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.
Positive isotropic curvature (PIC) is a natural condition for the Riemann curvature tensor in dimensions four and above, introduced by Micallef and Moore in their study of minimal two-spheres in Riemannian manifolds. Variants of the PIC condition were instrumental in the proof of the Differentiable Sphere Theorem by Brendle and Schoen.
The Ricci flow approach to this problem originates in the work of Hamilton, who developed a surgery program for four-dimensional manifolds with positive isotropic curvature, exploiting the special structure of the curvature operator to obtain a pinching estimate toward positive curvature operator. A central problem in Ricci flow for higher-dimensional manifolds with positive isotropic curvature is to establish suitable pinching properties of the Riemann curvature tensor.
In this talk, I wil explain Brendle's breakthrough classification of manifolds with positive isotropic curvature. His technique relies on a pinching estimate which forces high-curvature regions toward the PIC2 cone, and may be viewed as a higher-dimensional counterpart of the Hamilton-Ivey phenomenon. It also develops a higher-dimensional version of Perelman's program with an analysis of ancient solutions, a Canonical Neighborhood Theorem, and a surgery construction.
The mean curvature of a set is an object with an intrinsic geometric meaning but also a deep variational interpretation. In this spirit, it is quite natural to define a non-local mean curvature in a variational sense.
In this talk, we present the two notions of classical and fractional mean curvature and discuss similarities and differences between the two.
Then, we focus on the problem of sets with prescribed mean curvature and how this problem is related to the asymptotic behavior of solution of a mass-conserved Allen-Cahn equation.
Quite surprisingly, in the non-local setting, a new variational object, which we refer to as "non-local hybrid mean curvature", naturally appears.
The study of nonlocal minimal surfaces—driven by fractional perimeters and nonlocal operators—was pioneered by Luis Caffarelli, Jean-Michel Roquejoffre, and Ovidiu Savin in their groundbreaking 2009 work. These surfaces arise naturally in contexts ranging from phase transition models with long-range interactions to stochastic control problems.
In this talk, we will explore the foundational theory of nonlocal minimal surfaces and their associated nonlocal mean curvature. We will carefully examine key analytical tools that mirror classical minimal surface theory, including uniform density estimates, monotonicity formulas, blow-up analysis, and the improvement of flatness property. These techniques form the cornerstone for understanding the regularity and structure of nonlocal boundaries.