Geometric measure theory — rectifiability, integral currents, minimal surfaces. Big promises, sexy words, and tons of technical machinery to back it up. There’s some serious stuff here, have fun :)

Topics really should include:

• Hausdorff measure and dimension
• rectifiability and geometric regularity (i.e. generalizations of manifold structure)
• the integral currents program for minimizing over geometric structures
• minimal surfaces and PDE generalizations/extensions
• fractal stuffs
• also maybe some diff geo or alg geo, wherever they may arise

list of papers

• [ ] "A Sharp Strong Maximum Principle and a Sharp Unique Continuation Theorem for Singular Minimal Hypersurfaces" - Neshan Wickramasekera
• [ ] The Maximum Principle for Minimal Varieties of Arbitrary Codimension — Brian White
• [ ] Which Ambient Spaces Admit Isoperimetric Inequalities for Submanifolds — Brian White
• [ ] The Barrier Principle for Minimal Submanifolds of Arbitrary Codimension — Jorge & Tomi
• [ ] "A sharp bound on the Hausdorff dimension of the singular set of a uniform measure" - A. Dali Nimer
• [ ] "Topology of Surfaces with Finite Willmore Energy" - Jie Zhou
• [ ] "GEOMETRIC AND ANALYTIC STRUCTURES ON METRIC SPACES HOMEOMORPHIC TO A MANIFOLD" - Basso, Marti, and Wenger
• [ ] "Characterization of Rectifiability via Lusin-Type Approximation" - Marchese & Marlo
• [ ] "A Square Function Involving the Center of Mass and Rectifiability" - Villa
• [ ] "A Universal Law of Robustness via Isoperimetry" - Sebastien Bubeck and Mark Sellke
• [ ] The Regularity Theory for the Area Functional — Camillo
• [x] Geometric Properties of Disintegration of Measures — Possobon, Rodriguez
• [x] Coarea integration in metric spaces — Maly
• [x] Polarity and Separation of Cones — Valeriu Soltan
• [x] Moreau-type characterizations of polar cones — Valeriu Soltan