Introduction to Geometric Measure Theory (L24) Spencer Becker-Kahn

Geometric Measure Theory began with the use of analysis and measure theory to generalize many fundamental geometric concepts from the classical setting of smooth surfaces to much wider classes of sets. The motivation to do so came primarily from the study of area-minimizing surfaces in higher dimensions but the powerful (and sometimes technical) frameworks that were developed were also inspired by and have been applied to PDE, harmonic analysis, algebraic topology, differential geometry and a host of other geometric variational problems. This is very much an analysis-based course though and we will introduce the subject by first building up the necessary tools from analysis and measure theory (Radon measures, covering theorems, a deeper study of Lipschitz functions) and then introducing rectifiable sets and integral varifolds (the name ‘varifold’ comes from the words ‘variational’ and ‘manifold’, so-called because of their applicability in the geometric calculus of variations).

Pre-requisites

Essential: The core parts of real analysis, functional analysis and measure theory at an advanced undergraduate level. Being comfortable with the material in the first half of the Part II course ‘Analysis of Functions’ (which has as prerequisites both ‘Linear Analysis’ and ‘Probability & Measure’) is definitely an appropriate background. Occasionally some computations will require non-trivial linear algebra, but a standard undergraduate background will suffice.

Helpful: In the latter part of the class there will be some points at which it will be helpful to know what an “embedded C1 submanifold of Rn” is.

No background in differential geometry or PDE is necessary.

Literature

1. L. C. Evans & R. F. Gariepy Measure Theory and Fine Properties of Functions. Chapman and Hall/CRC, 2015.

2. L. Simon Lectures on Geometric Measure Theory. The Australian National University, Mathematical Sciences Institute, Centre for Mathematics & its Applications, 1983. Draft of revised version available at: https://web.stanford.edu/class/math285/ts-gmt.pdf

Evans and Gariepy [1] will be most useful for the first half of the class and parts of Simon’s notes [2] will be most useful for the latter parts of the class.

Additional support

Four examples sheets will be provided and four associated examples classes will be given. There will be a one-hour revision class in the Easter Term.

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