DescriptionThis topic is at the interface between two mathematical worlds: the geometry of the hyperbolic space and the analysis of a fundamental operator motivated by physics: the Laplacian. Our aim is to study beautiful connections between these two mathematical worlds. Here are some ideas illustrating what we want to explore.You may have seen the construction how to obtain a topological torus by taking a quotient R^2/Z^2 or by identifying sides of a parallelogram. While the bagel has one hole in the middle (and is an eatable version of a torus), you may also have seen pretzels which are topologically more complicated than bagels and have more holes (mathematicians say that pretzels have a higher genus than a bagel). To model pretzels and other "higher genus" surfaces mathematically, we need to leave the Euclidean world and have to enter another geometry called the hyperbolic plane. The hyperbolic plane has constant negative curvature and it is possible to create surfaces of higher genus again as quotients of this space. In the hyperbolic plane the angle sum of a triangle is no longer equal to 2 pi and the areas of metric balls no longer grow quadratically with the radius but instead exponentially. So the first part of the project will be to become familiar with important properties of this strange geometry of negative curvature. The second protagonist of the story is the Laplacian. It is in some sense the generalisation of the second derivative and it doesn't only exist in Euclidean space but has a generalisation to the hyperbolic plane and its quotients, the higher genus hyperbolic surfaces. There are various important objects associated to this linear operator: the heat kernel, motivated by Physics and useful to describe mathematically the evolution of heat in time and the eigenvalues, a concept which you have already seen in Linear Algebra. However, in Linear Algebra the vector spaces were usually finite dimensional, whereas our Laplacian is a linear operator on an infinite dimensional Hilbert space. There are many natural questions about the interplay between these analytical data of the Laplacian and the geometry of the underlying hyperbolic surface. In this project you will eventually investigate some of these interesting and beautiful connections between the hyperbolic surfaces and their associated Laplace operators. There are many books about the geometry of the hyperbolic plane and its quotients, the higher genus hyperbolic surfaces. Instead of listing them, we like to mention one book which is particularly concerned with the interplay between hyperbolic geometry and the Laplacian, and this book will be our main reference. Prerequisites and CorequisitesDifferential Geometry III is essential.Analysis III/IV or PDE III/IB or Riemannian Geometry IV would be useful but are not essential.
Main Resource
email: N Peyerimhoff |