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Matrixfaktorisierungen
Im Folgenden soll ein kurzer Abriss des Themas Matrixfaktorisierungen gegeben werden. Wir werden darlegen, warum dieses recht simple Konzept zu erstaunlich tiefen mathematischen Gedankengängen führt und auch in der modernen theoretischen Physik wichtige Anwendungen hat.
Billiards and flat surfaces
Billiards, the study of a ball bouncing around on a table, is a rich area of current mathematical research. We discuss questions and results on billiards, and on the related topic of flat surfaces.
Estimating the volume of a convex body
Sometimes the volume of a convex body needs to be estimated, if we cannot calculate it analytically. We explain how statistics can be used not only to approximate the volume of the convex body, but also its shape.
Geometry behind one of the Painlevé III differential equations
The Painlevé equations are second order differential equations, which were first studied more than 100 years ago. Nowadays they arise in many areas in mathematics and mathematical physics. This snapshot discusses the solutions of one of the Painlevé equations and presents old results on the asymptotics at two singular points and new results on the global behavior.
Spaces of Riemannian metrics
Riemannian metrics endow smooth manifolds such as surfaces with intrinsic geometric properties, for example with curvature. They also allow us to measure quantities like distances, angles and volumes. These are the notions we use to characterize the "shape" of a manifold. The space of Riemannian metrics is a mathematical object that encodes the many possible ways in which we can geometrically deform the shape of a manifold.
Minimizing energy
What is the most efficient way to fence land when you’ve only got so many metres of fence? Or, to put it differently, what is the largest area bounded by a simple closed planar curve of fixed length? We consider the answer to this question and others like it, making note of recent results in the same spirit.
The Willmore Conjecture
The Willmore problem studies which torus has the least amount of bending energy. We explain how to think of a torus as a donut-shaped surface and how the intuitive notion of bending has been studied by mathematics over time.
The codimension
In this snapshot we discuss the notion of codimension, which is, in a sense, “dual” to the notion of dimension and is useful when studying the relative position of one object insider another one.
Topological Complexity, Robotics and Social Choice
Topological complexity is a number that measures how hard it is to plan motions (for robots, say) in terms of a particular space associated to the kind of motion to be planned. This is a burgeoning subject within the wider area of Applied Algebraic Topology. Surprisingly, the same mathematics gives insight into the question of creating social choice functions, which may be viewed as algorithms for making decisions by artificial intelligences.
Closed geodesics on surfaces and Riemannian manifolds
Geodesics are special paths in surfaces and so-called Riemannian manifolds which connect close points in the shortest way. Closed geodesics are geodesics which go back to where they started. In this snapshot we talk about these special paths, and the efforts to find closed geodesics.