Student Analysis Seminar (Fall 2026): Geometric variational problems and PDE

Abstract

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.