# Karen Uhlenbeck and Global Analysis – what is a manifold?

# BLOG: Heidelberg Laureate Forum

As part of this year’s HLF, two alumni – Janelle C. Mason and Jaqueline Godoy Mesquita – interviewed 2019 Abel prize winner Karen Uhlenbeck. The first woman to receive the prize, Uhlenbeck talked about her career and the opportunities she’s had, as well as her work. When asked about her research, she explained how she got interested in the subject of global analysis.

Early in her career, Karen Uhlenbeck was interested in a lot of different kinds of mathematics – “I found almost everything very interesting – it doesn’t matter what it was.” But when she started her MA and PhD studies at Brandeis University with her supervisor Richard Palais, she discovered the new and emerging field of global analysis and found it appealed to her. It crosses between two areas of maths – analysis, the study of functions, and topology, the study of manifolds.

A manifold is roughly an object which, when you look at it close up, looks flat. For example, a sphere is very much not a flat object, but if you zoom in on one part of a sphere, you can see what looks like a flat surface – ‘locally’, as a topologist would say, it looks and behaves like a normal 2-dimensional flat plane.

This can be seen in the way that we map the surface of the earth: some people might use an atlas – a collection of maps or charts, each showing one small flat portion of the surface; others might use a globe to see how the whole thing connects up. Topologists use these same words – they literally describe a manifold as being defined using an ‘atlas’ of ‘charts’ and having a ‘global’ structure.

A sphere is an example of a 2-dimensional manifold (also called a ‘surface’). Other manifolds might locally look like ordinary 3-dimensional space – but globally they might join up in a way that can’t be seen or understood just by looking at one small part. We call a manifold that’s locally n-dimensional, an n-dimensional manifold or n-manifold.

We also require that manifolds have this ‘locally flat’ property everywhere – for example, a circle is a 1-manifold, as every part of it looks like a 1-dimensional piece of straight line, but a figure-of-eight shape isn’t a 1-manifold, as the crossing point in the middle doesn’t look like a straight line. This property allows manifolds to be studied and understood at the simple, local level, even if their global structure is very complex.

In the same way we can define a function on a set of numbers, or on points in a plane, we can define functions on the surface of a manifold, which means we can apply things like calculus to them in order to study how they behave (and this might also tell us something about the manifold itself).

Once we have n-dimensional manifolds, it’s possible to extend this notion to an infinite-dimensional manifold, which would be something that locally behaves like infinite-dimensional space – such as a space of functions. For example, the set of all maps between the interval [0,1] and itself – which map numbers between 0 and 1 onto other numbers between 0 and 1 – can be considered as an infinite dimensional space. Global analysis, which Karen Uhlenbeck found herself studying back in the 1960s, is what happens when infinite dimensional manifolds meet functional analysis.

Describing it as a “faddish and very exciting field”, Uhlenbeck recalls how people discovered that a lot of the calculus you did on finite dimensional manifolds could be done on infinite dimensional manifolds instead. At a time when the field of topology had recently gone through a large amount of development and expansion, people were looking for new ways to apply these ideas, and global analysis used techniques in infinite-dimensional manifold theory to classify the behavior of differential equations – ODEs and PDEs. This allows you to solve optimisation problems, and has applications in physics and quantum field theory.

At the time, this field was emerging, and Uhlenbeck was able to be part of the team which did a lot of the early work in establishing it. “I just got really excited,” she says – and it’s good to know that someone who is now considered one of the founders of modern geometric analysis did so with the glee of getting to play with a subject they’re passionate about.

Katie Steckles wrote (02. Oct 2020):

>

[…] A manifold is roughly an object which, when you look at it close up, looks flat. […] We also require that manifolds have this ‘locally flat’ property everywhereDoes the surface of a cube constitute a (2-dimensional) manifold ? …

It’s complicated 🙂 A cube is a topological manifold, because it can be stretched out into e.g. a sphere without cutting or gluing (what we’d call homomorphic to a sphere); however, most topologists are interested in things called ‘smooth manifolds’, where the maps all have to be continuous and satisfy some conditions that mean they’re well-behaved (we’d say something with this property is diffeomorphic to a sphere, like a bigger sphere, or an ellipsoid) – but a cube isn’t, because of the corners and edges where it doesn’t work. Hope that helps!