, October 2022.

• Compactness and regularity for a generalized Aviles-Giga functional. Minisymposium on Geometric Variational Problems and their Applications, SIAM Annual Meeting, July 2022.

• Compactness and regularity for a generalized Aviles-Giga functional. Special Session on Nonlinear Partial Differential Equations from Variational Problems and Complex Fluids, AMS Spring Central Sectional Meeting, March 2022 (virtual).

• Regularity of unit vector fields related to the Aviles-Giga functional. Special Session on Calculus of Variation, Nonlinear Waves and their Numerical Realizations, AMS Fall Southeastern Sectional Meeting, November 2021 (virtual).

• On the first critical field of a 3D anisotropic superconductivity model. Minisymposium on Singular Solutions to Geometric Problems in Continuum and Discrete Mechanics, SIAM Conference on Analysis of PDEs, December 2019.

• Null Lagrangian measures in subspaces with applications to a system of conservation laws. 82nd Midwest PDE Seminar, Purdue University, October 2018.

• Regularity of the Eikonal equation with two vanishing entropies. Minisymposium on Variational Problems from Materials Science, SIAM Conference on Mathematical Aspects of Materials Science, July 2018.

• On the first critical field of a 3D anisotropic superconductivity model. Special Session on Singularities and Phase transitions in Nonlinear PDE's, 2018 CMS Summer Meeting, June 2018.

• Gamma-convergence for an anisotropic superconductivity model with magnetic fields near $H_{c_1}$. Minisymposium on Modeling and Analysis of Condensed Matter Systems, SIAM Conference on Analysis of PDEs, December 2017.

• Regularity of the Eikonal equation with two vanishing entropies. Special Session on Analysis of Variational Problems and Nonlinear Partial Differential Equations, AMS Spring Central Sectional Meeting, Indiana University, April 2017.

• Regularity of the Eikonal equation with two vanishing entropies. Special Session on Nonlinear PDEs in Material Science and Mathematical Biology, AMS Fall Eastern Sectional Meeting, Bowdoin College, September 2016.

• Analysis of minimizers of the Lawrence-Doniach model for layered superconductors in magnetic fields. Minisymposium on the Ginzburg-Landau Theory and Related Topics, SIAM Conference on Analysis of PDEs, December 2015.

• Analysis of the Lawrence-Doniach energy for layered superconductors in magnetic fields. Minisymposium on the Ginzburg-Landau Model and Related Topics, The 8th International Congress on Industrial and Applied Mathematics, August 2015.

• Analysis of the Lawrence-Doniach model for layered superconductors in magnetic fields. Special Session on Calculus of Variations, Nonlinear PDEs, and Applications, AMS Spring Central Sectional Meeting, Michigan State University, March 2015.

Invited talks at seminars and colloquia

• Regularity of the Eikonal equation with $L^p$ entropies. Analysis, Logic and Physics Seminar, Virginia Commonwealth University, March 2024.

• Regularity of the Eikonal equation and its generalizations: a differential inclusion approach. Colloquium, Old Dominion University, September 2023.

• The Aviles-Giga functional and its generalization: challenges and some recent progress. Analysis, Logic and Physics Seminar, Virginia Commonwealth University, April 2023.

• Compactness and regularity for a generalized Aviles-Giga functional. PDE and Differential Geometry Seminar, University of Connecticut, April 2023. 

• The Aviles-Giga functional and its generalization: challenges and some recent progress. Colloquium, Old Dominion University, November 2022.

• Regularity of unit vector fields related to the Aviles-Giga functional. PDE and Differential Geometry Seminar, University of Connecticut, October 2021 (virtual).

• Rigidity of a non-elliptic differential inclusion. Differential Equations Seminar. University of Missouri, November 2020 (virtual).

• On the Aviles-Giga functional and regularity of its zero energy states. Colloquium, New Mexico State University, August 2018.

• Regularity of the Eikonal equation with two vanishing entropies. PDE-Analysis Seminar, University of Kentucky, November 2016.

• A lower bound for the Lawrence-Doniach energy with magnetic fields in the lower regime. PDE Seminar, Purdue University, April 2015.

• Properties of minimizers of the Lawrence-Doniach energy in perpendicular magnetic fields. Harmonic Analysis and Differential Equations Seminar, University of Illinois, October 2013.

Upcoming events

• Compactness and regularity for a generalized Aviles-Giga functional. Minisymposium on Analytical and Computational Methods in Models of Soft Matter, SIAM Conference on Mathematical Aspects of Materials Science, May 2024 (upcoming).