
The Discrete Laplace-Beltrami Operator

Usually in geometry we work with smooth objects in the sense that we define our shapes by differentiable or continuous functions. In practice however, we have to define shapes by gluing together a finite number of triangles or some other flat object (think of how a picture is pixelated or how the 'Gherkin' building in London looks curved but is made of flat pieces of glass.)
An important area of current mathematical interest in this subject is the investigation of the discrete Laplace-Beltrami operator. This versatile operator allows us to describe shapes so that we can compare them, deform them and look for areas of interest on them. It can be used in many real life applications for example product design or brain scans in medical imaging. Typical material would include the range of possible discrete Laplace-Beltrami operators available, heat kernel type signatures and the comparison of shapes.

Prerequisites: MATH2051 Geometry of Curves and Surfaces
If the student has some knowledge of programming, then the project could involve a computing element if they wish but this is not compulsory.

Books
Martin Reuter, Laplace spectra for shape recognition, Books on Demand GmbH, 2006.
Mario Botsch et al, Polygon mesh processing, A. K. Peters, 2010.