MATJ5127 MA2: Geometric Measure Theory for the Evolution of Dislocations (JSS35) (2 cr)

The motion of dislocations, i.e. topological defects in a crystal lattice, constitutes the predominant microscopic mechanism enabling the plastic deformation of crystalline solids (such as metals). Because only low regularity is to be expected of these lines and because topological changes may occur along the flow, integral and normal currents have long been identified as the natural mathematical objects to model these phenomena. One can think of them as measure-theoretic versions of geometric objects like curves or surfaces. While the stationary (time-independent) theory has been well understood for some time, the study of evolutions is more recent. In particular, over the last couple of years such questions have been investigated via a variational approach in space-time as well as through the so-called geometric (Lie) transport equation. These lectures will give an introduction to these approaches with a view to unsolved theoretical challenges as well as applications from the realm of material science.

Topics:

1. Introduction and physical motivation

2. Integral and normal currents

3. Variational theory of space-time integral currents

4. The geometric derivative

5. Applications

- to see how low-regularity geometric objects arise naturally in material science modelling

- to learn some basics of the theory of currents

- to understand how the evolution of geometric objects can be described variationally and via the geometric transport equation

- to be able to use these methods in the study of applied problems

Just a good undergraduate knowledge of analysis and measure theory - no knowledge of currents or geometric analysis will be required!