Research interests

• Nonlinear partial differential equations (PDEs)

• Calculus of variations

• Applied mathematics

I am interested in applying techniques from nonlinear PDEs and calculus of variations to understand complex singularity structures in problems arising from materials science and continuum mechanics. These problems are highly interdisciplinary, and their study often requires the development of new mathematical tools. Such mathematical investigations also contribute to a deeper fundamental understanding in related scientific fields. 

• The main focus of my current research is a second-order singular perturbation problem known as the Aviles-Giga functional and related nonelliptic differential inclusions. The Aviles-Giga functional is connected to the theories of smectic liquid crystals and thin film blisters, and it lies at the intersection of variational problems and scalar conservation laws. I am particularly interested in the variational analysis of this problem, employing methods from differential inclusions and scalar conservation laws. Additionally, I explore related questions concerning nonelliptic differential inclusions and scalar conservation laws.

• Another line of my research explores the connections between various notions of convexity in matrix spaces—such as polyconvexity, quasiconvexity, and rank-one convexity—and the qualitative properties of nonlinear PDE systems that can be formulated as differential inclusions. The geometric aspects inherent in these notions of convexity have intriguing connections to the compactness and uniqueness of certain nonlinear PDEs, including systems of conservation laws.

• Previously, I have worked on the variational analysis of a three-dimensional anisotropic superconductivity model known as the Lawrence-Doniach model, leading to a deeper understanding of defect structures in such superconductors with magnetic fields in various physical regimes.

Research support

• NSF DMS-2206291 (PI): From Differential Inclusions to Variational Problems: Theory and Applications, 2022–2025.

Invited talks at conferences and workshops

• A regularizing property of the 2D Eikonal equation. Session on Modeling, Analysis, and Computation of Variational Problems, Canadian Mathematical Society (CMS) Winter Meeting, December 2024.

• Compactness and regularity for a generalized Aviles-Giga functional. Minisymposium on Modeling and Analysis of Multi-Constituent Systems, SIAM Annual Meeting Online, July 2024 (virtual). 

• 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. 

• 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.