Student Analysis Seminar (Fall 2026): Geometric variational problems and PDE
Student Analysis Seminar (Fall 2026): Geometric variational problems and PDE
Organizers: Raphael Tsiamis, Mark Kunyi Ma
Time: Mondays 5:30 - 7:30 pm
Location: Columbia Math Department, Room 528
This seminar will study geometric variational problems and partial differential equations, with an emphasis on the interactions between geometry, nonlinear PDE, and the calculus of variations. Many central objects in geometric analysis arise either as critical points of geometric energies or as solutions of natural geometric PDE. Examples include minimal and constant mean curvature surfaces, special Lagrangian submanifolds, solutions of geometric flows, and critical points of anisotropic or nonlocal energies. Despite their different origins, these problems share many common themes, including existence and regularity, stability and rigidity, singularity formation, blow-up analysis, and the classification of special solutions..
A central goal of the seminar will be to highlight connections between different areas of geometric analysis. For instance, minimal surfaces and cones arise from the classical area functional, while anisotropic and nonlocal variants lead to related but substantially different notions of curvature and minimality. Geometric flows such as mean curvature flow provide a dynamical approach to variational problems and naturally lead to questions about singularities and their models. Special Lagrangian geometry connects calibrated geometry with fully nonlinear elliptic equations, while free-boundary and phase-transition problems provide further links between geometric measure theory and nonlinear PDE. Techniques such as monotonicity formulas, variational arguments, maximum principles, compactness and blow-up methods, and stability inequalities appear throughout these subjects.
The seminar will cover both foundational results and recent developments in these areas, with topics depending on the interests of the participants. Possible themes include minimal surfaces and minimal cones, mean curvature and related geometric flows, special Lagrangian equations and calibrated geometry, anisotropic variational problems, nonlocal minimal surfaces, geometric inequalities, free-boundary problems, and regularity and singularity theory. The broad scope is intended to allow participants working in different parts of geometry and analysis to contribute while emphasizing the common structures and techniques that connect these problems.
The seminar will discuss a variety of topics. We collect references on each topic below.
The special Lagrangian equation
Bhattacharya, A. and Ogden, W.J., Nonuniqueness of solutions to the Lagrangian mean curvature equation. Preprint arXiv:2608.22114 (2026)
In the first meeting, we will set out the goals of the seminar and discuss various aspects of the references, schedule, and content of the talks.
The special Lagrangian equation is a second-order elliptic PDE governing the potentials of minimal gradient graphs. It originates in the beautiful subject of calibrated geometry, introduced by Harvey and Lawson. Beginning from calibrated geometry, we introduce the special Lagrangian equation and provide an overview of its regularity theory and associated Bernstein problems. I will then present joint work with Yu Yuan establishing a constant rank principle for the special Lagrangian equation and discuss its applications to regularity and rigidity. Time permitting, we will discuss joint work with Arunima Bhattacharya demonstrating surprising phenomena for the related Hamiltonian stationary equation and Lagrangian mean curvature equation, including singularity and nonuniqueness, which contrast with the special Lagrangian equation.
A well-known question of Nadirashvili, Tkachev, and Vlăduţ asks whether the solutions of uniformly elliptic equations in three dimensions satisfy a uniform interior W^{1,1} estimate depending only on the ellipticity constants. In higher dimensions, this property is known to not hold, with counterexamples built from the isoperimetric foliation of the Cartan cubic. We study the recent construction of Nam Q. Le, Qi Sun, and Hung V. Tran, who obtained a counterexample to this conjecture in dimension three.
Following Jacob's talk, we will discuss more properties of complete special Lagrangians and graphical Lagrangians. Bernstein-type results on such objects will be derived.