Topics in geometric analysis I - 2012


Topics in geometric analysis I


To lay some foundations in modern geometric analysis.



The subject of Geometric Analysis has appeared some 60 years ago although the name is more recent. The subject deals with differential geometry and its relation to global analysis, partial differential equations, geometric measure theory and variational principles to name a few. Geometric Analysis is at force whenever strong mathematical analysis is used to solve problems in differential geometry. The Calabi conjecture, the Yamabe conjecture and most spectacular the Poincaré conjecture all have been solved by methods from geometric analysis. The goal of this course is to introduce the student to the basic techniques of geometric analysis. The subject covered vary each year. Typical subjects will be: global analysis, Hodge theory, PDE's on Manifolds, advanced Riemannian geometry etc.     


Required prior knowledge

Basic culture in geometry, smooth manifolds, tensors, measure theory and functional analysis.


In spring 2012, the course should take place on Wednesday morning, 10:15-12:00  room MAA-110
The seminar will start on Wednseday March 7.

Subject for spring 2012

During this semester we will study some topics in picewise (polyhedral) geometry. We will start by studying the basic notions of piecewise linear (PL) topology and then go one with notions from discrete differential geometry. In particuler we will study conditions under which geometric properties of polyhedral surfaces in Euclidean-s-space converge to their smooth counterpart under Hausdorff convergence. We will then define and study a discrete Laplace-Beltrami operator for simplicial surfaces and discrete notions of harmonic functions and minimal surfaces. Depending on time and interest of the audience we may continue with a study of  differential forms on simplicial complexes and/or the theory of Riemannian polyhedra.


  Basic on PL topology: 
  • I. M. Singer and J. A. Thorpe, Lecture Notes on Elementary Topology and Geometry (chapter 4).
  •  C. P. Rourke and B. J. Sanderson, Introduction to Piecewise-Linear Topology, SpringerVerlag, Berlin, 1972.
  • J.L Bryant. Piecewise linear topology. Handbook of geometric topology, 219–259, North-Holland, Amsterdam, 2002  (to be downloaded here).
  • ...and many other references.

 On discrete differential geometry:

  •  Klaus Hildebrandt, Konrad Polthier and Max Wardetzky, On the convergence of metric and geometric properties of polyhedral surfaces Geo, Dedicata, Volume 123, Number 1, 89-112, 2006. link here.
  • Alexander I. Bobenko and Boris A. Springborn  A Discrete Laplace–Beltrami Operator for Simplicial Surfaces Discrete & Computational Geometry  Volume 38, Number 4, 2007 link here.
  • More references will be given in due time.


Contact me for any question concerning this activity,



Marc Troyanov