Single Review on past events

What does he do when he doesn’t smile? That was the exciting question that moderator Panteleimon Eleftheriou and office mate Ben Lambert posed at the beginning of Ben Lambert´s presentation on “Topology, Curvature, and Flows”. His objects of study in this talk were geometric ones such as 1-dimensional curves sittings in the plane or 2-dimensional surfaces sittings in 3-dimensional space. These shapes are called manifolds. The mathematician wants to consider the structures inherent in these manifolds. Two shapes are considered to be topologically the same if there is a continuous deformation which takes one to another. It is allowed to stretch and bend the manifolds but not tear or glue them. “In topology it is necessarily true that connectedness (number of pieces), boundary (edge of a topological space) as well as topological invariants (mathematical structures that can be attached but remain the same regardless of deformation) are preserved”, he explained.

One phenomenon which is important in mathematics is the Euler Characteristic: “We deform our shape into one made up of polygons, e.g. the sphere. We now count up the number of vertices (points), edges (lines), and faces (sides). Then we define the Euler characteristic by x = V - E + F.” The Euler Characteristic does not depend on how you choose the polygons. Any two spaces that are topologically equivalent have the same Euler Characteristic.

In the next step Ben Lambert introduced the idea of curvature of a curve, and then used this to describe several notions of curvature of surfaces. This included ideas of mean curvature and Gaussian curvature. He then described several classical questions in mathematics, for example: How do the topology and the curvature of a surface interact? If we impose restrictions on the curvature, what kinds of topological surfaces are allowed? How do curvatures relate to other geometric ideas such as area?

To make his investigations more visible for the audience Ben Lambert presented an example: “I am stuck on a hill and want to get to the bottom. But unfortunately I have no map and it is so misty that I can only see the ground at my feet. If I reach the bottom, how will I know I am there? And in what direction should I go to get to the bottom?” The answers: There is no slope at the bottom of the hill. You are at a stationary point. And 2nd: Follow the direction of greatest slope! This is called a gradient flow.

He then used this idea of a stationary point to define minimal surfaces – surfaces of least possible area. These surfaces are defined by the vanishing of the mean curvature. These surfaces occur naturally in a variety of places, such as bubbles.

And finally he explained what a curve shortening flow is by showing an animation demonstrating that by analysing the singularity of curve shortening flow we saw that any smooth embedded closed curve has the same topological type

In the end Ben Lambert summarized that topology deals with fundamental properties of shapes “number of holes”. Curvatures are useful properties on geometric surfaces and curves which interact with the topology. Surfaces with restricted curvatures occur in nature and are useful in mathematics. Curvature flows, the objects of Ben Lambert´s research, are useful methods of deforming shapes which give us new information on how the curvature and topology interact.