Autumn 2025 (Stanford): Frequency Functions and Branched Minimal Surfaces
Time/Location: Monday & Friday, 2:30–3:50pm, Room 380D, starting Monday Sep 29.
The aim of this course is to explain, in a modern light, Almgren’s so-called ‘big regularity theorem’ concerning the partial regularity of area minimising currents. The key idea is how Almgren’s frequency function can be used to prove quantitative unique continuation statements in the absence of regularity.
We will start from the basic PDE setting and gradually build up to Almgren’s theorem. Topics covered are:
- Frequency for harmonic functions and elliptic PDEs
- Multi-valued functions and their Sobolev theory
- Partial regularity for Dirichlet minimisers
- Branch set dimension bounds for multi-valued minimal graphs
- Partial regularity for area minimisers
Our approach follows Almgren’s but differs from it in one crucial way, namely by making use of the planar frequency function introduced by Krummel–Wickramasekera. This allows a much clearer analogy between Almgren’s dimension bound on the singular set of an area minimiser and the (more classical) dimension bound on the touching set of two harmonic functions. It also significantly simplifies the discussion surrounding Almgren’s center manifold construction.
Update: Course now complete. Typed notes will appear here once I’ve had the time to write them… (Expected April 2026)
Spring 2024 (Princeton): Regularity Techniques in Geometric Measure Theory
The aim of this course is to cover several key results within geometric measure theory, in particular the regularity theory of minimal surfaces. One key idea is that of excess decay, arising from a suitable blow-up (i.e. linearisation) procedure. Topics covered are:
- Tangent cones and singular set stratification
- Schoen–Simon’s regularity theory for stable minimal hypersurfaces (via Bellettini)
- Allard’s regularity theorem for stationary varifolds
- Simon’s cylindrical tangent cones
- Wickramasekera’s regularity theory for stable minimal hypersurfaces
One key topic which is not covered in the course is the use of frequency functions and center manifolds to establish dimension bounds on branch sets.
Completed Notes (last updated: Dec 30, 2024)
Spring 2023 (Princeton): MAT 425 (Analysis III: Integration Theory and Hilbert Spaces). More details on the course can be found here.
Fall 2022 (Princeton): MAT 103 (Calculus I). I made a “formula/main result” booklet to help students study, which can be found here: Formula book for MAT 103.
Compactness 